Alessandro Audrito

Alejandro Garriz

Fernando Quirós

Alejandro G Árriz

Fernando Quir Ós

CONVERGENCE IN RELATIVE ERROR FOR THE POROUS MEDIUM EQUATION IN A TUBE

Keywords: 2010 Mathematics Subject Classification. 35K57, 35K65, 35C07, 35K55 Porous medium diffusion in tubes, long-time behaviour, convergence in relative error, traveling waves

whether they are published or not. The documents may come

Introduction

We investigate the long-time behaviour of nonnegative solutions to the Cauchy-Dirichlet problem for the Porous Medium equation (PME) (1.1)

     ∂ t u = ∆u m
in Ω × (0, ∞), u = 0 in ∂Ω × (0, ∞), u(x, 0) = u 0 (x)

in Ω, posed in a tubular domain Ω ⊂ R N +1 :

Ω = D × R,
where D ⊂ R N is a bounded domain with smooth boundary, which is assumed without loss of generality to contain the origin O, and the diffusion parameter satisfies m > 1 (degenerate diffusion framework). The initial data are assumed to be nontrivial and to satisfy (1.2)

u 0 ≥ 0 in Ω, u 0 ∈ C 0 (Ω), u m 0 ∈ C 1 (Ω).
The most peculiar aspect of this setting is the cylindrical shape of the spatial domain Ω, that we call tube. When the PME equation is posed either in a bounded domain or in the whole space, the long-time behaviour of nonnegative solutions has been widely investigated and the theory has been fully developed, see for instance [START_REF] Aronson | Large time behaviour of solutions of the porous medium equation in bounded domains[END_REF][START_REF] Vázquez | Asymptotic behaviour for the porous medium equation posed in the whole space[END_REF][START_REF] Vázquez | The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior[END_REF][START_REF] Vázquez | The porous medium equation: mathematical theory[END_REF][START_REF] Stan | Asymptotic behaviour of the doubly nonlinear diffusion equation ut = ∆pu m on bounded domains[END_REF]. In both frameworks, if the initial data belong to a reasonable class, the corresponding solutions converge to some special solutions to the PME for large times: the Friendly Giant in the bounded domain setting (see (1.14) and [START_REF] Aronson | Large time behaviour of solutions of the porous medium equation in bounded domains[END_REF][START_REF] Vázquez | The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior[END_REF][START_REF] Stan | Asymptotic behaviour of the doubly nonlinear diffusion equation ut = ∆pu m on bounded domains[END_REF]), while to the Barenblatt solution when the equation is posed in the whole R N (see [START_REF] Vázquez | Asymptotic behaviour for the porous medium equation posed in the whole space[END_REF][START_REF] Vázquez | The porous medium equation: mathematical theory[END_REF]).

Even though the tubular setting is intermediate between the above two and presents significant novelties in the description of the long-time behaviour, it has been less studied: to the best of our knowledge, the only papers on the topic are [START_REF] Vázquez | The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior[END_REF][START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF] by Vázquez and [19] by Gilding and Goncerzewicz (see also [START_REF] Audrito | Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains[END_REF] in the p-Laplacian diffusion setting). Nonnegative solutions to (1.1) exhibit a traveling wave (TW) behaviour as t → +∞, when computed at the correct rescaled variables (see [START_REF] Gilding | The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains[END_REF]Theorems 4.1 and 4.2], our main Theorem 1.4 and the change of variables (1.3)). In this paper we establish new results in this direction and we refine some of the previous ones. As we will explain later, our main theorem is sharp.

TW fronts arise frequently in the study of reaction-diffusion equations (see for instance [START_REF] Fisher | The advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique[END_REF][START_REF] Aronson | Nonlinear diffusion in population genetics, combustion and nerve propagation[END_REF][START_REF] Bramson | Convergence of Solutions of the Kolmogorov Equation to Travelling Waves[END_REF] and [START_REF] De Pablo | Travelling Waves and Finite Propagation in a Reaction-Diffusion Equation[END_REF][START_REF] De Pablo | Travelling wave behaviour for a Porous-Fisher equation[END_REF][START_REF] Biró | Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type[END_REF][START_REF] Gilding | Travelling waves in nonlinear diffusion-convection reaction[END_REF][START_REF] Audrito | The Fisher-KPP problem with doubly nonlinear diffusion[END_REF][START_REF] Audrito | The Fisher-KPP problem with doubly nonlinear "fast" diffusion[END_REF][START_REF] Audrito | Bistable and monostable reaction equations with doubly nonlinear diffusion[END_REF][START_REF] Du | Logarithmic corrections in Fisher-KPP type Porous Medium Equations[END_REF][START_REF] Gárriz | Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour[END_REF][START_REF] Du | Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations[END_REF] for nonlinear diffusion models) and are not expected in purely diffusive models. The reason why TW solutions appear in our context is the geometry of the domain: when Ω is a tube, solutions spread along the longitudinal direction, with a loss of mass through the fixed boundary ∂Ω which, in turn, is compensated by a linear reaction term appearing in the rescaled problem. As a consequence, the asymptotic behaviour for t → +∞ is in fact a sort of combination of two modes: in the middle of the tube solutions behave like the Friendly Giant corresponding to the section D ⊂ R N , while far away they look like a TW and their front moves with constant speed for large times.

Before entering into details, we recall that degenerate nonlinear diffusion plays a crucial role in the entire theory (as explained in [START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF]Section 7] solutions in the range m ≤ 1 behave quite different). In particular, when m > 1 solutions u to (1.1) (with compactly supported initial data) have a free boundary, i.e., for every t > 0, u(•, t) is identically zero outside a set having compact closure in Ω and the set {u(•, t) > 0} covers eventually the whole Ω when t → +∞. The study of the geometry of such set and its boundary is one of the most interesting (and complicated!) issues of the theory. As proved in [START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF][START_REF] Gilding | The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains[END_REF], when t is large enough, the free boundary of solutions to (1.1) is made of two sets described by two bounded and continuous functions y = Γ ± u (z, t) (we set x = (z, y) with z ∈ D and y ∈ R for points x ∈ Ω), moving towards both ends of the tube with logarithmic law: Γ ± u (z, t) ∼ ±c * ln t for t ∼ +∞, where c * = c * (m, D) > 0 is the critical wave speed (cf. [START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF]Theorem 3.1] and/or Theorem 1.2). As mentioned above, this pattern strongly deviates from the asymptotics of the PME posed either in bounded domains or the whole space. In our main theorem, we will give a new proof of the logarithmic law for the free boundary.

In the remaining part of the introduction we recall the most important results of [START_REF] Vázquez | The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior[END_REF][START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF] and [START_REF] Gilding | The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains[END_REF] and we state our main theorem. According to the main ideas of [START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF], it is convenient to rescale and renormalize the solution u to (1.1) as explained in the following paragraph.

The transformed problem. As just mentioned, the long-time behaviour can be equivalently described in terms of the rescaled and renormalized solution

(1.3) v(x, τ ) = (t + t 0 ) 1 m-1 u(x, t), τ = ln(t + t 0 ),
where u is a weak solution to problem (1.1) and t 0 > 0 is fixed. At the initial time t = 0, we have

(1.4) v(x, τ 0 ) = t 1 m-1 0 u 0 (x) := v 0 (x), τ 0 := ln t 0 .
It is not difficult to check that v is a weak solution to the reaction-diffusion problem

(1.5)      ∂ τ v = ∆v m + v m-1 in Ω × (τ 0 , ∞), v = 0 in ∂Ω × (τ 0 , ∞), v(x, τ 0 ) = v 0 (x)
in Ω, and thus problems (1.1) and (1.5) are equivalent, and we can pass from one to the other by means of the change of variables (1.3). Notice that, when t 0 = 0, v is defined for all times τ ∈ R and satisfies

(1.6) ∂ τ v = ∆v m + v m-1 in Ω × R, v = 0 in ∂Ω × R.
Both problems play a role in the analysis: the first is crucial in order not to loose the initial condition, while the second is used to construct the special wave solutions (see [START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF]).

The work of Aronson and Peletier. As anticipated in the above paragraphs, the long-time behaviour of solutions to (1.1) has strong connections with the PME posed in bounded domains, with homogeneous Dirichlet boundary conditions, namely

(1.7)      ∂ t u = ∆u m in D × (0, ∞), u = 0 in ∂D × (0, ∞), u(z, 0) = u 0 (z) in D,
where D ⊂ R N is a bounded domain with smooth boundary and u 0 belongs to a suitable class of nontrivial initial data. Rescaling as in (1.3)-(1.4), we obtain a solution v of (1.5) posed in D

(1.8)      ∂ τ v = ∆v m + v m-1 in D × (τ 0 , ∞), v = 0 in ∂D × (τ 0 , ∞), v(x, τ 0 ) = v 0 (x) in D.
We rephrase below the main result of the seminal paper [START_REF] Aronson | Large time behaviour of solutions of the porous medium equation in bounded domains[END_REF] by Aronson and Peletier.

v 0 ∈ C 0 (D), v m 0 ∈ C 1 (D). Then (1.9) lim τ →∞ sup z∈D v(z, τ ) Φ(z) -1 = 0,
where Φ = Φ(z) is the unique nonnegative (and nontrivial) weak solution to the stationary problem

(1.10) -∆Φ m = 1 m-1 Φ in D, Φ = 0 in ∂D.
Furthermore,

(1.11) Φ ∈ C ∞ (D) ∩ C(D), Φ > 0 in D, ∂Φ ∂ν < 0 in ∂D,
where ν = ν(z) denotes the outward unit normal to ∂D at z.

Notice that, in terms of the solution u to (1.7) with nontrivial and nonnegative initial data

u 0 satisfying u 0 ∈ C 0 (D), v m 0 ∈ C 1 (D), the above statement asserts that (1.12) lim t→∞ sup z∈D u(z, t) U (z, t) -1 = 0,
where U = U (z, t) is the unique weak solution to

(1.13) ∂ t U = ∆U m in D × (0, ∞), U = 0 in ∂D × (0, ∞), in self-similar form (1.14) U (z, t) = t -1 m-1 Φ(z),
called the Friendly Giant1 . The limits in (1.9) and (1.12) show that u converges to U in relative error in D for large times: this means that not only u(•, t) → U (•, t) uniformly in D as t → +∞, but also that u and U are comparable up to the boundary ∂D, a highly nontrivial information since both u and U are zero in ∂D.

Keeping this fact in mind, we proceed by presenting the main result of Vázquez, about the existence of a wave solution in the tubular domain framework.

The work of Vázquez. The starting point of our study is the paper [START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF] (see also [START_REF] Vázquez | The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior[END_REF]), where Vázquez proved the existence of a wave solution to problem (1.5) and established a first result concerning the asymptotic behaviour for large times of solutions to (1.1). We rephrase below the main result in [START_REF] Vázquez | The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior[END_REF][START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF]:

(1.15) ṽ(z, y, τ ) = ϕ(z, y -c * τ ),
and the wave profile ϕ = ϕ(z, ξ) satisfies

(1.16) ∂ ξ ϕ ≤ 0, lim ξ→-∞ sup z∈D ϕ(z, ξ) Φ(z) = 1, ϕ(z, ξ) = 0 for all z ∈ D, ξ ≥ ξ 0 ,
for some ξ 0 ∈ R (depending only on m and D), where Φ = Φ(z) is the unique nonnegative weak solution to problem (1.10). Furthermore, for every compact set Ω ⊂ Ω, the solution v to

(1.4)-(1.5) satisfies lim τ →+∞ sup (z,y)∈Ω |v(z, y, τ ) -Φ(z)| = 0,
that is, in compact subsets of the tube, v forgets about the longitudinal direction and behaves (in absolute value) as the solution to the PME posed in D.

Some comments are now in order. First, notice that plugging the ansatz (1.15) into (1.6), we easily see that the pair (c * , ϕ) is a weak solution to the elliptic problem

(1.17) ∆ϕ m + c * ∂ ξ ϕ + ϕ m-1 = 0 in Ω, ϕ = 0 in ∂Ω.
It thus follows that ϕ is invariant under shifts along the longitudinal coordinate. Further, the reflection ξ → -ξ gives a profile φ with speed -c * , that is, the new profile travels towards the bottom end of the tube, and satisfies

(1.18) ∂ ξ φ ≥ 0, lim ξ→+∞ sup z∈D φ(z, ξ) Φ(z) = 1, φ(z, ξ) = 0 for all z ∈ D, ξ ≤ -ξ 0 .
Second, by the last relation in (1.16), we have that the support of ϕ is contained in D×(-∞, ξ 0 ] ⊂ Ω and so ϕ has a free boundary, which is a locally Hölder hypersurface in Ω far from ∂Ω (see for instance [START_REF] Caffarelli | Regularity of the free boundary for the one-dimensional flow of gas in a porous medium[END_REF][START_REF] Caffarelli | Regularity of the free boundary of a gas flow in an n-dimensional porous medium[END_REF][START_REF] Caffarelli | Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation[END_REF][START_REF] Koch | Non-Euclidean Singular Integrals and the Porous Medium Equation[END_REF][START_REF] Kienzler | Flatness implies smoothness for solutions of the porous medium equation[END_REF]). It can be parametrized setting [START_REF] Vázquez | Porous Medium flow in a tube: traveling waves and KPP behaviour[END_REF]Proposition 4.1], it turns out that the function Γ + ϕ is bounded and continuous in D. Finally, notice that coming back to the usual variables, we obtain a family of special solutions to the PME (posed in Ω with homogeneous Dirichlet boundary conditions) defined by

(1.19) ξ = Γ + ϕ (z) := inf{η : ϕ(z, η) = 0}, z ∈ D. By
ψ (z, y, t) := t -1 m-1 ϕ(z, y -c * ln t -), ∈ R, t > 0. D as t → 0 + .
2 Actually, ϕ is C α (Ω) for some α ∈ (0, 1). See for instance [START_REF] Aronson | Large time behaviour of solutions of the porous medium equation in bounded domains[END_REF][START_REF] Vázquez | The porous medium equation: mathematical theory[END_REF].

The work of Gilding and Goncerzewicz. Later, in [START_REF] Gilding | The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains[END_REF], Gilding and Goncerzewicz investigated the long-time behaviour of solutions to (1.1) and further properties of the wave solutions introduced above, such as uniqueness and stability. Their arguments are based on a powerful invariance principle (see [START_REF] Gilding | The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains[END_REF]Theorem 3.1]) that allows to characterize the critical speed of propagation. Exploiting the same invariance principle, Gilding and Goncerzewicz showed that, in an appropriate scale, the solution to (1.1) converges to a suitable displacement (in dependence of the initial data) of the wave solution (1.15) as t → +∞. We present below the main results in [START_REF] Gilding | The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains[END_REF], rephrasing them coherently with our notations. (i) Up to longitudinal shifts, the pair (c * , ϕ) is the unique solution to problem (1.17) satisfying (1.16) and, furthermore,

c * = 1 (m -1) λ 1 (D) ,
where λ 1 (D) > 0 is the first eigenvalue of the Dirichlet Laplacian in D.

(ii) There exists 0 ∈ R depending only on m, D and the initial data such that for every α ∈ R, there holds

(1.20) lim τ →+∞ sup z∈D,|y|≥α+c * τ |v(z, y, τ ) -ϕ(z, y -c * τ -0 )| = 0.
The shift 0 is given explicitly in [19, Formula (4.1)].

(iii) For every z ∈ D, there holds3 lim inf

τ →+∞ Γ + v (z, τ ) -c * τ ≥ Γ + ϕ (z) + 0 .
(iv) Finally, there holds lim sup

τ →+∞ Γ + v (z, τ ) -c * τ ≤ Γ + ϕ (z) + 0 ,
uniformly w.r.t z ∈ D, where Γ + ϕ denotes the concave envelope of Γ + ϕ .

Also this statement deserves some additional comments. Part (i) completes the wave analysis by showing the uniqueness of the pair (c * , ϕ) and gives an explicit expression for the critical speed. The presence of the first eigenvalue of the Dirichlet Laplacian in D is a consequence of the invariance principle mentioned above (see [START_REF] Gilding | The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains[END_REF]Theorem 3.2]). From now on, in order to fix the notations, we will implicitly assume that ϕ is the unique solution to (1.16)-(1.17) satisfying sup{Γ + ϕ (z) : z ∈ D} = 0. Part (ii) proves that the solution to (1.4)-(1.5) converges (in absolute error) to the wave solution displaced by a suitable shift 0 along the longitudinal direction. Remarkably, such shift is explicit in terms of the initial data (again this is a consequence of the invariance principle), but one has to note too that the convergence in absolute error gives no information about the solution near the boundary of the tube, since both the solution and the limit take the value 0 there. Our result not only provides a different approach to the study of the large time behaviour but also fixes this and, furthermore, provides a speed of convergence in compact sets.

Finally, parts (iii) and (iv) give a quite robust description of the long-time behaviour of the free boundary of (1.4)-(1.5) in terms of the free boundary of the wave solution. Notice that, if the set of {ϕ > 0} is convex then (iii) and (iv) imply that Γ

+ v (•, τ ) -c * τ -0 → Γ + ϕ (•) uniformly on compact subsets of D, as τ → +∞.
Our main result. We can now state our main result.

Theorem 1.4. Let c * > 0 be as in Theorem 1.2 and let v 0 be defined as in (1.4). Then there exists t 0 > 0 such that the solution v to (1.5) with initial data v 0 satisfies the following two assertions:

(i) For every c ∈ (0, c * ), (1.21) lim τ →+∞ sup z∈D, |y|≤cτ v(z, y, τ ) Φ(z) -1 = 0,
where Φ is the unique nonnegative and nontrivial weak solution to (1.10).

(ii) For every c > c * , there exists τ c > 0 such that

(1.22) v(z, y, τ ) = 0 in D × {|y| ≥ cτ }, ∀ τ ≥ τ c .
Furthermore, there exist C, T > 0 depending on m, D and the initial datum, such that for all τ ≥ T , the set

Γ + v (τ ) := {(z, Γ + v (z, τ )) : z ∈ D} is a locally Hölder 4 hypersurface, and the function y = Γ + v (z, τ ) satisfies (1.23) lim τ →+∞ sup z∈D Γ + v (z, τ ) τ = c * ,
and

(1.24) sup τ ≥T sup z∈D |Γ + v (z, τ ) -Γ + ϕ (z) -c * τ | ≤ C.
Before moving forward, it is important to rephrase the above statement and highlight a couple of significant facts. First of all, rewriting (1.21) and (1.22) To conclude, we notice that, as an immediate consequence of (1.21) and (1.22), the limit in (1.23) follows, while the relation in (1.24) gives more precise information about the location of the free boundary in the spirit of Gilding and Goncerzewicz, see Theorem 1.3, parts (iii)-(iv). Our estimate is rougher, but the bound from below is uniform in D.

Structure of the paper. The remaining part of the paper is divided in sections as follows.

In Section 2 we improve the statement of [START_REF] Vázquez | The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior[END_REF]Theorem 3.2] by showing that the solution to problem (1.4)-(1.5) converges to the stationary solution Φ not only in absolute error, but also in relative error in any compact subset of Ω.

In Section 3 we show Theorem 1.4 under the assumption that D is star-shaped. In this setting the proof is more direct and is based on a comparison argument with a rescaled version of the TW (1.15). The proof is independent from the one we give for general bounded domains.

In Section 4 we treat the general case, by constructing new families of barriers modeled on the usual wave solution. To compare, we exploit Corollary 2.4, in which we quantify the rate of convergence in relative error in compact subsets of Ω.

Convergence in relative error in bounded cylinders

In this section we show convergence in relative error in sets of the form D × (-A, A), where A > 0 is arbitrarily fixed and, as consequence, in Corollary 2.4, we prove a second technical result, where we quantify the speed of convergence in relative error. Proof. Let us fix A > 0 and ε ∈ (0, 1). Our proof exploits a technical proposition from [START_REF] Aronson | Large time behaviour of solutions of the porous medium equation in bounded domains[END_REF]: for this reason, we follow Aronson and Peletier notations and we prove the statement for the solution u to (1.1), that is, we show that

(2.1) (1 -ε)U (z, t) ≤ u(z, y, t) ≤ U (z, t) in D × (-A, A) × (T, ∞),
where U = U (z, t) is the unique solution to (1.13) in self-similar form defined in (1.10) and (1.14), and T ≥ 0 must be suitable chosen depending only on m, D, u 0 , A and ε.

Step 1: The upper bound. To show the upper bound in (2.1), we follow the ideas of [2, Proof of Theorem 1]. First we notice that, since U is a weak solution to (1.13) and it is independent of y, then it is a weak solution to the same problem posed in Ω. So, if there is t > 0 such that

(2.2) u 0 (z, y) ≤ U (z, t) in Ω, then, by comparison, u(z, y, t) ≤ U (z, t + t) = (t + t) -1 m-1 Φ(z) ≤ t -1 m-1 Φ(z) = U (z, t) in Ω × (0, ∞),
which is exactly our upper bound with T = 0 and A = +∞. Assume by contradiction that (2.2) does not hold for any t > 0. Then there exists a sequence

{(z k , y k )} k∈N ⊂ supp u 0 ⊂ Ω such that 0 ≤ Φ m (z k ) ≤ 1 k u m 0 (z k , y k )
, for every k. Since supp u 0 is compact, there is (z, y) ∈ supp u 0 such that (z k , y k ) → (z, y) as k → +∞ (up to passing to a suitable subsequence) and, since u 0 is bounded and Φ > 0 in D, it must be z ∈ ∂D. Consequently, dividing both sides of the above inequality by dist(z k , ∂D) and using the third assumption in (1.2), we pass to the limit as k → +∞ to find

∂ ν Φ m (z) = 0,
where ν is the outward unit normal to ∂D, in contradiction with [2, Proposition 1] (see also (1.11)).

Step 2: A subsolution. To prove the lower bound we construct an explicit subsolution and we exploit a technical result proved in [START_REF] Aronson | Large time behaviour of solutions of the porous medium equation in bounded domains[END_REF] in the bounded domain framework.

To construct the barrier from below, we fix α, λ > 0 and consider the function

f (y) := [λ cos(αy)] 1 m , y ∈ I α := (-π 2α , π 2α
), and we set Ψ(z, y) := Φ(z)f (y), (z, y) ∈ Ω α := D × I α . We claim that

(2.3) -∆Ψ m ≤ 1 m-1 Ψ in Ω α , Ψ = 0 in ∂Ω α ,
for every λ, α > 0 satisfying

(2.4) λ m-1 m 1 + α 2 (m -1) Φ m-1 L ∞ (D) ≤ 1. To see this, we notice that since ∇ z,y Ψ m (z, y) = (f m (y)∇ z Φ m (z), Φ m (z)(f m ) (y))
and Φ is a weak solution to (1.10), it follows

-∆Ψ m (z, y) = -f m (y)∆ z Φ m (z) -Φ m (z)(f m ) (y) = Φ(z) m-1 f m (y) + α 2 Φ m (z)f m (y) = Φ(z) m -1 f m (y) 1 + α 2 (m -1)Φ m-1 (z) in Ω α . Consequently, Ψ satisfies (2.3) if (2.5) f m (y) 1 + α 2 (m -1)Φ m-1 (z) ≤ f (y) in Ω α .
Now, writing down f explicitly and using that m > 1, we find

f m (y) 1 + α 2 (m -1)Φ m-1 (z) ≤ λ cos(αy) 1 + α 2 (m -1) Φ m-1 L ∞ (D)
, and thus (2.5) is automatically satisfied if

[λ cos(αy)] m-1 m 1 + α 2 (m -1) Φ m-1
L ∞ (D) ≤ 1 in Ω α , which, in turn, holds true whenever λ, α > 0 satisfy (2.4).

Step 3: The lower bound in thin cylinders. In this step we prove the existence of T > 0 (depending only on m, D, u 0 , and ε) and δ > 0 (depending only on m, D, and ε) such that

(2.6) u(z, y, t) ≥ (1 -ε)U (z, t) in D × (-δ, δ) × (T, ∞).
First we notice that, if λ ≤ 1, then

-∆(aΨ m ) ≤ a m-1 Ψ ≤ a m-1 Ψ L ∞ (Ωα) ≤ a m-1 Φ L ∞ (D) ≤ 1 in Ω α , for every 0 < a ≤ (m -1)/ Φ L ∞ (D)
, and thus, if H denotes the unique solution to

(2.7) -∆H = 1 in Ω α , H = 0 in ∂Ω α ,
we obtain -∆(aΨ m ) ≤ -∆H in Ω α , with aΨ m = H = 0 in ∂Ω α , and so Combining the above inequality with (2.8), we deduce u m (T * ) ≥ abΨ m in Ω α , that is (2.9) u(z, y, T * ) ≥ (ab)

1 m Ψ(z, y) in Ω α . Now, let t * > 0 be fixed and consider the function

w(z, y, t) := (t * + t) - 1 m-1 Ψ(z, y).
Since Ψ satisfies (2.3), w satisfies

∂ t w ≤ ∆w m in Ω α × (0, ∞), w = 0 in ∂Ω α × (0, ∞),
and thus, choosing t * > 0 such that (ab)

1 m ≥ (t * + T * ) - 1 m-1 , inequality (2.9) yields u(z, y, t) ≥ (t * + t) - 1 m-1 Ψ(z, y) in Ω α × (T * , ∞),
by parabolic comparison. Now, take λ := 1 -ε 2 and notice that, by continuity, there exists δ 0 ∈ (0, 1) depending only on ε and m, such that cos(αy)

≥ (1 -ε 2 ) m-1
if and only if y ∈ (-δ 0 α , δ 0 α ). Consequently, we may choose α small enough (depending only ε, D and m) such that (2.4) is satisfied and so, as a consequence of the last two inequalities, it follows that

(2.10) u(z, y, t) ≥ (1 -ε 2 )(t * + t) - 1 m-1 Φ(z) in D × (-δ 0 α , δ 0 α ) × (T * , ∞). Choosing T ≥ T * large enough (depending on ε) and δ = δ 0 α , we have u(z, y, t) ≥ (1 -ε 2 ) t t * + t 1 m-1 U (z, t) ≥ (1 -ε)U (z, t) in D × (-δ, δ) × (T, ∞),
and (2.6) follows.

Step 4: The lower bound in finite cylinders. To complete the proof of the lower bound, we fix A > 0 and we choose α > 0 satisfying both (2.4) and

δ = δ 0 α ≥ A.
Since δ 0 depends only on m and ε, by (2.4) we have that δ depends only on m, D, ε and A. At this point, we may repeat the argument above: using [2, Proposition 5], we find T * , b > 0 (now depending also on A) for which (2.9) is satisfied, we choose t * > 0 exactly as above, deducing that (2.10) holds true in D × (-A, A) × (T * , ∞). Finally, taking T ≥ T * large enough (depending on m, D, u 0 , A and ε), the lower bound in (2.1) follows.

Remark 2.2. Notice that Step 1 of the proof yields the existence of a time t > 0 such that u(z, y, t) ≤ U (z, t + t) in Ω × (0, ∞). From now on, we will always assume that t 0 ∈ (0, t), where t 0 is the parameter appearing in (1.3)- (1.4). In this way, we automatically have the universal upper-bound for the solution to (1.5):

v(z, y, τ ) ≤ Φ(z) in Ω × (τ 0 , ∞),
where we recall that τ 0 := ln t 0 .

The last part of this section consists on proving the exponential speed of convergence of the solution v to the friendly giant, a key result in order to stablish comparison on a parabolic boundary later on when comparing with the sub and supersolutions that converge to travelling waves. An intermediate result prior to this is the following lemma. Lemma 2.3. Given A, k > 0, and ε ∈ (0, 1/2), let f be the solution of

(2.11)      f τ = k(f m ) yy + f m-1 (1 -f m-1 ), (y, τ ) ∈ (-A, A) × R + , f (y, 0) = 1 -ε, y ∈ [-A, A], f (±A, τ ) = 1 -ε, τ ∈ R + . (i) (f m ) yy ≤ 0 in [-A, A] × R + .
(ii) There exists ε * = ε * (m) ∈ (0, 1/2) such that, if ε ∈ (0, ε * ), then, given B ≥ 0, B < A,

(2.12) f (y, τ ) ≥ 1 -δe -λτ for all y ∈ [-B, B], τ > T, for some constants δ, λ > 0, T ≥ 0.

Proof. (i) Let h := (f m ) yy . We want to check that h ≤ 0 in (-A,

A) × R + , i.e., f m is concave. Since f ≥ 1 -ε in [-A, A] × R + (
by comparison with the constant subsolution 1 -ε), we have that f is smooth. Consequently, we may change the order of differentiation to obtain

h τ = (f m ) yy τ = (f m ) τ yy = mf m-1 f τ yy ,
and thus, using the equation of f ,

h τ = m kf m-1 h + 1 m -1 (f m -f 2m-1 ) yy .
Using that

(f 2m-1 ) yy = (f m ) 2-1/m yy = 2 -1 m f m-1 h + 2 -1 m 1 -1 m ((f m ) y ) 2 f -1 ,
we obtain the equation

h τ = mk f m-1 h yy + m m-1 1 -2 -1 m f m-1 h -2 -1 m ((f m ) y ) 2 f -1 .
Since f ∈ (1 -ε, 1), the previous equation can be written in the form

h τ = mk (a(t, y)h) y + m m-1 b(t, y)h -c(t, y),
where a is a C 2 , bounded and positive function, b is continuous and bounded and c ≥ 0. As a consequence, we obtain by our initial assumption

         h τ = mk (a(y, τ )h) yy + m m -1 b(y, τ )h -c(y, τ ), y ∈ (-A, A), τ > 0, h(y, 0) = 0, y ∈ [-A, A], h(±A, τ ) = (1 -ε) m -(1 -ε) m -1 < 0, τ > 0,
where we have used that f satisfies the equation in (2.3) at the boundary y = ±A to compute the value of h there. Since c ≥ 0 the function h = 0 is a supersolution and so h ≤ 0 follows by comparison.

(ii) The exponential speed of convergence of f (0, τ ) to 1 is a consequence of the "flatness" of the solution near the value 1, which implies that the solution resembles that of the ordinary differential equation f = f m-1 (1-f m-1 ), which in turn can be approximated by its linearization around the level 1, f = (1 -f ). However, we are loosing something in each approximation, and the rigorous proof requires to work a bit. Nevertheless, since it analogous to the one of [START_REF] Du | Logarithmic corrections in Fisher-KPP type Porous Medium Equations[END_REF]Lemma 3.2], we only sketch it.

We consider the change of variables v = f m . Hence, v satisfies the equation

1 m v 1/m-1 v τ = kv yy + 1 m-1 (v 1/m -v). Notice that (1 -ε) m ≤ v < 1.
In particular, we can make v arbitrarily close to 1, by taking ε close to 0. As a consequence, if ε is small enough (how small depending only on m),

v 1/m -v ≥ m -1 2m (1 -v), so that v is a supersolution to (2.13) 1 m v 1/m-1 v τ = kv yy + 1 2m (1 -v).
Let ψ be the solution to

       ψ τ = km 2 ψ yy + 1 4 (1 -ψ), (y, τ ) ∈ (-A, A) × R + , ψ(y, 0) = 1 -ε, y ∈ [-A, A], ψ(±A, τ ) = 1 -ε, τ ∈ R + .
Solutions of this problem are known to converge exponentially to 1 in [-B, B],

ψ(y, τ ) ≥ 1 -δe -λτ for all y ∈ [-B, B], τ > T, for some constants δ, λ > 0, T ≥ 0, see [START_REF] Du | Locally uniform convergence to an equilibrium for nonlinear parabolic equations on R N[END_REF]Lemma 2.6]. Therefore, if we prove that ψ is a subsolution to (2.13) we are done.

On the one hand, 1

-ε ≤ ψ < 1, so that 1 ≥ ψ 1 m -1 /2, since ε ∈ (0, 1/2). On the other hand, g = ψ τ satisfies          g τ = km 2 g yy - g 4 , (y, τ ) ∈ (-A, A) × R + , g(y, 0) = ε 4 , y ∈ [-A, A], g(±A, τ ) = 0, τ ∈ R + ,
where we have used the equation satisfied by ψ to compute the initial value of g. Hence, a direct comparison argument shows that ψ τ ≥ 0. Combining both things,

ψ 1 m -1 2 ψ τ ≤ ψ τ = km 2 ψ yy + 1 4 (1 -ψ),
from where we immediately get, multiplying by 2/m, that ψ is a subsolution to (2.13), as desired.

Corollary 2.4. Let v be the nonnegative weak solution to problem (1.4)- (1.5). Then there exist δ, λ > 0 and T ≥ 0 such that

1 -δe -λ(τ -T ) Φ(z) ≤ v(z, 0, τ ) in D × [2T, ∞),
where Φ = Φ(z) is the unique nonnegative weak solution to the stationary problem (1.10). A,A) × [T, ∞), for A, T > 0 that we will choose properly. To prove our statement, we compare the solution v with a barrier from below defined as

Proof. Let us fix a domain D × (-

w(z, y, τ ) := f (y, τ )Φ(z),
where f satisfies (2.11) with k = Φ m-1 L ∞ (D) . In order to do that, we need to check that w is a subsolution of the problem, that w and v are ordered at the initial time and on the lateral boundary ∂(D × (-A, A)) × (T, ∞), and apply the comparison principle as above.

Given ε ∈ (0, 1/2), Proposition 2.1 implies that for every A > 0, there exists T > 0, which may be assumed to be bigger than the value of T in (2.12), such that

v(z, y, τ ) ≥ (1 -ε)Φ(z) in D × [-A, A] × [T, ∞).
It is clear then, from Lemma 2.3, that w ≤ v in the parabolic boundary of the problem. Next, we compute

w τ -∆w m - w m -1 = Φ f τ -Φ m-1 (f m ) yy - f -f m m -1 . Take k = Φ m-1 L ∞ (D) . Since (f m ) yy ≤ 0, we have that -Φ m-1 (f m ) yy ≤ -k(f m ) yy
and one readily finds that ). By comparison we obtain

w τ -∆w m - w m -1 ≤ Φ f τ -k(f m ) yy - f -f m m -1 = 0 and thus w is a subsolution in D × [-A, A] × [T, ∞
w(z, y, τ -T ) ≤ v(z, y, τ ) in D × (-A, A) × (T, ∞),
and so our statement follows in light of (2.12).

Proof of Theorem 1.4 when D is star-shaped

We are ready to proceed with the proof of Theorem 1.4. We begin by assuming that the domain D is star-shaped with pole at the origin O: thanks to this assumption, the dilation λD satisfies D ⊂ λD for λ > 1, and we can accordingly perturb the TW (1.15) to construct a subsolution for (1.4)-(1.5). We mention that this proof does not use Corollary 2.4, but only Proposition 2.1.

Proof of Theorem 1.4 (D star-shaped). Assume that D is star-shaped with pole O. Let u 0 be nontrivial satisfying (1.2), v 0 satisfying (1.4), v = v(z, y, τ ) the corresponding solution to (1.5) and c * > 0 as in Theorem 1.2.

We first prove the following claim: for every ε ∈ (0, 1) and every c ∈ (0, c * ), there exists T > 0 depending only on m, D, u 0 , c and ε, such that

(3.1) v(z, y, τ ) ≥ (1 -ε)Φ(z) in D × [0, cτ ] × [T, ∞).
The proof of (3.1) is the hardest part and it is divided in several steps as follows.

Step 1. Let us fix ε ∈ (0, 1) and define

D ε := λ ε D, λ ε := (1 -ε) -m-1 2 > 1.
As mentioned above, since D is star-shaped, D ⊂ D ε . Now, let Φ ε = Φ ε (z) be the unique nonnegative weak solution to (1.10) posed D ε and let

Φ(z) := (1 -ε)Φ ε (λ ε z), z ∈ D.
Then, using the equation of Φ ε and the definition of λ ε , we have

-∆ Φm (z) = -(1 -ε) m λ 2 ε ∆Φ m ε (λ ε z) = (1-ε) m λ 2 ε m-1 Φ ε (λ ε z) = (1-ε) m-1 λ 2 ε m-1 Φ(z) = 1 m-1 Φ(z) in D, and thus, since Φ = Φ = 0 in ∂D (this follows since Φ ε = 0 in ∂D ε ), we obtain (3.2) Φ(z) = (1 -ε)Φ ε (λ ε z) in D,
by uniqueness. Now, let us write

1 -ε = 1 -a ε 1 -b ε , a ε := ε(3 -ε) 2 , b ε = ε 2 .
Notice that a ε , b ε ∈ (0, 1) with a ε , b ε → 0 as ε → 0 + and, in addition, a ε ∈ (ε, 1) and b ε ∈ (0, a ε ).

In particular, as a consequence of (3.2), it follows

(3.3) (1 -b ε ) Φ(z) = (1 -a ε )Φ ε (λ ε z), z ∈ D.
Further, by Proposition 2.1 (with A = 1), there is τ ε > 0 (depending only on m, D, u 0 and ε) such that

(3.4) v(z, y, τ + τ ε ) ≥ (1 -b ε ) Φ(z) in D × [0, 1) × (0, ∞).
Step 2. To prove (3.1), we build a subsolution of wave type. Our candidate is the function

ψ(z, y, τ ) := (1 -a ε ) ϕ ε (λ ε z, λ ε y -c ε τ + ), z ∈ D, y, τ ≥ 0, ≥ 0,
where

ϕ ε = ϕ ε (z, ξ) is the nonnegative weak solution to ∆ϕ m ε + c ε * ∂ ξ ϕ ε + ϕε m-1 = 0 in Ω ε , ϕ ε = 0 in ∂Ω ε , satisfying (3.5) ∂ ξ ϕ ε ≤ 0, lim ξ→-∞ sup z∈D ϕ ε (z, ξ) Φ ε (z) = 1, ϕ ε (z, ξ) = 0 for all z ∈ D, ξ ≥ ξ ε ,
for some ξ ε ∈ R depending only on m, D and ε (cf. Theorem 1.2). The speed c ε * > 0 is the unique number for which the above problem has a nontrivial weak solution, while

Ω ε := D ε × (0, ∞).
The speed c ε > 0 is defined as

(3.6) c ε := c * 1 -a ε 1 -ε m-1 . Since ε < a ε , we have c ε < c * with c ε → c * as ε → 0 (this follows since a ε → 0 as ε → 0). Let us fix c ∈ (0, c * ).
In light of the above observations and recalling that λ ε → 1 as ε → 0, we deduce the existence of ε 0 ∈ (0, 1) small enough depending on c, such that

(3.7) c < λ ε c < c ε < c * , ∀ ε ∈ (0, ε 0 ).
From now on, we fix ε ∈ (0, ε 0 ). Notice that, since D ⊂ D ε , we have c * < c ε * with c ε * → c * as ε → 0 (cf. [28, Corollary 5.5] 6 ) and so δ ε := c ε * -c * > 0 for every ε ∈ (0, 1), with δ ε → 0 as ε → 0. Now, since ∂ ξ ϕ ε ≤ 0, we have by direct differentiation and (3.6):

(3.8)

∂ τ ψ = -c ε (1 -a ε )∂ ξ ϕ ε = -c * (1 -a ε ) m (1 -ε) m-1 ∂ ξ ϕ ε = -c ε * (1 -a ε ) m (1 -ε) m-1 ∂ ξ ϕ ε + δ ε (1 -a ε ) m (1 -ε) m-1 ∂ ξ ϕ ε ≤ -c ε * (1 -a ε ) m (1 -ε) m-1 ∂ ξ ϕ ε ,
where we have also used the definition of δ ε . On the other hand, by definition of λ ε ,

∆ψ m = (1 -ε) m λ 2 ε ∆ϕ m ε = (1 -a ε ) m (1 -ε) m-1 ∆ϕ m ε ,
and thus, combining (3.8) with the equation of ϕ ε and recalling that a ε > ε, it follows

∂ τ ψ -∆ψ m -1 m-1 ψ ≤ -c ε * (1 -a ε ) m (1 -ε) m-1 ∂ ξ ϕ ε - (1 -a ε ) m (1 -ε) m-1 ∆ϕ m ε -1 m-1 (1 -a ε )ϕ ε ≤ - (1 -a ε ) m (1 -ε) m-1 c ε * ∂ ξ ϕ ε + ∆ϕ m ε + 1 m-1 ϕ ε = 0 in D × [0, ∞) × (0, ∞).
Since ψ = 0 in ∂D × [0, ∞) × (0, ∞) by construction, ψ is a subsolution, as claimed at the beginning of this step.

Step 3. On the other hand, we have 6 The speed c * is a monotone and continuous function of the domain.

(1 -a ε )Φ ε (λ ε z) ≥ ψ(z, y, τ ) in D × [0, ∞) × (0, ∞),
by construction and the upper bound in (2.1) and thus, by (3.3) and (3.4), we obtain (3.9) v(z, y, τ + τ ε ) ≥ ψ(z, y, τ ) in D × [0, 1) × (0, ∞).

Further, since the support of ϕ ε is bounded to the right, we may choose = ε large enough such that

v(z, y, τ ε ) ≥ ψ(z, y, 0) in D × [0, ∞).
Moreover, by (3.9) we easily get v(z, 0, τ + τ ε ) ≥ v(z, 0, τ ) for any z ∈ D, τ ≥ 0 and, since ψ = 0 in ∂D × [0, ∞) × [0, ∞), we deduce by comparison

v(z, y, τ + τ ε ) ≥ (1 -a ε ) ϕ ε (λ ε z, λ ε y -c ε τ + ε ) in D × [0, ∞) × [0, ∞).
Now, in view of the first limit in (3.5), we know that there is ξ ε ∈ R such that

ϕ ε (λ ε z, ξ) ≥ (1 -ε 2 )Φ ε (λ ε z) in D × (-∞, ξ ε ],
and thus, combining the above two inequalities with (3.3) and the definition of b ε , we obtain

(3.10) v(z, y, τ + τ ε ) ≥ (1 -ε 2 )(1 -a ε )Φ ε (λ ε z) = (1 -ε ) 2 Φ(z) ≥ (1 -ε)Φ(z),
for every z ∈ D and every y, τ ≥ 0 satisfying λ ε y -c ε τ + ε ≤ ξ ε . Finally, by the second inequality in (3.7), whenever y ≤ cτ , we have

λ ε y -c ε τ + ε ≤ -(c ε -λ ε c)τ + ε ≤ ξ ε , for every τ ≥ τ ε := ε-ξ ε cε-λεc ,
v 0 (z, y) := t 1 m-1 0 u 0 (z, y) ≤ t 0 t 1 m-1 Φ(z) in Ω,
for some t > 0 depending only on m, D and u 0 . Thus, choosing t 0 ≤ 2 -(m-1) t, we obtain v 0 (z, y) ≤ 1 2 Φ(z) in Ω. Consequently, in view of (1.16), we may choose ≥ 0 large enough so that ψ(z, y, 0) = ϕ(z, y -) ≥ v 0 (z, y) in Ω, and so the comparison principle yields

ψ(z, y, τ ) = ϕ(z, y -c * τ -) ≥ v(z, y, τ ) in Ω × (τ 0 , ∞),
where τ 0 := ln t 0 according to (1.4) (notice that the comparison at the boundary ∂Ω × (0, ∞) is immediate since v = ψ = 0 there). Recalling that ϕ(z, ξ) = 0 in D × [ξ 0 , ∞) for some ξ 0 ∈ R, we immediately see that

(3.11) 0 ≤ v(z, y, τ ) ≤ ϕ(z, y -c * τ -) = 0 in D × [cτ, ∞) × [T, ∞),
where T := ξ 0 + c-c * > 0.

Step 5. To conclude the proof, it is sufficient to repeat the above arguments working in D × (-∞, 0] and using the reflected wave solutions instead of ϕ (cf. (1.18)): in the very same way, we obtain that (3.1) holds true in D × [-cτ, 0] × [T, ∞) whenever c < c * , while (3.11)

in D × (-∞, -cτ ] × [T, ∞), for every c > c * .

Proof of Theorem 1.4 when D is a bounded domain

In this section we present the proof of Theorem 1.4 in its full generality, that is, dropping the assumption that D is star-shaped. To do so, we construct new barriers by perturbing the TW solution in (1.15). Inspired by the work of Biró [START_REF] Biró | Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type[END_REF] (see also [START_REF] Gárriz | Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour[END_REF]), the main idea is to look for solutions to (1.5) with form (4. [START_REF] Aronson | Nonlinear diffusion in population genetics, combustion and nerve propagation[END_REF] w(z, y, τ

) := f (τ )ϕ(z, y -g(τ )),
where ϕ is the wave profile satisfying (1.17), while f : R + → R + and g : R → R are smooth functions satisfying f (0) = f 0 , g(0) = g 0 . The functions f and g will be suitably chosen to build a subsolution and a supersolution, respectively. In this perspective, we plug the ansatz (4.1) into the equation in (1.5): a direct computation and (1.17) show that (4.2)

∂ τ w -∆w m -1 m-1 w = f (τ )ϕ -f (τ )g (τ )∂ ξ ϕ -f m (τ )∆ϕ m -f (τ ) m-1 ϕ = f (τ )ϕ -f (τ )g (τ )∂ ξ ϕ + c * f m (τ )∂ ξ ϕ + f m (τ ) m-1 ϕ -f (τ ) m-1 ϕ = f (τ )∂ ξ ϕ c * f m-1 (τ ) -g (τ ) + ϕ f (τ ) + f m (τ ) m-1 -f (τ ) m-1 .
With this computation in mind, we proceed in two main steps. As already anticipated, we construct a subsolution and a supersolution using ODEs techniques, and then we use them to trap the solution to (1.4)-(1.5). The crucial fact is that we can choose the functions f and g in such a way f (τ ) ∼ 1, g(τ ) -c * τ ∼ c 0 for τ ∼ +∞, for some constant c 0 depending only on m, D, f 0 and g 0 . Thanks to these properties, the statement of Theorem 1.4 will easily follow. Construction of a subsolution. To build a subsolution, we consider the solutions f and g of the following Cauchy problems

(4.3) f = f m-1 (1 -δ(τ ) -f m-1 ), τ > 0, f (0) = f 0 g = c * f m-1 , τ > 0, g(0) = g 0 ,
where f 0 ∈ (0, 1) and g 0 ∈ R will be suitably chosen and

(4.4) δ(τ ) := 2(m -1)δ 0 (1 + τ ) 2 ,
for some δ 0 ∈ (0, 1).

Lemma 4.2. Let m > 1, f 0 ∈ (0, 1) and let f satisfying (4.3), with δ as in (4.4). Then, for every δ 0 > 0, we have

(4.5) lim τ →+∞ f (τ ) = 1.
Further, there exists δ ∈ (0, 1) depending only on m and f 0 such that such that for every δ 0 ∈ (0, δ)

(4.6) f (τ ) ≤ 1 - δ(τ ) 2(m -1)
, ∀τ ≥ 0.

Proof. We first notice that 0 < f (τ ) < 1, for all τ > 0. This follows from the comparison principle for first order ODEs: the assumptions f 0 ∈ (0, 1) and δ(τ ) > 0 imply that the constants 0 and 1 are sub and supersolutions, respectively.

To check the validity of (4.5) it is enough to prove that the limit lim τ →+∞ f (τ ) := ∈ (0, 1] exists (the fact that = 1 follows by passing to the limit into the equation and using that δ(τ ) → 0 as τ → +∞). With this goal in mind, we integrate the equation of f to deduce

(4.7) (m -1) ln(f (τ )/f 0 ) = τ 0 1 -f m-1 (s) ds - τ 0 δ(s) ds,
and thus, since both the integral functions in the r.h.s. are non-decreasing and the second is bounded by definition of δ, it follows that ∈ (0, 1] exists.

To prove (4.6), we construct a supersolution for f . We consider the function

h(τ ) := 1 - δ 0 (1 + τ ) 2 = 1 - δ(τ ) 2(m -1)
,

and we fix δ ∈ (0, 1) (depending only on m and f 0 ) such that for every δ 0 ∈ (0, δ], we have

h(0) = 1 -δ 0 ≥ 1 -δ ≥ f 0 and (1 -δ) m-1 ≥ 1 -2(m -1) δ, ∀ δ ∈ (0, δ 0 ].
Then, using the above relation and the definition of δ, we obtain

h - h m -1 (1 -δ(τ ) -h m-1 ) = 2δ 0 (1 + τ ) -3 + h m -1 δ(τ ) + [1 -δ 0 (1 + τ ) -2 ] m-1 -1 > h m -1 δ(τ ) + 1 -2(m -1)δ 0 (1 + τ ) -2 -1 = 0,
for all τ > 0, that is, h is a (strict) supersolution for f . We deduce f (τ ) ≤ h(τ ) for all τ ≥ 0 by comparison, that is (4.6).

Remark 4.3. We remark that the control over the growth of f in (4.6) is crucial in order to apply the comparison principle: to the one hand, we know that the solution v to (1.4)-(1.5) grows exponentially fast in compact sets of Ω (cf. Corollary 2.4) while, on the other, the subsolution w in (4.1) approaches the Friendly Giant Φ = Φ(z) much slower. This will allow us to compare w and v on the "vertical" part of the boundary D × {y = 0} × (0, ∞).

Lemma 4.4. Let m > 1, δ 0 ∈ (0, 1), f 0 ∈ (0, 1), g 0 ∈ R, and let f, g as in (4.3). Then for all τ > 0. Passing to the limit as τ → +∞ and using (4.5), (4.8) follows.

Lemma 4.5. Let t 0 > 0, τ 0 := ln t 0 and let v be the weak solution to (1.4)-(1.5). Then for every f 0 ∈ (0, 1) and g 0 ∈ R, there exist δ 0 ∈ (0, 1) and T > 0 such that if f and g satisfy (4.3), then (4.9) v(z, y, τ + τ 0 + T ) ≥ f (τ )ϕ(z, y -g(τ )) in Ω × (0, ∞).

Proof. Let us fix f 0 ∈ (0, 1) and g 0 ≥ 0 and choose δ 0 ∈ (0, 1) such that δ 0 < min δ, 1 2c * (m-1) , where δ ∈ (0, 1) is as in Lemma 4.2 and set w(z, y, τ ) := f (τ )ϕ(z, y -g(τ )).

Using that ϕ has bounded support to the right and that f 0 ∈ (0, 1), we may apply Proposition 2.1 to deduce the existence of T > 0 such that w(z, y, 0) = f 0 ϕ(z, y -g 0 ) ≤ v(z, y, τ 0 + T ) in D × R + . Furthermore, by (4.6) and Corollary 2.4, it follows w(z, 0, τ ) = f (τ )ϕ(z, -g(τ )) ≤ 1 -δ(τ ) 2(m -1) Φ(z) ≤ v(z, 0, τ 0 + τ + T ) in D, for all τ ≥ 0, taking eventually T > 0 larger. Since w = v in ∂Ω × (0, ∞) and w is a subsolution by construction (it is enough to combine (4.2) and (4.3)), the inequality in (4.9) follows by the comparison principle.

Construction of a supersolution. To build a supersolution, we follow the argument above by considering solutions to the following Cauchy problems Proof. The proof is similar to the one of Lemma 4.2.

Lemma 4.7. Let t 0 > 0, τ 0 := ln t 0 and let v be the weak solution to (1.4)-(1.5). Then, for every f0 ∈ (0, 1), there exist ḡ0 ∈ R and t 0 > 0 such that if f and ḡ satisfy (4.3), then (4.13) v(z, y, τ + τ 0 ) ≤ f (τ )ϕ(z, y -ḡ(τ )) in Ω × (0, ∞),

Proof. The proof follows the same idea of Lemma 4.5. Given f0 ∈ (0, 1) and setting w(z, y, τ ) := f (τ )ϕ(z, y -ḡ(τ )), we choose ḡ0 ∈ R such that supp v 0 ⊂⊂ supp w(•, 0). Then, we take t 0 > 0 small enough depending on f0 such that v 0 (z, y) = t 1 m-1 0 u 0 (z, y) ≤ w(z, y, 0) in Ω.

Recalling that v = w = 0 on ∂Ω and noticing that w is a supersolution by construction, (4.13) follows by comparison.

We are ready to complete the proof of Theorem 1.4.

Proof of Theorem 1.4. Let us fix ε ∈ (0, 1). By Lemma 4.5 and Lemma 4.7, there are τ 0 < τ 1 such that (4.14) f (τ -τ 0 )ϕ(z, y -g(τ -τ 0 )) ≤ v(z, y, τ ) ≤ f (τ -τ 1 )ϕ(z, y -g(τ -τ 1 )) in Ω × (0, ∞). Now, fix c < c * and assume 0 ≤ y ≤ cτ , τ > 0. By (4.5), (4.8) and the first two relations in (1.16), we have v(z, y, τ ) ≥ f (τ -τ 0 )ϕ(z, y -g(τ -τ 0 )) ≥ f (τ -τ 0 )ϕ(z, cτ -g(τ -τ 0 ))

≥ (1 -ε) 1/2 ϕ(z, cτ -g(τ -τ 0 )) ≥ (1 -ε)Φ(z) in D × [0, cτ ]
for large τ 's. Recalling that Φ is an upper-bound for v and repeating the same argument when -cτ ≤ y ≤ 0 with the reflected wave solutions (cf. Remark 4.1), the limit in (1.21) follows. Now, fix c > c * and assume y ≥ cτ , τ > 0. Then by (4.11), (4.12) and (1.16), we obtain v(z, y, τ ) ≤ f (τ -τ 1 )ϕ(z, y -g(τ -τ 1 )) ≤ ϕ(z, cτ -g(τ -τ 0 )) = 0 in D × [cτ, ∞), for large τ 's. Exactly as above, we may repeat the same argument in the second half of the tube by Remark 4.1, and the proof of (1.22) is completed.

Theorem 1 . 1 . ([ 2 ,

 112 Proposition 1, Theorems 1 and 2]) Let v = v(z, τ ) be the nonnegative weak solution to problem (1.8) with nontrivial and nonnegative initial data v 0 , satisfying

Theorem 1 . 2 .

 12 ([28, Theorems 3.1 and 5.4] and [26, Theorem 3.2]) There exists a unique c * > 0 depending only on m and D such that problem (1.6) has a continuous 2 nonnegative weak solution with wave form

Theorem 1 . 3 . ([ 19 ,

 1319 Section 4]) Let v be the solution to (1.4)-(1.5) with initial datum satisfying (1.2) and let (c * , ϕ) the wave solution satisfying (1.16)-(1.17). Then the following assertions hold true:

Proposition 2 . 1 .

 21 Let v be the nonnegative weak solution to problem (1.4)-(1.5) with nontrivial initial data v 0 . Then, for every A > 0, lim τ →∞ sup z∈D, |y|≤A v(z, y, τ ) Φ(z) -1 = 0, where Φ = Φ(z) is the unique nonnegative weak solution to the stationary problem (1.10).

(2. 8 )

 8 aΨ m ≤ H in Ω α , by virtue of the (elliptic) comparison principle. Now, let u be the solution to problem (1.7) posed in Ω α (that is, D = Ω α in (1.7)). By comparison, we have u ≤ u in Ω α × (0, ∞). On the other hand, [2, Proposition 5] 5 assures the existence of T * , b > 0 depending only on m, u 0 , D and α such that bH(z, y) ≤ u m (z, y, T * ) ≤ u m (z, y, T * ) in Ω α .

  and (3.1) follows by (3.10), choosing T := τ ε + τ ε . Step 4. Now, we fix c > c * and prove (1.22). Let ϕ the unique wave solution to (1.5) with speed c * satisfying (1.16) and let ψ(z, y, τ ) = ϕ(z, y -c * τ -), z ∈ D, y, τ ≥ 0, ≥ 0. By (2.2), we have

Remark 4 . 1 .

 41 It is important to mention that the arguments below can be easily adapted to construct barriers with form (4.1) traveling towards the bottom-end of the tube by perturbing the reflected wave solution (cf. (1.18)) instead of ϕ.

0 g

 0 ) -c * τ = g 0 -2c * (m -1)δ 0 + c * (m -1) ln f 0 .Proof. Combining the equation of g with (4.7), we easily see thatc * (m -1) ln(f (τ )/f 0 ) = c * τ -τ (s) ds -c * τ 0 δ(s) ds = c * τ -g(τ ) + g 0 -2c * (m -1)δ 0 τ 1 + τ ,

( 4 .

 4 10) f = f m-1 (1 -f m-1 ), τ > 0, f (0) = f0 ḡ = c * f m-1 , τ > 0, ḡ(0) = ḡ0 ,for some suitable choice of f0 ∈ (0, 1) and ḡ0 ∈ R.Lemma 4.6. Let m > 1, f0 ∈ (0, 1), ḡ0 ∈ R and let f and ḡ satisfying (4.10). Then, f is increasing and ) -c * τ = ḡ0 + c * (m -1) ln f0 .

Finally, ( 1 .

 1 [START_REF] Koch | Non-Euclidean Singular Integrals and the Porous Medium Equation[END_REF] follows by the arbitrariness of c in (1.21) and(1.22), whilst (1.24) by combining (4.14) with (4.8) and (4.12).

This name comes from the fact that U is a global upper bound for solutions to (1.7), since U (•, t) → +∞ in

According to(1.19), we define Γ + v (z, τ ) := sup{y ∈ R : v(z, y, τ ) > 0}.

Actually, under suitable conditions on the initial data, Γ + v (τ ) is locally smooth far away from ∂Ω (cf.[START_REF] Caffarelli | Regularity of the free boundary of a gas flow in an n-dimensional porous medium[END_REF][START_REF] Caffarelli | Lipschitz continuity of solutions and interfaces of the N -dimensional porous medium equation[END_REF][START_REF] Kienzler | Flatness implies smoothness for solutions of the porous medium equation[END_REF][START_REF] Koch | Non-Euclidean Singular Integrals and the Porous Medium Equation[END_REF]).

This result holds in domains with smooth boundary: to bypass this technical difficulty, it is enough to solve problem (2.7) in a bounded domain Ω with smooth boundary, satisfying Ωα ⊂ Ω ⊂ Ω.

Acknowledgments.

A. Audrito has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement 892017 (LNLFB-Problems).

A. Gárriz received finantial support from the ANR project DEEV ANR-20-CE40-0011-01.

F. Quirós was funded by MCIN/AEI (Spain) through projects MTM2017-87596-P, PID2020-116949GB-I00, RED2018-102650-T and the ICMAT-Severo Ochoa grant CEX2019-000904-S.

Email address: alessandro.audrito@math.ethz.ch (A. Gárriz) Institut de Mathématiques de Toulouse. 1R3, Université Paul Sabatier, 118 Rte de Narbonne, 31400 Toulouse, France

Email address: alejandro.garriz@math.univ-toulouse.fr (F. Quirós) Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain, and, Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UCM-UC3M), 28049 Madrid, Spain

Email address: fernando.quiros@uam.es


Laurent Barbiero
email: laurent.barbiero@ird.sn

Vincent Valles

Annick Regeard

Claude Cheverry

Residual alkalinity as tracer to estimate the changes induced by forage cultivation in a non-saline irrigated sodic soil

Keywords: Sodic soils, Echinochloa stagnina, Residual Alkalinity, de-alkalinisation, Niger

Soil alkalinisation generally constitutes a major threat to irrigated agriculture in the semiarid regions of West Africa. The improvement of sodic soils is generally difficult and expensive. However, a recent study in the Niger valley in NIGER, reveals that a natural dealkalinisation is possible under natural conditions in a semi-arid climate. Transformation of non-saline sodic soil into brown steppe soil type was recorded. On the same site, the cultivation of a Sahelian fodder grass, locally known as "Burgu" was used on the sodic soil/brown steppe soil transition zone to accelerate this natural de-alkalinisation and characterise its mechanisms. The geochemical properties of both soil types were monitored before cultivation and one year after continuous crop cultivation. After cultivation and regular irrigation, the chemical properties of the former sodic soils were close to those of the surrounding brown steppe soils, which are better suited for agriculture. This modification of the sodic soil properties can be attributed to (i) the large amount of water supplied during cultivation that induced salt leaching. This is the main phenomenon responsible for the changes observed; (ii) the root activity that modified the acid-base equilibrium and consumes alkalinity.

The Residual Alkalinity (RA) concept was used to select chemical tracers of the concentration/dilution of the soil solution. Here, sodium amount and calcite + fluorite residual alkalinity (RA calcite+fluorite ) were the most adequate ones. These two tracers decreased proportionally under the influence of leaching, but the exchanges between cations and protons changed the RA calcite+fluorite , without modifying the sodium amount. Their combined use allowed us to separate and quantify the uptake of the plant from the leaching in the dealkalinisation process. This study highlighted that reclamation of this type of sodic soils is feasible. The use of the RA concept is advisable to design a sustainable management system for irrigated sodic or saline soils.

Introduction

In arid or semi-arid regions, valley bottoms are often covered by saline or sodic soils.

Irrigation generally leads to a secondary salinisation or alkalinisation. In the west-African region, salinisation often occurs north of the Sahara Desert whereas alkalinisation appears in the Sahelian zone. The physical properties of a non-saline sodic soil reduce their agricultural potential. Sodic soil solutions poor in calcium lead to the swelling and deflocculation of clay particles when the proportion of sodium increases on the exchange complex [START_REF] Shainberg | Response of soils to sodic and saline conditions[END_REF][START_REF] Daoud | Influence of particle size and clay organization on hydraulic conductivity and moisture retention of clays from saline soils[END_REF][START_REF] Sumner | Sodic soils", new perspectives[END_REF]. Reclamation of sodic soils is generally difficult and expensive. Several studies have recently been carried out on the effect of certain tolerant grasses in amending their soil [START_REF] Abrol | Salt-affected soils and their management[END_REF]Qadir et al., 1996a & b). [START_REF] Bozza | Mise en valeur et exploitation des terrasses du Niger[END_REF] observed that the agriculture potential of a non-saline sodic soil in the Niger valley can be significantly improved with the cultivation of a Sahelian fodder grass (Echinochloa stagnina Retz. Beauv.) locally known as "Burgu" (Francois et al., 1989). The objective of this study is not to evaluate this reclamation but to understand the mechanisms associated with the apparent remediation of these soils.

Changes in the alkalinity of the soil solution may have various origins: 1 -leaching of alkalinity by the fresh irrigation during the cultivation; 2 -the plant activity that modifies the acid-base equilibrium through alkalinity production or consumption. This work is therefore carried out in three steps. In a first time, the comparison between the initial and final states of the soil (before and after cultivation) will make it possible to appreciate the global effect of the forage cultivation. In order to distinguish the effect of leaching from the effect of the plant physiology, the residual alkalinity is used as a tracer. Lastly, the ash alkalinity is calculated in order to obtain an independent second estimate of alkalinity consumption related to the plant.

Material

Location

The experiment was located at Lossa experimental station (13°54N -1°35E) INRAN (Nigerien National Institute of Agricultural Research). The 90 ha irrigated area lies at the bottom of a 10-km² catchment on the left bank of the river Niger. The climate belongs to the type III semi-arid of the African zonation [START_REF] Griffiths | Climates of Africa[END_REF] with a short rainy season from June to September (400 mm rainfall). During the 8-month dry season, dry winds from the Northeast raise the annual potential evapo-transpiration to 2400 mm. The mean annual temperature is 33°C with maximum daily temperature up to 47°C. The water table is located at 30 to 40 m below the soil surface and is not subject to evaporation. It does not contribute to changes in the soil solution.

The experimental field (1-hectare) had not been cultivated or irrigated since 1977, when the Lossa station has been developed. The soil has a sandy clay texture, and has developed on a calco-alkaline gneiss with biotite, generally observed near 1.2 m depth. The parent rock is part of the regional Birrimian substrata, which is locally called ″Liptako bedrock″. Two major differences of the soil are observed in the experimental plot; 45% of the plot is occupied by non-saline sodic soil (pH s = 8.5 to 9.8 (s = on saturated paste), EC s = 0.22 to 0.32 dS m -1 , S.A.R. = 12 to 28 (mmol L -1 ) 1/2 , E.S.P. = 15 to 40 %) classified as Solonetz [START_REF] Fao | Draft World Reference Base for Soil Ressources[END_REF]; the remaining 55% is covered by non sodic brown steppe soil (cambic Calcisol, pH s < 8.4, EC s = 0.07 to 0.22 dS m -1 , E.S.P. = 0.5 to 2%). The limit between the two soils is abrupt with a transition zone around 1 m wide, which has long been interpreted as an active alkalinisation front. However, recent studies demonstrate the active transformation of the sodic soil into a brown steppe soil [START_REF] Barbiéro | Mise en évidence d'une désalcalinisation naturelle des sols en région tropicale. Transformation de sols alcalins en sols bruns subarides sur un bas fond sahélien du Niger[END_REF]. This natural dealkalinisation will not be discussed in this paper. From a chemical point of view, calcite (CaCO 3 ) and fluorite (CaF 2 ) precipitation were identified. Those precipitations are responsible for the lateral changes observed in the soil chemistry between sodic soils and brown steppe soils [START_REF] Barbiéro | Contribution de la fluorine au contrôle géochimique du calcium sur un bas fond sahélien du Niger. Conséquences pour une estimation quantitative de l'évolution des sols[END_REF][START_REF] Barbiéro | The alkaline soils of the Niger valley. Origins, formation and present evolution[END_REF].

The infiltration rate of sodic soils is very low and its resistance to penetration (hardsetting) is quite high. Surface crusts develop that have the ability to impede seedling emergence. [START_REF] Marlet | Soil alkalinization and irrigation in the Sahelian of Niger II. Agronomic consequences of alkalinity and sodicity[END_REF] has shown also that plant nutrition is perturbed on the sodic soil. The uptake of phosphorus, zinc, and more generally all the metals with no reduced forms, is blocked. The water for irrigation comes from a temporary channel of the Niger River. Its chemical profile is a bicarbonate-calcic/sodic type at the end of the rainy season, and becomes bicarbonate-sodic during the dry season (Table I), due to the formation of calcite, during concentration by evaporation in the channel [START_REF] Ribolzi | Contrôle géochimique des eaux par la formation de calcite en milieu méditerranéen et en milieu tropical. Arguments d'équilibre et argument de bilan[END_REF]. Such water exhibits a positive ″calcite Residual Alkalinity″ (RA calcite ) and evolves in a ″carbonate alkaline″ way by concentration when evaporation occurs [START_REF] Vallès | Ivariant criteria for irrigation water quality assessment in arid and semi-arid regions[END_REF], hence exhibits an alkalinising power. However, this water is fresh (the total electrolyte concentration ranges from 1.3 to 13.2 meq/l during the dry season), and can leach the soil by percolation if the balance between percolation and evaporation is favourable.

Methods

The field was pre-irrigated, tilled, harrowed, and divided into 80 plots of 20 m² separated by small dykes. Each plot was irrigated with 50mm of water twice a week during seven months and fertilised once a month during the growing season essentially with KCl and Urea (N 265 kg ha -1 yr -1 , K 255 kg ha -1 yr -1 ). Cuttings of fodder grass were transplanted by hand in November 1991 with a 0.3-m spacing between plants on 60 plots. The 20 remaining plots were similarly irrigated and amended without fodder grass transplantation. The semiaquatic fodder grass used, the "Burgu" Echinochloa stagnina Retz. Beauv [START_REF] François | Le pâturage semi-aquatique à Echinocloa stagnina (Retz.) Beauv. Etude approfondie de la plante "Bourgou" et des bourgoutières situées en zone lacustre du Mali[END_REF], is common in the Sahelian rivers and used locally to feed the cattle. This plant is well suited to clayey depressions that are periodically flooded. It has a very strong fasciculated root system that is able to spread quickly into the soil profile. It is a highly productive C4 plant with an optimum growth temperature above 30°C. The forage has been cut and harvested three time during the experiment on all the plot and wet and dry organic matter were recorded.

The sodic soil/brown steppe soil distribution has been delineated on the experimental field using pH measurements according to a regular 10-m squared grid. Soil samples were collected with an auger at a depth of 40 cm before cultivation and after one year of crop growing. The pH values were measured in the laboratory on saturated pastes. These data underwent an analyse of variance.

The evolution of the wetting front was followed during the cultivation using a 2cmdiameter drill. The drill penetrates in the wet soil and ridges in the dry soil.

Two large pits were excavated over the abrupt transition zone between the two soil types. 72 soil samples were collected along the first pit before cultivation, and 50 samples along the second one after one year of crop growing.

Extracts from the saturated pastes were analysed by ion chromatography (DIONEX) to measure the concentrations of the following elements: Ca ++ , Mg ++ , Na + , K + , NH 4

+ , F -, Cl -, NO 2 - , NO 3 - , PO 4 3- , SO 4 2-
, oxalate, formate and acetate. A satisfactory quantification of fluoride was achieved using sodium tetraborate as eluant. The pH was measured first on the paste for comparison with the range of pH values measured in the field. It was afterwards measured on the extracts. The carbonate alkalinity (i.e., (HCO 3

- ) + (CO 3 2-
)), which makes up the major anionic compound, was derived from the equation of electro-neutrality of the solution because the sample volumes were too small for titration. However, an acid titration (Gran method, 1952) was carried out on three samples, giving similar results as the estimation by the ion balance. Equilibria computations were performed using the "Aqua" ion pair model [START_REF] Vallès | Elaboration d'un logiciel de géochimie appliqué à l'étude de la qualité des eaux[END_REF].

According to Francois et al. (1989), the ash content varies little during the season or during the biological cycle of the plant "Burgu". The ash alkalinity of the plant material, an indirect measure of the net proton excretion by the plant, has been estimated from an average ash analysis of the plant given in Table II.

Residual alkalinity and properties

Some authors have used alkalinity as an algebraic parameter to quantify the alkalisation of the nutrient solution by plants [START_REF] Vallès | Alkalization of nutrient solutions by sorghum[END_REF] or to calculate proton budgets or H + transfer processes involving vegetation (Van Breemen et al., 1983;[START_REF] Bourrié | Formalization of proton balances in elementary basins based upon the alkalinity concept and graph properties[END_REF][START_REF] Grimaldi | Importance de l'hydrolyse acide dans les systèmes pédologiques des régions tropicales humides[END_REF]. Residual alkalinity ( [START_REF] Beek | The alkalinity of alkali soils[END_REF], which is derived from the residual carbonate concept [START_REF] Eaton | Signifiance of carbonates in irrigation waters[END_REF] and generalised by [START_REF] Droubi | Generalized residual alkalinity concept; Application to prediction of the chemical evolution of natural waters by evaporation[END_REF], has been used successfully for recent quantitative studies, and particularly for hydrograph deconvolutions [START_REF] Christophersen | Modelling streamwater chemistry as a mixture of soil-water end-member. A step towards second generation acidification models[END_REF][START_REF] Ribolzi | Comparison of hydrograph deconvolutions using residual alkalinity, chloride and oxygen 18 as hydrochemical tracers[END_REF]. In this study, the residual alkalinity is used as a tracer in the interpretation and the quantification of observed changes in the soil chemistry during the forage production.

The alkalinity equals the algebraic sum of the strong base cation and strong acid anion equivalents [START_REF] Stumm | Aquatic chemistry -An introduction emphasing chemical equilibria in natural waters[END_REF]. In other words, it is the sum of the products of the alkaline compound concentration by the number of protons each compound can neutralise, minus the proton concentration [START_REF] Bourrié | Relations entre le pH, l'alcalinité, le pouvoir tampon et les équilibres de CO 2 dans les eaux naturelles[END_REF]:

Alk. = (HCO 3 - ) + (CaHCO 3 + ) + (MgHCO 3 + ) + ... ... + (CO 3 2- ) + (CaCO 3 0 ) + ... + (OH -) -(H + ) (1) or Alk. = (HCO 3 - ) + (CO 3 -- ) + (OH -) -(H + )
(2) where ( ) in equation 2 denote the total concentrations in equivalents. In most soils, the alkalinity equivalent is equal to the carbonate alkalinity because the proton and hydroxide ion concentrations and non-carbonate alkalinity are negligible (Voro'yeva andZamana, 1984, Keller et al., 1987). Alkalinity is conservative inasmuch as it is affected either by the addition or loss of CO 2 or by the dissolution/precipitation of a salt of strong acid or base [START_REF] Stumm | Aquatic chemistry -An introduction emphasing chemical equilibria in natural waters[END_REF]. However, in arid regions, the concentration of the solutions generally leads to the formation of calcite, usually the first mineral to precipitate. Calcite residual alkalinity (RA calcite ), is defined as the alkalinity of the solution after complete precipitation of the calcite :

RA calcite = Alk. -(Ca 2+ ) -(CaHCO 3 + ) -(CaCO 3 0 ) -... (3) Or RA calcite = Alk. -(Ca 2+ )
(4) Where (Ca 2+ ) in equation 4 denote the total calcium in the solution, i.e. the free calcium and the calcium ions due to mixed salts. This quantity is just as conservative during concentration as during dilution of solution affected by calcite precipitation or dissolution. This conservative aspect of the concept is shown below.

When an initial solution under-saturated with respect to calcite concentrates, as long as the equilibrium with the mineral is not reached, calcium content and alkalinity increase proportionally to the concentration of the solution. When equilibrium is reached, we have:

(Alk.) e = ξ (Alk.) i (5) and:

(Ca ++ ) e = ξ (Ca ++ ) i (6) where ξ denotes the concentration ratio of the solution, () i refers to the initial solution and () e the solution at equilibrium.

The difference between these two equations gives ( 6) -( 5) : (RA calcite ) e = ξ (RA calcite ) i (7) If the concentration increases (up to ξ'), calcite precipitates and a quantity "q" of equivalents of calcium and alkalinity is removed from the solution: (Alk.) f = ξ' ((Alk.) e -q) (8) (Ca ++ ) f = ξ' ((Ca ++ ) e -q) (9) Where () f denotes the final concentration. The difference between these two equations gives (8) -( 9) : (RA calcite ) f = ξ' (RA calcite ) e (10) Therefore, using ( 7) and ( 10) we can calculate:

(RA calcite ) f = ξ . ξ' (RA calcite ) i (11)
The RA calcite increases proportionally to the concentration ratio and therefore demonstrates the "conservative" property, even if it is obtained by the difference of two nonconservative parameters [START_REF] Ribolzi | Contrôle géochimique des eaux par la formation de calcite en milieu méditerranéen et en milieu tropical. Arguments d'équilibre et argument de bilan[END_REF]. This concept was extended to the successive formation of various minerals [START_REF] Droubi | Generalized residual alkalinity concept; Application to prediction of the chemical evolution of natural waters by evaporation[END_REF] and is known as Generalised Residual Alkalinity. For example, as we will see on our site, calcite + fluorite Residual Alkalinity (RA calcite+fluorite ) is the residual alkalinity after complete precipitation of calcite and fluorite:

RA calcite+fluorite = Alk. -(Ca ++ ) + (F -) (12)
This quantity exhibits the same conservative properties in solution affected by calcite and fluorite precipitation or dissolution. After the fluorite reaches saturation (equilibrium e'), of calcite and fluorite occur together. The formation of fluorite will remove calcium and fluoride from the solution with no consumption of alkalinity. If "t" moles of fluorite are formed: (RA calcite ) f' = ξ" ((RA calcite ) e' +t) (13) (F -) f' = ξ" ((F -) e' -t) ( 14) with f' refers to the final solution after fluorite precipitation (up to ξ"). ( 13) and ( 14) can be rewritten as (RA calcite+fluorite ) f' = ξ" (RA calcite+fluorite ) e' (15) which can be combined with (11) to give: (RA calcite+fluorite ) f' = ξ .ξ '.ξ" (RA calcite+fluorite ) i ( 16) The conservative property of RA calcite disappears but the RA calcite+fluorite remains conservative. This demonstration can also be applied for dilution processes. Moreover, the residual alkalinity is expressed in total concentration but the soil complex is sensitive to the ratio of free cations in the soil solution. However, this formulation takes into account the ion pairs due to mixed salts, which are eliminated by difference in the expression of the residual alkalinity. The increase in CO 2 due to the roots respiration can locally induce a decrease in the pH value, but the alkalinity is not changed [START_REF] Bourrié | Relations entre le pH, l'alcalinité, le pouvoir tampon et les équilibres de CO 2 dans les eaux naturelles[END_REF]. If this temporary decrease in pH is accompanied with calcite solubilisation, the calcium and alkalinity increase in the same proportion, but the difference RA = Alk. -(Ca 2+ ) is constant. The residual alkalinity is here used to describe and quantify the geochemical changes during cultivation for the two soil types.

Results

The pH values measured on saturated pastes before and after the cultivation are presented on Figure 1. A decrease in the pH of the field is locally observed indicating that the cultivation induced a acidification of the soil in the plot. The analyse of variance indicates that the decrease in soil pH value is not significative when the two soil types are analysed on the same sampling. It is however highly significative when the sodic soils are studied on a separated sampling. The decrease in pH value seems affect the soil which were the most alkaline before the "Burgu" cultivation. Anyway, this result shows that the acidification induced by the forage cultivation may be questioned, and set the necessity of a better assessment of this evolution, hence to resort to conservative parameters as the Residual Alkalinity. The wetting front quickly reaches the horizon of bedrock weathering in the brown steppe soils. 140 days are necessary for the same result in the sodic soils (Fig. 2). The soil leaching is then possible in the two types of soil.

The range of pH of the samplings in the pit (7.4 -9.5) is similar to the range of pH of the whole field before and after cultivation. The chemistry of the sampling appears therefore representative of the chemistry of the field. Moreover, these data are also very similar to the values presented by several authors (unpublished) on this site on the surrounding fields, for several years.

The main geochemical processes identified at the transition from brown steppe soil to sodic soil, and responsible for the changes in the soil chemistry, are calcite and fluorite precipitations [START_REF] Barbiéro | The alkaline soils of the Niger valley. Origins, formation and present evolution[END_REF]. In this case, the calcite + fluorite Residual Alkalinity (RA calcite+fluorite ) shows theoretically a conservative behaviour. All the soil solutions are widely under-saturated with respect to the Na-salts. Therefore, the sodium content, too, evolves conservatively, because the sodium ions are not involved in any precipitation. The effect of sodium adsorption on the exchange complex can disturb this conservative evolution and will be discussed below. The quantity of sodium and RA calcite+fluorite have been used to quantify the chemical evolution of the soil during cultivation.

Before cultivation, a linear relationship is observed between sodium and RA calcite+fluorite (Fig. 3). After cultivation, this relationship has changed (Fig. 4). The concentrations are lower and the relationship is no longer linear. Some points slightly depart from the straight line.

Discussion

Cultivation modified the sodic soil chemistry in different ways. The sodium and RA calcite+fluorite concentrations are significantly lower after cultivation. This suggests a dilution process. If this dilution is only due to leaching, a constant ratio between sodium and RA calcite+fluorite is expected and a linear relationship between the two (Fig. 4). Parts of the measurement depart from the straight line, suggesting that the conservative behaviour of one or both of the indicators be affected. Two others processes can affect the conservative property of the indicators. 1 Drastic changes of the chemical facies of a soil solution can modify the proportions of exchangeable cations. However, these exchanges concern essentially Ca and Na because the proportion of exchangeable Mg does not significantly change in this environment [START_REF] Barbiéro | Les sols alcalinisés sur socle dans la vallée du fleuve Niger. Origine de l'alcalinisation et évolution des sols sous irrigation[END_REF]. An increase/decrease in exchangeable sodium produces an equivalent decrease/increase in exchangeable calcium, i.e., an equivalent decrease/increase in the RA calcite+fluorite of the soil solution. So, this process can not disturb the relationship that links sodium content and RA calcite+fluorite . 2 The ionic root uptake induces an exchange between cations and protons, which also disturbs the conservative property of RA calcite+fluorite . The change of the pH in the rhizosphere, as well as in the nutrient solution, during plant growth, is well known [START_REF] Smiley | Rhizospere pH as influenced by plants, soils and nitrogen fertilizers[END_REF][START_REF] Marshner | In vivo measurement of root-induced pH changes at the soil root interface. Effect of plant species and nitrogen source[END_REF][START_REF] Bernardo | Nitrate/ammonium ratio effects on mineral element uptake by sorghum[END_REF]. This change depends on the nutrient source and on the plant species (Van Breemen et al., 1983) and is due in part to uptake of ammonium and potassium, and transfer of protons to the soil solution. Changes in pH due to CO 2 consumption by the root respiration do not influence alkalinity [START_REF] Bourrié | Relations entre le pH, l'alcalinité, le pouvoir tampon et les équilibres de CO 2 dans les eaux naturelles[END_REF] and cannot be responsible for the relation observed.

The samples that depart from the initial linear relationship were taken from a horizon, between 50 and 100 cm depth, which was intensely occupied by the roots and easy to delineate on the profiles. In compensation, in the superficial horizons, the root density was low or moderate and the points remain on the former relationship between the two indicators. Consequently, the non-linearity of the ratio between sodium and RA calcite+fluorite after cultivation seems to be related to the plant contribution in the de-alkalinisation process. The plant more easily affects the geochemical facies of the soil solution, as the concentration of the soil solution is low in this non-saline sodic soil. In detail, this change expresses a decrease of alkalinity and fluoride amount without a noticeable increase of calcium amount.

Chemical balance estimate on a 20 m²-plot: An estimate of the alkalinity uptake by the plant is: Alk. Uptake = K + Na + Ca + Mg -H 2 PO 4 + 0.3 N (15) Where the ion concentrations are expressed in equivalent and if the nitrogen source is ammonium (NH 4 + ) [START_REF] Vallès | Alkalization of nutrient solutions by sorghum[END_REF]. Therefore an underestimate is: Alk. Uptake > K + Na + Ca + Mg -H 2 PO 4 (16) 30 kg of dry matter of forage were produced on the 20 m²-plot during the cultivation. According to the ash analysis of the plant (Table II), the alkalinity up-taken by the plant is > 25 eq. This value must be increased with the possible acid generation through mineralisation of urea. In the roots zone, the average decrease in the RA calcite+fluorite from the initial relationship between RA calcite+fluorite and Na is 1.53 meq l -1 . This correspond to 25.3 eq alkalinity for a 20 m²-plot, a 0.5 m thick root-prospected horizon, a soil density of 1.65, and a soil/water ratio of 1/1 (the average soil/water ration of the saturated pastes). These values (25 eq and 25.3 eq) are very close. It is probably a coincidence, because of the uncertainty generated by this fast calculation, but we can take off that they have the same rough estimate. The non-linearity in the relation between Na and RA calcite+fluorite can therefore be attributed to the net acid contribution of the cultivation in the de-alkalinisation process.

On the other hand, an estimate of the part of leaching in the de-alkalinisation process during cultivation is given by the evolution of the upper limit of the RA calcite+fluorite value, which decreases from 17.1 before to 8.2 meq l -1 after cultivation, i.e. 8.9 meq l -1 . This correspond to 147 eq leached on a 20 m²-plot. This emphasises that leaching is the main phenomenon responsible for the chemical changes measured before and after cultivation (85%)

Conclusion:

The alkalinisation process is often difficult to reverse, as an impermeable soil makes it difficult to leach sodium and alkalinity. In the "terraces" of the Niger Valley, where a natural de-alkalinisation is observed, there was an opportunity to accelerate this natural process. The non-saline sodic soils result from the precipitation of calcite and fluorite in the soil solution. The cultivation of a local semi-aquatic fodder induces chemical changes, which have been attributed to two phenomena: 1. the large amount of water supplied during cultivation produces salt leaching and it is the main phenomena, responsible for 85% of the de-alkalinisation. 2. the root activity modifies the acid-base equilibrium and consumes alkalinity. This study reveals that root activity can affect significantly such diluted soil solutions in a very short time. It is responsible for 15% of the de-alkalinisation measured in the studied plot using RA calcite+fluorite .

When the minerals precipitating during evaporation of the soil solution are identified, it is possible to select chemical tracers using the residual alkalinity concept. In this case, the quantity of sodium and calcite + fluorite residual alkalinity (RA calcite + fluorite ) were found to be effective tracers and decreased proportionally under the influence of salt leaching. However, exchanges between cations and protons change the RA calcite + fluorite , without modifying the sodium amount. Their combined use allows separation and quantification of the action that was attributed to the plant from the leaching in the de-alkalinisation process. The use of the Residual alkalinity is advisable to design a sustainable management of irrigated sodic soils or saline soils.

(2)


Laurent Barbiero
email: laurent.barbiero@ird.sn

Sébastien Cunnac

Landing Mané

Caroline Laperrousaz

Claude Hammecker

Jean-Luc Maeght

Landing Mane

MAEGHT Jean Luc

L Barbiéro

B P 1386 Dakar

Sénégal

Salt distribution in the Senegal middle valley. Analysis of a saline structure on the future irrigation schemes

Keywords: Salinity, Electromagnetic induction, Irrigation Schemes, Senegal

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Senegal river valley has been recently managed by the surrounding countries taking advantage of the construction of two major dams, "Diama" in the delta and "Manantali" upriver in Mali [START_REF] Omvs | The objectives and the main outlines of the integrated development strategy of the senegal river basin[END_REF]. On the Senegalese side, the development of irrigation projects concerns about 240,000 ha. About 60,000 ha are today under full irrigation control. Most of the irrigation infrastructure was constructed by the donors as a response to the droughts of the 70s [START_REF] Verheye | Impact of climate and soil conditions on conception and implementation of irrigation schemes in the Senegal River basin[END_REF]. The ecological, economical and social changes induced by this new way of water management are studied in a common IRD/ISRA research program, which main objective is to define the conditions for the establishment of a sustainable irrigated agriculture in the Senegal valley.

A synthesis of the studies concerning the environmental side of the program is given by [START_REF] Boivin | Nianga, laboratoire de l'agriculture irriguée en moyenne vallée du Sénégal[END_REF]. The situation is mainly characterised by a lack of information prior to irrigation and absence of impact studies or keeps track of the environment.

The soil salinity is one of the major constraints for the development of irrigated agriculture. For example, in the Nianga irrigation scheme, which has been managed in 1974-1975, a large area has been deserted by the farmers due to excess of salinity. This is not due to a secondary salinisation but to the expression of the salt already present in the soil prior to the establishment of the rice culture. An inventory of the most suitable sites and particularly of the salt distribution, which is dangerously lacking, is a first prerequisite for a successful implementation of sustainable irrigated agriculture in these expensive infrastructures. Many irrigation schemes are today under construction along the creek N'Galenka, in the middle Senegal valley. This area has never been cultivated and the occurrence of saline soil can not be related to former land-use. Preliminary studies [START_REF] Laval | Rapport d'activités. Travaux d'expérimentation ORSTOM-FED dans la moyenne du Fleuve Sénégal[END_REF][START_REF] Zanolin | Etude de site pour l'implantation d'un réseau piézométrique sur périmètres irrigués (Vallée du fleuve Sénégal)[END_REF] have revealed that the salt distribution is not related with the present faint topography or geomorphological units. First, a saline area is here studied to better understand its variability, to describe its structure and spatial arrangement. Then the results are made profitable to attempt a regional survey of the saline soil distribution along the N'Galenka.

Site

The N'Galenka creek is located in the region of Podor (Northern Senegal) from 16°26 to 16°30 N and from 14°50 to 15°05 W (Fig. 1). The climate is of the semiarid type, characterised by a wet season (approximately 200-mm of rainfall) from July to September, a cold dry season from October to February and a hot dry season from March to June. Dry and warm winds increase evaporative demands and create a disequilibrium in the plant water balance. Average Class A pan evaporation exceeds average rainfall in each month and by over 2000 mm annually.

The Senegal valley was submitted to a recent transgression, called Nouakchottian (or Duinkerke) transgression, which started about 4300 B.P. [START_REF] Faure | Geoidal changes and shorelevel tilt along holocene estuaries: Senegal river area, West Africa[END_REF] up to Boghé, today located 380 km inland. According to [START_REF] Michel | Les bassins des fleuves Sénégal et Gambie. Etude géomorphologique[END_REF], this transgression is responsible for salt incorporation (neutral chemical facies) in the sediments into which the soil sequences were formed [START_REF] Maymard | The residual effect of submersion on soil structure (Demonstrated by applying S. Henin's index of structural instability)[END_REF]. The soils have developed after the post-Nouakchottian (or Taffolian) regression, which was accompanied by a mangrove vegetation, in the roots of which, considerable amounts of pyrite were formed by sulphur fixing bacteria. The salts and the acidity have subsequently been evacuated in part by the river swelling, but traces of salinity are still observed up to 300 km inland. The saline soils are mainly characterised by high variability in the chemical facies of the salinity at small distance. It can be of sodium/chloride, sodium/sulfate, magnesium/sulfate or calcium/sulfate type [START_REF] Barbiéro | Aeolian origine of some saline soil stripes in the Senegal middle valley. Morphological and geochemical considerations[END_REF].

Two major geomorphological units are distinguished in this region: depression and former river banks [START_REF] Fao-Sedagri | Etude hydro-agricole du bassin du fleuve Sénégal[END_REF], respectively occupied by Vertic Xerofluvents and Typic Xeropsamments [START_REF] Usda | Soil Taxonomy. A basic system for soil classification for making and interpreting soil survey[END_REF], locally known as Hollaldé and Fondé [START_REF] Maymard | The residual effect of submersion on soil structure (Demonstrated by applying S. Henin's index of structural instability)[END_REF]. Fondé soils exhibit a clay content around 30%, and a saturated hydraulic conductivity ranging from 10 -6 to 10 -5 m s -1 [START_REF] Meyer | Etude du transport des sels au cours des écoulements et de l'évolution des propriétés physiques du sol au contact de différentes solutions salines[END_REF]. Hollaldé soils are characterised by clay contents around 65% (smectite and interstratified: 60%, kaolinite: 30%, illite: 5%, chlorite: 5%) and saturated hydraulic conductivity about 10 -7 m s -1 . These clay soils constitute the best areas for irrigated rice cropping because of their adequate physical and chemical properties.

In the N'Galenka floodplain, where irrigation has never been performed, the water table occurs generally at about a depth of 7 m, but close to the surface under the irrigation schemes. The salt concentration in the groundwater in the middle Senegal valley varies from place to place ranging from 0.2 to 85 g l -1 [START_REF] Laval | Rapport d'activités. Travaux d'expérimentation ORSTOM-FED dans la moyenne du Fleuve Sénégal[END_REF]. In the N'Galenka region the electrical conductivity (EC) in the water table ranges from 0.15 to 1 dS m -1 .

The irrigation water has a very low Total Dissolved Content (EC ranges from 0.5 to 0.8 dS m -1 ) and does not therefore reduce the quality of irrigation in terms of secondary salinisation hazard. However, it exhibits a positive calcite residual alkalinity, as it is often the case in continental waters (Valles et al., 1991). Therefore, it could lead to secondary alkalisation if the carbonates brought at each irrigation remained in the soil profiles. The calcite residual alkalinity is lower than in the river Niger, where secondary alkalisation has occurred. Therefore the alkalisation hazard is lower in the Senegal valley than in the Niger valley. Specific researches about this eventual aspect of soil degradation by alkalisation and about the monitoring of this phenomenon are actually carried out [START_REF] Hammecker | Contribution of modelling solute and water transfer to the irrigation of saline soils in the valley of river Senegal[END_REF].

The study was carried out prior to the establishment of 48 irrigation schemes along the creek N'Galenka. These perimeters will be constructed on the former floodplain locally known as "walo" areas (Fig. 2). They are already alloted to the surrounding villages.

Material and method

Field salinity measurements

The soil salinity was measured using a portable electromagnetic conductivity meter (Geonics EM38). This portable device is today currently used to monitor spatial changes in soil salinity, hence to delineate saline areas. This instrument measures apparent soil electrical conductivity in millisiemens per metre (mS m -1 ). The calibration of the EM38 according to soil texture, moisture, and to the type of salinity was subjected to a specific study [START_REF] Laperrousaz | Influence de la variabilité du faciès chimique sur la conductivité électromagnétique[END_REF].

This work has been carried out in three stages. In a first stage, an area of 73 ha, located at the eastern part of the N'Galenka region around a large winding of the creek (sector VI, Fig. 3), has been prospected using vertical measurements according to a 25 x 25 m regular grid. In the vertical mode, 75% of the signal is estimated to come from the top 1.8 m of the soil [START_REF] Mcneill | Survey Interpretation Techniques: EM38[END_REF]. These data underwent a geostatistical treatment. In a second stage, 2 ha-area located in the first prospected area (Fig. 3), was subjected to a more detailed survey of the salinity (10 x 10 m regular grid), coupled with topography measurements. Finally, the saline area delineated on the computed map was compared to the aerial photograph of the site [START_REF] Geotronics | Couverture aérienne, mission Teledyne Geotronics (USA)[END_REF] in order to identify the relationship between the salt distribution and the actual or former geomorphological units. The regional survey of the whole N'Galenka was attempt from ECm measurements transects, located so as to maximise the intersection with the structures identified on the aerial photograph and that are suspected to be saline. ECm values over 100 mS m -1 are used to delineate the saline areas.

Geostatisical treatments

A chi-squared test showed that the data may not be assumed to have a normal distribution. Therefore, the calculation was performed on a theoretical distribution of the data by lognormal transformation as recommended by [START_REF] Dowd | Lognormal geostatistical[END_REF],

) 05 . 0 ) ( ln( ) ( + = i i x s x z (1)
where s(x i ) is the ECm data at x i , z(x i ) is the log-transformed data, and 0.05 is a constant because some s(x i ) were equal to 0. An estimate of the sample variogram is given by the formula :

γ ( ) ( ) ( ( ) ( )) ( ) h N h z x z x h i i i N h = - + = ∑ 1 2 2 1 (2)
where N(h) is the number of pairs of points and z(x i ) and z(x i+h ) are the log-transformed EC m value at x i and x i+h . Raw and directional variogram were calculated to detect an eventual anisotropy in the field salinity. The kriged map is built from a spherical model fitted on the sample variogram. This automatic computation was performed using the GEOSTAT-PC software [START_REF] Boivin | Geostat-PC. Logiciel interactif pour le calcul géostatistique[END_REF].

Results and discussion

Structure of the saline area on the prospected zone

High ECm values are observed, the maximum values being 390 mS/m, with an average value of 55.95 mS/m, and a standard deviation of 71.94 mS/m. The variation coefficient (128.6%) indicates a high dispersion of the data around this average value.

The experimental variogram built from the ECm data is presented in Fig. 4. The range is about 300 m and the nugget effect is low, indicating that the density of measurements is sufficient to describe the spatial structures at small distance. A slight anisotropy is detected by comparing the raw and directional variogram, but it has not been taken into account for the computation. It indicates a high dependence of the ECm values in the direction 45° (North East/South West) up to a distance of 300 m.

The computed ECm kriged map is presented in Fig. 5 and confirms the heterogeneity of the studied site in terms of salinity. A saline area clearly appeared, exhibiting the form of a stripe, which is about 200 m wide, and 800 m long, in the NE-SW direction. This ECm values distribution is consistent with the anisotropy detected by the directional variograms. Some isolated small saline areas are detected in the south of this main stripe.

The characteristics of this stripe are presented on Figure 6. The detail examination of the saline major stripe reveals that it is actually composed of two parallel minor stripes, which exhibit higher ECm values. A asymmetry can be detected in the boundary of the saline major strip, which is abrupt in the north and more diffuse in the south. This asymmetry is reinforced by the topographic data. The minor stripe located at north lies in a depression, whereas that located at the south is on a higher ground (Fig. 6 and7). This distribution of the saline area thus seems to depend on the site morphology, which will be studied from the aerial photographs.

Comparison with the aerial photograph

The salt distribution can be related with the trace of a former creek not easily perceptible in the field but detected on the aerial photograph. The northern minor stripe is located in the former creek bed whereas the southern one is fringing it on the southern bank (Fig. 8). This former creek can be followed on the aerial photograph, which can thus be used as support for a regional survey.

Survey of the whole N'Galenka area

The results of the regional survey, elaborated from ECm measurements on 11 transects, are presented on the Fig. 8. The saline stripe detected on the prospected area in sector VI is continuous in the north of the present N'Galenka creek bed in the sectors V and IV. This stripe is intersected by the present bed and the most recent non saline deposits of the N'Galenka in two points, the first along the large winding on sector VI, and the second one along another winding located between sector III and IV. These two points are set off by circles on Fig. 8. This organisation of the saline soil distribution indicates that it existed prior to the present N'Galenka and does not correspond to a recent remobilisation of the salt.

The saline stripe appears again down to the sector II, on the other side of the winding, where it is intersected by other deposits coming from the north in which 3 saline stripes were detected. These stripes coming from the north are in continuity with the Pont Gary site where Gascuel-Odoux and [START_REF] Gascuel-Odoux | Variability of variograms and spatial estimates due to soil sampling: a case study[END_REF] had also detected a saline stripe. Another saline stripe is detected in sector III on the other side of the N'Galenka. In compensation, the soil in sector VII, located upstream of the futur managed area, is non saline. After this work and the elaboration of the map (Fig. 8), several verifications on the field revealed that all the saline areas have been detected from these 11 transects, located so as to intersect the former geomorphological units. This confirms the efficiency of the method which was implemented on this site. During this regional prospecting, it appeared that the morphological characteristics of the stripe detected in sector VI were frequently observed and therfore, they are representative of the whole N'Galenka region. In particular, the major stripe is frequently composed of two minor stripes exhibiting higher ECm values, the northen one located in a depression and the southern one on higher ground. This detail is not presented on Fig. 8 where only the global saline structure is drawn.

The survey revealed that a large proportion, almost 10% of the future irrigation schemes, is occupated by saline soils. Each irrigation scheme is allotted to the surrounding villages before its establishment. Thus, according to this allotation, it can be noted that some villages will be particularly prejudicated by this saline soils distribution. In detail, the farmers of Nianga, Thiewle (sector V) and Pendaou (sector III) will be strongly prejudicated whereas those of Djamel (sector III) are not concerned with this problem.

Conclusion

As a result of climatological hazard in the past decades, the traditional agriculture in the Senegal valley has shifted to irrigated agriculture in the former floodplain areas. Many irrigation schemes are presently under construction along the creek N'Galenka. One of the first prerequisite, generally lacking, is an inventory of the salt distribution in this area, which appears as a major constraint for a sustainable irrigated agriculture. The systematic survey of the soil salinity using an electromagnetic conductivity meter (EM38), on a 73-ha prospected area, reveals a singular organisation. Indeed, the salt is distributed as a stripe in the SW-NE direction. This saline stripe, inherited from the past, is not correlated with the present but with the former geomorphology, which is detected on the aerial photographs. This relationship being established, it made it possible to survey the whole N'Galenka region from the prospecting on selected transects for ECm measurements. The regional study confirms that the difference in salinity are quite pronounced and that the saline soil distribution followed a regular pattern as stripes that are more than 10 km long and about 100 to 200 m wide, and related to the former geomorphology. Similar saline stripes were revealed by other prospecting studies in the lower middle valley, for example in Ouromadiou, Nianga, Pont Gary, Guia, M'Boyo [START_REF] Gascuel-Odoux | Variability of variograms and spatial estimates due to soil sampling: a case study[END_REF][START_REF] Laval | Rapport d'activités. Travaux d'expérimentation ORSTOM-FED dans la moyenne du Fleuve Sénégal[END_REF], Barbiéro et al., 1998). This structure appears therefore representative of the salt distribution in the lower middle valley. The specific characteristics of the saline stripes, which are actually composed of two parallel stripes, one in a depression and the other one on a higher ground, must be taken into account to propose a model for the origin of this salt distribution.

This method developed here, far from expensive in time and device, should be recommended for a fast mapping of the saline soil distribution before the establishment of new irrigated areas.

Fig. 1 .

 1 Fig. 1. Location of the study site in the middle Senegal Valley Fig. 2. Distribution of the future irrigation schemes along the N'Galenka creek. Fig. 3. Location of the prospected area along the large winding of the N'Galenka. Fig. 4. Raw and directional sample variogram. Fig. 5. ECm kriged map of the prospected area. Fig. 6. Distribution of ECm and topography values in the major saline stripe. Fig. 7. Relationship between ECm and topography. Fig. 6. Distibution of the saline soils in the whole N'Galenka region.

Acknowledgements

This work was partially supported by the regional pole for research on sudano-sahelian irrigated systems (PSI-CORAF).

Saint Louis