Guy Barles
email: guy.barles@lmpt.univ-tours.fr

Philippe Laurençot
email: philippe.laurencot@math.univ-toulouse.fr

Christian Stinner
email: christian.stinner@uni-due.de

Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation *

Keywords: convergence to steady state, degenerate parabolic equation, viscosity solutions, gradient source term AMS Classification: 35K65, 35B40, 35J70, 49L25, 35B05

HAL is

Introduction

We investigate the large time behaviour of non-negative and radially symmetric solutions to the initial-boundary value problem

     ∂ t u = ∆ p u + |∇u| q , x ∈ B, t ∈ (0, ∞), u = 0, x ∈ ∂B, t ∈ (0, ∞), u(x, 0) = u 0 (x), x ∈ B, (1.1)
where B := {x ∈ R N : |x| < 1} is the unit ball in R N , N ≥ 2, and the p-Laplacian operator is defined by ∆ p u = div(|∇u| p-2 ∇u).

1

We further assume the initial condition

u 0 ∈ W 1,∞ 0 (B)
is radially symmetric and non-negative and u 0 ≡ 0, (1.2) while the parameters p and q satisfy p ≥ 2 and 0 < q < p -1.

(1.

3)

The partial differential equation in (1.1) is a second-order parabolic equation featuring a diffusion term (possibly quasilinear and degenerate if p > 2) and a source term |∇u| q counteracting the effect of diffusion and depending solely on the gradient of the solution. The competition between the diffusion and the source term is already revealed by the structure of steady states to (1.1). Indeed, while it follows from [4, Theorem 1] that zero is the only steady state in C( B) when p ≥ 2 and q ≥ p -1, several steady states may exist when p ≥ 2 and q ∈ (0, p -1) [START_REF] Barles | Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non lipschitz nonlinearity[END_REF][START_REF] Ph | Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions[END_REF][START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF]. Another typical feature of the competition between diffusion and source is the possibility of finite time blow-up in a suitable norm, and this phenomenon has been shown to occur for (1.1) when p = 2 and q > 2 [START_REF] Ph | Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions[END_REF]. More precisely, it is established in [START_REF] Ph | Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions[END_REF] that, when p = 2 and q > 2, there are classical solutions to (1.1) for which the L ∞ -norm of the gradient blows up in finite time, the L ∞ -norm of the solution remaining bounded. These solutions may actually be extended to all positive times in a unique way within the framework of viscosity solutions [START_REF] Barles | On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF][START_REF] Tchamba | Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian[END_REF], the boundary condition being also satisfied in the viscosity sense. According to the latter, the homogeneous Dirichlet boundary condition might not always be fulfilled for all times, a property which is likely to be connected with the finite time blow-up of the gradient.

Coming back to the case where p and q fulfil (1.3) and several steady states may exist, a complete classification of steady states seems to be out of reach when B is replaced by an arbitrary open set of R N . Nevertheless, there are at least two situations in which the set of stationary solutions can be described, namely, when N = 1 and B = (-1, 1) [START_REF] Ph | Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions[END_REF][START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF] and when N ≥ 2 under the additional requirement that the steady states are radially symmetric and non-increasing, the latter being the first result of this paper. More precisely, we show that (1.1) has a one-parameter family of stationary solutions and that each stationary solution is characterized by the value of its maximum.

Theorem 1.1 Assume (1.3). Let w ∈ W 1,∞ (B) be a radially symmetric and non-increasing viscosity solution to -∆ p w -|∇w| q = 0 in B satisfying w = 0 on ∂B. Then there is ϑ ∈ [0, 1] such that w = w ϑ , where

w ϑ (x) := c 0 1 max{|x|,ϑ} ρ -ϑ β ρ -(β-1) 1/(p-1-q) dρ, x ∈ B, (1.4
)

for ϑ ∈ [0, 1] with β := 1 + (N -1)(p -1 -q) p -1 > 1 and c 0 := p -1 -q (p -q)β 1/(p-1-q) > 0. (1.5)
In particular, we have w 0 (x) = (c 0 /α) (1 -|x| α ) for x ∈ B, where α := (p -q)/(p -1 -q) > 1.

Remark 1.2 As already mentioned, for any M ∈ [0, c 0 /α] there is one and only one ϑ

∈ [0, 1] such that w ϑ L ∞ (B) = M as w ϑ L ∞ (B) is a decreasing function of ϑ ∈ [0, 1]
. This property plays an important role in the forthcoming analysis of the large time behaviour of solutions to (1.1).

Having a precise description of the set of steady states of (1.1) at our disposal, it is natural to investigate whether they attract the dynamics of (1.1) for large times. In other words, given a solution to (1.1), does it converge to a steady state as t → ∞? A positive answer to this question is given in [START_REF] Ph | Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions[END_REF][START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF] when N = 1, B = (-1, 1), and p and q fulfil (1.3). The one dimensional framework is fully exploited there as it allows the construction of a Liapunov functional by the technique developed in [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable[END_REF]. Such a nice tool does not seem to be available here and we instead use the theory of viscosity solutions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] and more precisely the relaxed half-limits method introduced in [START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF]. This approach has already been used in [START_REF] Barles | On the large time behaviour of solutions of Hamilton-Jacobi equations[END_REF][START_REF] Namah | Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations[END_REF][START_REF] Roquejoffre | Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations[END_REF] to investigate the large time behaviour of solutions to Hamilton-Jacobi equations and can be roughly summarized as follows: given a nonnegative and radially symmetric solution u to (1.1) which is bounded in W 1,∞ (B), the half-relaxed limits u * (x) := lim inf

(s,ε)→(t,0) u(x, ε -1 s) and u * (x) := lim sup (s,ε)→(t,0) u(x, ε -1 s), x ∈ B,
are well-defined, do not depend on t > 0, and are Lipschitz continuous viscosity supersolution and subsolution to -∆ p z -|∇z| q = 0 in B, z = 0 on ∂B, respectively, by [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]Lemma 6.1]. Clearly, u * ≤ u * on B but we cannot apply the comparison principle at this stage to conclude that u * ≥ u * on B. However, additional information are available in this particular case, namely that u * and u * are both non-negative, radially symmetric, non-increasing, and have the same maximal value. Extensive use of these properties allows us to prove that u * ≥ u * , from which we readily conclude that u * = u * is a Lipschitz continuous radially symmetric and non-increasing stationary solution to (1.1). Consequently, u * = u * = w ϑ for some ϑ ∈ [0, 1] by Theorem 1.1 and the assumption u 0 ≡ 0 prevents ϑ = 1. The convergence result we obtain actually reads as follows.

Theorem 1.3 Assume (1.2) and (1.3) and let u denote the (radially symmetric) viscosity solution to (1.1). Then there is a unique ϑ ∈ [0, 1) such that

lim t→∞ u(t) -w ϑ C( B) = 0.
Notice that Theorem 1.3 applies in particular in the semilinear case p = 2 with q ∈ (0, 1) according to (1.3). Still in the semilinear case p = 2, several results on the large time behaviour of solutions to (1.1) are also available when q ≥ 1 and B is replaced by an arbitrary open set Ω of R N [START_REF] Arrieta | Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena[END_REF][START_REF] Benachour | Decay estimates for a viscous Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions[END_REF][START_REF] Ph | Global solutions of inhomogeneous Hamilton-Jacobi equations[END_REF][START_REF] Tchamba | Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian[END_REF], including the convergence to zero of global solutions which are bounded in W 1,∞ (Ω).

The analysis in this paper being restricted to radially symmetric solutions, we define r := |x| and switch between the notation u = u(x, t) and u = u(r, t), whenever this is convenient.

For further use, we introduce the following notations:

F (s, X) := -|s| p-2 trace(X) -(p -2)|s| p-4 Xs, s -|s| q for (s, X) ∈ R N × R N ×N , (1.6)
its radially symmetric counterpart

f (r, µ, ζ) := -(p -1)|µ| p-2 ζ - N -1 r |µ| p-2 µ -|µ| q for (r, µ, ζ) ∈ (0, 1) × R × R, (1.7)
and the radially symmetric p-Laplacian operator

f 0 (r, µ, ζ) := -(p -1)|µ| p-2 ζ - N -1 r |µ| p-2 µ for (r, µ, ζ) ∈ (0, 1) × R × R. (1.8)
2 Radially symmetric and non-increasing stationary solutions

In this section, we prove Theorem 1.1, that is, if w is a radially symmetric, non-increasing, and Lipschitz continuous viscosity solution to the stationary equation

-∆ p w -|∇w| q = 0 in B, w = 0 on ∂B, (2.1)
then w = w ϑ for some ϑ ∈ [0, 1]. To this end, we first observe that, as a function of r = |x|, w is a viscosity solution to f (r, ∂ r w, ∂ 2 r w) = 0 in (0, 1) with w(1) = 0 (recall that f is defined in (1.7)). Next, as a preliminary step, let us first give a formal proof, assuming w to be in C 1 ( B) and solving (2.1) pointwise. In particular, we will derive an identity (see (2.3) below) which turns out to be valid for viscosity solutions as we shall see later on. As w is radially symmetric and in C 1 ( B), we have ∂ r w(0) = 0. In addition, by (2.1),

ϕ(r) := r N -1 (|∂ r w| p-2 ∂ r w)(r), r ∈ [0, 1],
fulfils ϕ ∈ W 1,∞ ((0, 1)) with ∂ r ϕ(r) = -r N -1 |∂ r w(r)| q ≤ 0 a.e. in (0, 1). Thus, ϕ is a nonincreasing function in [0, 1]. As moreover w is non-increasing with w(1) = 0, we have ∂ r w(1) ≤ 0. Now, either ∂ r w(1) = 0 and thus ϕ(1) = 0. Since ϕ is non-increasing with ϕ(0) = 0, we conclude that ϕ ≡ 0. This implies w = w 1 ≡ 0. Or ∂ r w(1) < 0, and the continuity and monotonicity of ϕ warrant that there is a unique ϑ ∈ [0, 1) such that ϕ = 0 in [0, ϑ] and ϕ < 0 in (ϑ, 1]. Hence, ∂ r ϕ(r) = -r [(N -1)(p-1-q)]/(p-1) |ϕ(r)| q/(p-1) = -r β-1 (-ϕ(r)) q/(p-1) in (ϑ, 1).

After integration we obtain

- p -1 p -1 -q (-ϕ(r)) (p-1-q)/(p-1) + 1 β r β = γ for r ∈ (ϑ, 1)
with some constant γ ∈ R. Introducing

χ(z) := p -1 p -1 -q |z| p-2-q z for z ∈ R, (2.2)
we end up with

r β-1 χ(∂ r w(r)) + 1 β r β = γ for r ∈ (ϑ, 1) (2.3)
as ∂ r w < 0 in (ϑ, 1). Letting r ց ϑ implies γ = ϑ β /β owing to ∂ r w(ϑ) = 0 and 0 < q < p -1. Furthermore, due to ∂ r w < 0 in (ϑ, 1), we have

- p -1 p -1 -q r (N -1)/(p-1) (-∂ r w(r)) p-1-q = 1 β ϑ β -r β for r ∈ (ϑ, 1).
Hence, we conclude

∂ r w(r) = - p -1 -q (p -1)β r -ϑ β r -(β-1) 1/(p-1-q)
for r ∈ (ϑ, 1).

Using w(1) = 0 and the definition of c 0 , a further integration implies

w(r) = c 0 1 r ρ -ϑ β ρ -(β-1) 1/(p-1-q) dρ = w ϑ (r) for r ∈ [ϑ, 1].
Furthermore, we get w(r) = w(ϑ) for any r ∈ [0, ϑ] since ∂ r w ≡ 0 in [0, ϑ] and we conclude that w = w ϑ .

We now turn to the proof of Theorem 1.1 and first establish some preliminary results. We recall that, by the Rademacher theorem, a Lipschitz continuous function v ∈ W 1,∞ ((0, 1)) is differentiable a.e. and the measure of the differentiability set

D(v) := {r 0 ∈ (0, 1) : ∂ r v(r 0 ) exists } is thus equal to one. Lemma 2.1 Let v ∈ W 1,∞ ((0, 1
)) be a non-negative and non-increasing viscosity supersolution to

f 0 (r, ∂ r z, ∂ 2 r z) = 0 in (0, 1), (2.4)
the Hamiltonian f 0 being defined in (1.8). Then, if r 1 ∈ D(v) and r 2 ∈ D(v) are such that r 1 < r 2 , we have r

(N -1)/(p-1) 2 ∂ r v(r 2 ) ≤ r (N -1)/(p-1) 1 ∂ r v(r 1 ).
Proof. Take 0 < r 1 < r 2 < 1 with r 1 , r 2 ∈ D(v) and assume for contradiction that

ξ 1 := r (N -1)/(p-1) 1 ∂ r v(r 1 ) < r (N -1)/(p-1) 2 ∂ r v(r 2 ) =: ξ 2 .
As v is non-increasing we have ξ 2 ≤ 0. Now take ξ 1 < η 1 < η 2 < ξ 2 ≤ 0 and define Φ by

r (N -1)/(p-1) ∂ r Φ(r) = η 1 + (η 2 -η 1 ) r -r 1 r 2 -r 1 , r ∈ [r 1 , r 2 ],
along with Φ(r 1 ) = 0. On the one hand, v-Φ is continuous in [r 1 , r 2 ] and thus attains its minimum at a point

r 0 ∈ [r 1 , r 2 ].
On the other hand, we have

∂ r (v -Φ)(r 1 ) = ξ 1 -η 1 r (N -1)/(p-1) 1 < 0 and ∂ r (v -Φ)(r 2 ) = ξ 2 -η 2 r (N -1)/(p-1) 2
> 0 so that we cannot have r 0 = r 1 or r 0 = r 2 . Thus, r 0 ∈ (r 1 , r 2 ) and, since v is a viscosity supersolution to (2.4), we have

- 1 r N -1 0 ∂ r r N -1 |∂ r Φ| p-2 ∂ r Φ (r 0 ) ≥ 0. Since r (N -1)/(p-1) ∂ r Φ(r) ≤ η 2 < 0 for r ∈ [r 1 , r 2 ] we obtain -r N -1 |∂ r Φ| p-2 ∂ r Φ (r) = r N -1 |∂ r Φ(r)| p-1 = -r (N -1)/(p-1) ∂ r Φ(r) p-1 = η 1 + (η 2 -η 1 ) r -r 1 r 2 -r 1 p-1
.

Differentiating and taking r = r 0 , we end up with

0 ≤ -∂ r r N -1 |∂ r Φ| p-2 ∂ r Φ (r 0 ) = (p -1) η 1 + (η 2 -η 1 ) r 0 -r 1 r 2 -r 1 p-3 η 1 + (η 2 -η 1 ) r 0 -r 1 r 2 -r 1 η 2 -η 1 r 2 -r 1 < 0,
and a contradiction. //// In order to show that a viscosity solution to (2.1) satisfies (2.3), we next prove that the left-hand side of (2.3) is non-increasing for a supersolution to (2.1).

Lemma 2.2 Let w ∈ W 1,∞ ((0, 1)) be a non-increasing viscosity supersolution to f (r, ∂ r z, ∂ 2 r z) = 0 in (0, 1) such that w L ∞ ((0,1)) > 0 and w(1) = 0, and define r 0 ∈ [0, 1] by r 0 := inf r ∈ (0, 1] : w(r) < w L ∞ ((0,1)) .

If r 1 ∈ D(w) and r 2 ∈ D(w) are such that r 0 < r 1 < r 2 , then

r β-1 1 χ(∂ r w(r 1 )) + r β 1 β ≥ r β-1 2 χ(∂ r w(r 2 )) + r β 2 β ,
the parameter β and the function χ being defined in (1.5) and (2.2), respectively.

Proof. The properties of w imply r 0 ∈ [0, 1). As w is non-increasing and Lipschitz continuous, the definition of r 0 yields that there is a sequence

(̺ n ) n≥1 such that ̺ n ∈ D(w), ∂ r w(̺ n ) < 0 and ̺ n ց r 0 as n → ∞. Pick r 1 ∈ D(w) ∩ (r 0 , 1).
For n large enough, we have r 1 > ̺ n . Since w is clearly also a supersolution to (2.4), we infer from Lemma 2.1 that

r (N -1)/(p-1) 1 ∂ r w(r 1 ) ≤ ̺ (N -1)/(p-1) n ∂ r w(̺ n ) < 0
for n large enough. Consequently,

r (N -1)/(p-1) 1 ∂ r w(r 1 ) < 0 for r 1 ∈ D(w) ∩ (r 0 , 1). (2.5)
Assume now for contradiction that there are r 1 , r 2 ∈ (r 0 , 1) ∩ D(w) such that r 1 < r 2 and

r β-1 1 χ(∂ r w(r 1 )) + r β 1 β < r β-1 2 χ(∂ r w(r 2 )) + r β 2 β .
As ∂ r w(r 1 ) < 0 by (2.5), we have χ(∂ r w(r 1 )) < 0 and we can choose two real numbers η 1 and η 2 such that

r β-1 1 χ(∂ r w(r 1 )) + r β 1 β < η 1 < η 2 < r β-1 2 χ(∂ r w(r 2 )) + r β 2 β , η 1 < r β 1 β ,
and

a := 1 - β(η 2 -η 1 ) r β 2 -r β 1 ∈ (0, 1). Indeed we first choose η 1 ∈ (r β-1 1 χ(∂ r w(r 1 )) + (r β 1 /β), r β 1 /β
) and then η 2 > η 1 close enough to η 1 in order to have a ∈ (0, 1). Setting now

A := η 1 -(1 -a) r β 1 β = η 2 -(1 -a) r β 2 β ,
let Φ denote the solution to

r β-1 χ(∂ r Φ(r)) + a r β β = A, r ∈ [r 1 , r 2 ], (2.6)
such that Φ(r 1 ) = 0. Observe that the choice of a and A ensure that and

r β-1 i χ(∂ r Φ(r i )) + r β i β = η i for i = 1, 2. (2.7) Due to A -a r β 1 β = η 1 - r β 1 β < 0 we conclude by (2.6) that χ(∂ r Φ(r)) = r -(β-1) A -a r β β ≤ r -(β-1) A -a r β 1 β < 0 for r ∈ [r 1 , r 2 ]. This implies that ∂ r Φ(r) < 0 for r ∈ [r 1 , r 2 ], so that Φ ∈ C 2 ([r 1 , r 2 ]
) by (2.6). In addition,

(-∂ r Φ(r)) p-1-q = p -1 -q p -1 a β r -Ar -(β-1) , r ∈ [r 1 , r 2 ], hence ∂ r Φ(r) = - p -1 -q p -1 a β r -Ar -(β-1) 1/(p-1-q) , r ∈ [r 1 , r 2 ].
Furthermore, due to (2.7) and the choice of η 1 , we obtain

r β-1 1 χ(∂ r w(r 1 )) + r β 1 β < η 1 = r β-1 1 χ(∂ r Φ(r 1 )) + r β 1 β .
This implies χ(∂ r w(r 1 )) < χ(∂ r Φ(r 1 )) and, since χ is increasing,

∂ r w(r 1 ) < ∂ r Φ(r 1 ).
Similarly, we conclude

∂ r w(r 2 ) > ∂ r Φ(r 2 ). Now w -Φ is a continuous function in [r 1 , r 2 ]
and thus attains its minimum at some

r m ∈ [r 1 , r 2 ].
The above two inequalities prevent r m to be equal to r 1 or r 2 and, since w is a viscosity supersolution to f (r, ∂ r v, ∂ 2 r v) = 0 in (0, 1), we have

- 1 r N -1 m ∂ r r N -1 |∂ r Φ(r)| p-2 ∂ r Φ(r) (r m ) -|∂ r Φ(r m )| q ≥ 0. But as ∂ r Φ < 0, (2.6) implies -∂ r r N -1 |∂ r Φ(r)| p-2 ∂ r Φ(r) = ∂ r r N -1 |∂ r Φ(r)| p-1 = ∂ r p -1 -q p -1 r β-1 χ(∂ r Φ(r)) (p-1)/(p-1-q) = -ar β-1 p -1 -q p -1 r β-1 χ(∂ r Φ(r)) [(p-1)/(p-1-q)]-2 p -1 -q p -1 r β-1 χ(∂ r Φ(r))
= ar (β-1)(p-1)/(p-1-q) p -1 -q p -1 χ(∂ r Φ(r))

[(p-1)/(p-1-q)]-1 = ar N -1 |∂ r Φ(r)| q for r ∈ [r 1 , r 2 ], (2.8)
so that

- 1 r N -1 m ∂ r r N -1 |∂ r Φ(r)| p-2 ∂ r Φ(r) (r m ) -|∂ r Φ(r m )| q = (a -1)|∂ r Φ(r m )| q < 0 since a < 1,
and a contradiction.

//// In a similar way we now establish that the left-hand side of (2.3) is non-decreasing for viscosity subsolutions to (2.1).

Lemma 2.3 Let w ∈ W 1,∞ ((0, 1
)) be a non-increasing viscosity subsolution to f (r, ∂ r z, ∂ 2 r z) = 0 in (0, 1) such that w L ∞ ((0,1)) > 0 and w(1) = 0, and define r 0 ∈ [0, 1] by

r 0 := inf r ∈ (0, 1] : w(r) < w L ∞ ((0,1)) . If r 1 ∈ D(w) and r 2 ∈ D(w) are such that r 0 < r 1 < r 2 , then r β-1 1 χ(∂ r w(r 1 )) + r β 1 β ≤ r β-1 2 χ(∂ r w(r 2 )) + r β 2 β .
Proof. The properties of w imply r 0 ∈ [0, 1). Assume for contradiction that there are r 1 , r 2 ∈ (r 0 , 1) ∩ D(w) such that r 1 < r 2 and

r β-1 1 χ(∂ r w(r 1 )) + r β 1 β > r β-1 2 χ(∂ r w(r 2 )) + r β 2 β .
We may then choose η 1 , η 2 ∈ R such that

r β-1 1 χ(∂ r w(r 1 )) + r β 1 β > η 1 > η 2 > r β-1 2 χ(∂ r w(r 2 )) + r β 2 β ,
and define

a := 1 + β(η 1 -η 2 ) r β 2 -r β 1 > 1 and A := η 1 + (a -1) r β 1 β = η 2 + (a -1) r β 2 β .
Let Φ denote the solution to

r β-1 χ(∂ r Φ(r)) + a r β β = A, r ∈ [r 1 , r 2 ], (2.9)
such that Φ(r 1 ) = 0. Thanks to the choice of a and A, we have

r β-1 i χ(∂ r Φ(r i )) + r β i β = η i for i = 1, 2, (2.10)
and the monotonicity of w implies that

A -a r β 1 β = η 1 - r β 1 β < r β-1 1 χ(∂ r w(r 1 )) ≤ 0.
Consequently,

χ(∂ r Φ(r)) = r -(β-1) A -a r β β ≤ r -(β-1) A -a r β 1 β < 0 for r ∈ [r 1 , r 2 ], hence ∂ r Φ(r) < 0 for r ∈ [r 1 , r 2 ].
We then conclude from (2.9) that Φ ∈ C 2 ([r 1 , r 2 ]). Furthermore, due to (2.10), the choices of η 1 and η 2 , and the monotonicity of χ, we obtain

∂ r w(r 1 ) > ∂ r Φ(r 1 ) and ∂ r w(r 2 ) < ∂ r Φ(r 2 ). Now w -Φ is a continuous function in [r 1 , r 2
] and thus attains its maximum at some point r m ∈ [r 1 , r 2 ]. The above two inequalities prevent r m to be equal to r 1 or r 2 and, since w is a viscosity subsolution to f (r, ∂ r v, ∂ 2 r v) = 0 in (0, 1), we have

- 1 r N -1 m ∂ r r N -1 |∂ r Φ(r)| p-2 ∂ r Φ(r) (r m ) -|∂ r Φ(r m )| q ≤ 0.
But, owing to ∂ r Φ(r) < 0, (2.9) and a > 1, we conclude similarly to (2.8) that

- 1 r N -1 m ∂ r r N -1 |∂ r Φ(r)| p-2 ∂ r Φ(r) (r m ) -|∂ r Φ(r m )| q = (a -1)|∂ r Φ(r m )| q > 0
and end up with a contradiction.

////

We are now in a position to prove Theorem 1.1. The keystone of the proof is that, according to Lemma 2.2 and Lemma 2.3, any non-increasing viscosity solution to f (r, ∂ r v, ∂ 2 r v) = 0 in (0, 1) satisfying w(1) = 0 has to fulfil (2.3).

Proof of Theorem 1.1. Let w ∈ W 1,∞ ((0, 1)) be a non-increasing viscosity solution to f (r, ∂ r v, ∂ 2 r v) = 0 in (0, 1) satisfying w(1) = 0. Either w ≡ 0 = w 1 or M := w L ∞ ((0,1)) > 0 and we define r 0 ∈ [0, 1) by r 0 := inf{r ∈ (0, 1] : w(r) < M }. Now, owing to Lemma 2.2 and Lemma 2.3, there is a constant γ ∈ R such that

r β-1 χ(∂ r w(r)) + r β β = γ (2.11)
for any r ∈ (r 0 , 1) ∩ D(w) and thus a.e. in (r 0 , 1). Combining the monotonicity of w and χ with (2.11), we moreover deduce that

γ ≤ r β 0 β (2.12)
and

∂ r w(r) = - p -1 -q p -1 r β -γr -(β-1) 1/(p-1-q)
for a.e. r ∈ (r 0 , 1).

Integrating and using the boundary condition w(1) = 0, we obtain

w(r) = 1 r p -1 -q (p -1)β ρ -γβρ -(β-1)
1/(p-1-q) dρ for any r ∈ [r 0 , 1].

Recalling w(r) ≡ M for r ∈ [0, r 0 ] and the definition of c 0 , we conclude

w(r) = c 0 1 max{r,r0} ρ -γβρ -(β-1) 1/(p-1-q) dρ, r ∈ [0, 1]. (2.13)
It remains to show that γ = r β 0 /β in order to obtain that w = w r0 . Consider first the case r 0 = 0. Since β > 1, the Lipschitz continuity of w yields γ = 0 = r β 0 /β by letting r ց 0 in (2.11). Next, if r 0 ∈ (0, 1), we assume for contradiction that γ < r β 0 /β. Then we fix ϑ ∈ [0, r 0 ) such that γ < ϑ β /β and choose Λ > 1 such that

Λ p-1-q < 1 + ϑ β -γβ.
This choice of Λ implies that the function

g(r) := 1 -γβr -β -Λ p-1-q 1 -ϑ β r -β , r ∈ (r 0 , 1), satisfies g ′ (r) = β 2 r -β-1 γ -Λ p-1-q ϑ β β ≤ β 2 r -β-1 γ - ϑ β β < 0, r ∈ (r 0 , 1),
and thus

g(r) ≥ g(1) ≥ 1 -γβ -Λ p-1-q + ϑ β > 0, r ∈ [r 0 , 1]. Consequently, 1 -γβr -β > Λ p-1-q 1 -ϑ β r -β , r ∈ [r 0 , 1],
and it follows from (2.13) that

∂ r w(r) = -c 0 r 1/(p-1-q) 1 -γβr -β 1/(p-1-q)
< -c 0 r 1/(p-1-q) Λ 1 -ϑ β r -β 1/(p-1-q) = Λ∂ r w ϑ (r), r ∈ (r 0 , 1).

In particular, w(r) -Λw ϑ (r) ≤ w(r 0 ) -Λw ϑ (r 0 ) for r ∈ [r 0 , 1]. Furthermore,

w(r) -Λw ϑ (r) = w(r 0 ) -Λw ϑ (r) ≤ w(r 0 ) -Λw ϑ (r 0 ), r ∈ [0, r 0 ],
thanks to the monotonicity of w ϑ , and the function w -Λw ϑ has a global maximum at r 0 . Since w ϑ ∈ C 2 ((ϑ, 1)), ϑ < r 0 , and w is a viscosity subsolution to f (r, ∂ r v, ∂ 2 r v) = 0 in (0, 1), we conclude that f (r 0 , ∂ r (Λw ϑ )(r 0 ), ∂ 2 r (Λw ϑ )(r 0 )) ≤ 0. However, as Λ > 1 and ϑ < r 0 , we clearly have

f (r 0 , ∂ r (Λw ϑ )(r 0 ), ∂ 2 r (Λw ϑ )(r 0 )) = Λ p-1 -Λ q |∂ r w ϑ (r 0 )| q > 0,
and the contradiction. Therefore, γ = r β 0 /β and w = w r0 , which completes the proof. //// 3 Some properties of solutions to (1.1)

We now focus on time-dependent solutions to (1.1) and establish some qualitative properties of non-negative and radially symmetric viscosity solutions to (1.1) which are needed to analyse their large time behaviour.

Proposition 3.1 Assume that u 0 , p, and q fulfil (1.

2) and (1.3). There is a unique non-negative viscosity solution

u ∈ C( B × [0, ∞)) to (1.1
) such that u(x, t) = 0 for x ∈ ∂B and x -→ u(x, t) is radially symmetric and belongs to W 1,∞ (B) for all t ≥ 0. In addition, there is a constant A 0 > 0 depending only on p, q, and u 0 , and a decreasing function

W ∈ C 1 ([0, ∞)) such that 0 ≤ u(x, t) ≤ A 0 and -A 0 ≤ ∇u(x, t) • x |x| ≤ W (t) , (x, t) ∈ B × [0, ∞), (3.1)
and W (t) -→ 0 as t → ∞.

Proof.

We first derive the expected properties on suitable approximations to (1.1) which we introduce now. For ε ∈ (0, 1), let a ε ∈ C ∞ ([0, ∞)) and b ε ∈ C ∞ ([0, ∞)) be two functions such that • a ε is bounded and increasing and a ε (ξ

) := (ε 2 + ξ) (p-2)/2 for ξ ∈ [0, ε -1 ],
• b ε is increasing, Lipschitz continuous, and b ε (ξ) := (ε 2 + ξ) q/2 -ε q for ξ ∈ [0, ε -1 ].

In addition, owing to the properties (1.2) of u 0 , there exists a sequence (u 0ε ) ε∈(0,1) of non-negative and radially symmetric functions in C ∞ ( B) such that

u 0ε L ∞ (B) ≤ u 0 L ∞ (B) + ε , ∇u 0ε L ∞ (B) ≤ 2 ∇u 0 L ∞ (B) , (3.2)
and lim

ε→0 u 0ε -u 0 C( B) = 0.
Fix ε ∈ (0, 1). According to the properties of a ε , b ε , and u 0ε , it follows from [START_REF] Ladyženskaja | Linear and Quasi-linear Equations of Parabolic Type[END_REF] that the initialboundary value problem

     ∂ t u ε = div(a ε (|∇u ε | 2 )∇u ε ) + b ε (|∇u ε | 2 ), x ∈ B, t ∈ (0, ∞), u ε = 0, x ∈ ∂B, t ∈ (0, ∞), u ε | t=0 = u 0ε , x ∈ B, (3.3)
has a unique non-negative classical solution u ε . In addition, x -→ u ε (t, x) is radially symmetric for every t ≥ 0 and the comparison principle entails that

0 ≤ u ε (x, t) ≤ u 0ε L ∞ (B) ≤ u 0 L ∞ (B) + ε, (x, t) ∈ B × [0, ∞). (3.4)
We next derive some estimates on the gradient of u ε and begin with the normal trace ∂ r u ε (1, t). Let L ε be the parabolic operator

L ε z := ∂ t z - 1 r N -1 ∂ r r N -1 a ε |∂ r z| 2 ∂ r z -b ε |∂ r z| 2 , (r, t) ∈ (0, 1) × (0, ∞),
and fix

A 0 ∈ ( √ 3ε, ε -1/2 ) such that A 0 ≥ 2 1/(p-1-q) + 2 1 + u 0 L ∞ (B) + ∇u 0 L ∞ (B) . (3.5)
Then, thanks to the properties of a ε , b ε , and (3.5), the function ψ defined by ψ(r)

:= A 0 (1 -r) for r ∈ [0, 1] satisfies L ε ψ(r) = 1 r N -1 ∂ r r N -1 a ε A 2 0 A 0 -b ε A 2 0 = N -1 r a ε A 2 0 A 0 -b ε A 2 0 ≥ ε 2 + A 2 0 (p-2)/2 A 0 -ε 2 + A 2 0 q/2 + ε q ≥ ε 2 + A 2 0 (p-2)/2 ε 2 + A 2 0 -ε -ε 2 + A 2 0 q/2 ≥ ε 2 + A 2 0 (p-1)/2 1 - ε ε 2 + A 2 0 -ε 2 + A 2 0 q/2 ≥ 1 2 ε 2 + A 2 0 (p-1)/2 -ε 2 + A 2 0 q/2 ≥ 0, r ∈ (0, 1].
Furthermore, (3.2), (3.4), and (3.5) entail that

u ε 1 2 , t ≤ 1 + u 0 L ∞ (B) ≤ A 0 2 = ψ 1 2 , t ≥ 0 , and
u 0ε (r) = - 1 r ∂ r u 0ε (̺) d̺ ≤ 2 ∇u 0 ∞ (1 -r) ≤ ψ(r) , r ∈ 1 2 , 1 . Since L ε u ε = 0 in (1/2, 1) × (0, ∞), the comparison principle ensures that u ε (r, t) ≤ A 0 (1 -r) for (r, t) ∈ (1/2, 1) × (0, ∞). Since u ε (1, t) = 0, this implies in particular that 0 ≤ -∂ r u ε (1, t) ≤ A 0 for t ≥ 0.
Recalling that u ε (t) is radially symmetric and smooth, we thus have

-A 0 ≤ ∂ r u ε (1, t) ≤ 0 = ∂ r u ε (0, t) , t ≥ 0 . (3.6)
We next estimate the gradient of u ε in B. For that purpose, we introduce the parabolic operator

M ε z := ∂ t z -∂ r a ε z 2 + 2a ′ ε z 2 z 2 ∂ r z - N -1 r a ε z 2 + 2a ′ ε z 2 z 2 + 2b ′ ε z 2 z 2 ∂ r z + N -1 r 2 a ε z 2 z
for (r, t) ∈ (0, 1) × (0, ∞) and readily deduce from (3.3) that

M ε ∂ r u ε = 0 in (0, 1) × (0, ∞). (3.7)
Observe next that

∂ r u ε (r, 0) ≥ -2 ∇u 0 L ∞ (B) ≥ -A 0 by (3.2) and (3.5) and
M ε (-A 0 ) = - N -1 r 2 a ε A 2 0 A 0 ≤ 0 ,
which, together with (3.6), (3.7), and the comparison principle implies that

-A 0 ≤ ∂ r u ε (r, t), (r, t) ∈ [0, 1] × [0, ∞). (3.8)
Finally, let W ε ∈ C 1 ([0, ∞)) be the solution to the ordinary differential equation

dW ε dt + (N -1) a ε W 2 ε W ε = 0, W ε (0) = 2 ∇u 0 L ∞ (B) . (3.9)
Then W ε is positive and decreasing, W ε (0) ≥ ∂ r u ε (r, 0) for r ∈ (0, 1) by (3.2), and M ε W ε ≥ 0 in (0, 1) × (0, ∞) by (3.9). Recalling (3.7), we deduce from the comparison principle that

∂ r u ε (r, t) ≤ W ε (t), (r, t) ∈ [0, 1] × [0, ∞). (3.10)
Finally, we argue as in [START_REF] Gilding | The Cauchy problem for u t = ∆u + |∇u| q[END_REF]Lemma 5] to deduce from (3.3), (3.4), (3.8), and (3.10) that there is a constant C depending on ∇u 0 L ∞ (B) , p, q, and N , such that

|u ε (x, t 1 ) -u ε (x, t 2 )| ≤ C(|t 1 -t 2 | + |t 1 -t 2 | 1/2 ) (3.11)
for any x ∈ B, t 1 , t 2 ∈ [0, ∞) and ε ∈ (0, 1). Indeed, consider t 1 = t 2 and set τ := |t 1 -t 2 | 1/2 > 0 and L := max{A 0 , 2 ∇u 0 L ∞ (B) }. Since (3.8), (3.10), and the Dirichlet boundary conditions imply that

|u ε (x, t)| ≤ L dist(x, ∂B) for (x, t) ∈ B × [0, ∞), we have |u ε (x 0 , t 1 ) -u ε (x 0 , t 2 )| ≤ 2L dist(x 0 , ∂B) ≤ 2Lτ if dist(x 0 , ∂B) ≤ τ. (3.12)
If dist(x 0 , ∂B) > τ and ε ∈ (0, 1/L), we infer from (3.3), the properties of (a ε , b ε ), and

|∇u ε | ≤ L in B × [0, ∞) that |u ε (x 0 , t 1 ) -u ε (x 0 , t 2 )| = 1 |B|τ N {|x-x0|<τ } (u ε (x 0 , t 1 ) -u ε (x 0 , t 2 ))dx = 1 |B|τ N {|x-x0|<τ } (u ε (x, t) -u ε (x 0 , t))dx t=t2 t=t1 - t2 t1 {|x-x0|<τ } ∂ t u ε (x, t)dxdt ≤ 2L |B|τ N {|x-x0|<τ } |x -x 0 |dx + 1 |B|τ N t2 t1 {|x-x0|<τ } div(a ε (|∇u ε | 2 )∇u ε ) + b ε (|∇u ε | 2 ) (x, t)dxdt ≤ 2LN N + 1 τ + 1 |B|τ N t2 t1 {|x-x0|<τ } (ε 2 + |∇u ε | 2 ) q/2 (x, t)dxdt + 1 |B|τ N t2 t1 {|x-x0|=τ } a ε (|∇u ε | 2 )|∇u ε | (y, t)dSdt ≤ 2LN N + 1 τ + (1 + L 2 ) q/2 |t 1 -t 2 | + N τ (1 + L 2 ) (p-2)/2 L|t 1 -t 2 | ≤ 2LN N + 1 τ + (1 + L 2 ) q/2 τ 2 + N (1 + L 2 ) p/2 τ.
Combining (3.12) and the above estimate gives the claim (3.11).

We can now pass to the limit as ε → 0. Owing to (3.4)

dW dt + (N -1) |W | p-2 W = 0, W (0) = 2 ∇u 0 L ∞ (B) .
In fact, W (t) = W (0)

M ∞ := lim t→∞ u(t) L ∞ (B) > 0. (3.13)
Proof. Any positive constant being obviously a supersolution to (1.1), the time monotonicity of the L ∞ (B)-norm of u readily follows from the comparison principle. Next, since u 0 ≡ 0 by (1.2), there is x 0 ∈ B, ̺ > 0, and m > 0 such that

B ̺ (x 0 ) := {x ∈ R N : |x -x 0 | < ̺} ⊂ B and u 0 (x) ≥ m for x ∈ B ̺ (x 0 ).
Introducing v λ (x) := λ (p-q)/(p-1-q) w 0 (|x -x 0 |/λ) for x ∈ B λ (x 0 ) and λ ∈ (0, 1) (the function w 0 being defined in Theorem 1.1), a simple computation shows that v λ is a solution to

-∆ p v λ -|∇v λ | q = 0 in B λ (x 0 ) with v λ (x) = 0 ≤ u(x, t) for (x, t) ∈ ∂B λ (x 0 ) × (0, ∞).
Furthermore, if λ = λ m := min{1 -|x 0 |, (mα/c 0 ) (p-1-q)/(p-q) } , we have v λm (x) ≤ m ≤ u 0 (x) for x ∈ B λm (x 0 ). The comparison principle [START_REF] Giga | Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains[END_REF]Theorem 2.1] then warrants that u(x, t) ≥ v λm (x) for (x, t) ∈ B λm (x 0 ) × (0, ∞). In particular, u(t

) L ∞ (B) ≥ v λm L ∞ (B λm (x0)) for all t ≥ 0, whence M ∞ ≥ v λm L ∞ (B λm (x0)) > 0. ////

Convergence to steady states

We introduce the half-relaxed limits

u * (x) := lim inf (s,ε)→(t,0) u(x, ε -1 s), x ∈ B, and
u * (x) := lim sup (s,ε)→(t,0) u(x, ε -1 s), x ∈ B,
which are well-defined and do not depend on t > 0. Moreover, we infer from the stability theorem (see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]Lemma 6.1]) that u * is a viscosity subsolution to F (∇z,

D 2 z) = 0 in B, (4.1)
u * is a viscosity supersolution to F (∇z, D 2 z) = 0 in B.

(4.2)

Next we state some useful properties of the half-relaxed limits.

Lemma 4.1 The half-relaxed limits u * and u * enjoy the following properties:

u * ∈ W 1,∞ (B), u * ∈ W 1,∞ (B), (4.3) 0 ≤ u * (x) ≤ u * (x), x ∈ B, (4.4
) u * and u * are radially symmetric and non-increasing, (4.5)

u * (0) = u * (0) = M ∞ := lim t→∞ u(t) L ∞ (B) > 0, (4.6) u * (x) = u * (x) = 0 for x ∈ ∂B. (4.7)
Proof. By (3.1) there is

L := max {A 0 , W (0)} > 0 such that u(x, ε -1 s) ≤ u(y, ε -1 s) + L|x -y| for all (x, y, ε -1 s) ∈ B × B × [0, ∞) , (4.8)
from which we deduce that u * and u * are Lipschitz continuous in B by taking the lim sup or lim inf in ε and s. This proves (4.3), while (4.4) comes directly from the definition of u * and u * and the facts that u is non-negative, radially symmetric for any t ≥ 0 and vanishes identically on ∂B × (0, ∞). The proof of (4.7) uses, in addition, the uniform Lipschitz and C 0,1/2 -bounds we have for u in space and time respectively. In order to prove (4.5), we use Proposition

(x, t) ≤ u(0, t) + W (t)|x| ≤ u(0, t) + W (t) ≤ u(t) L ∞ (B) + W (t), x ∈ B whence u(t) L ∞ (B) ≤ u(0, t) + W (t) ≤ u(t) L ∞ (B) + W (t),
and (4.10) due to W (t) → 0 as t → ∞. Moreover, by the definition of the half-relaxed limits, we have u *

(0) = u * (0) = M ∞ and u * L ∞ (B) ≤ u * L ∞ (B) ≤ M ∞ .
This completes the proof of (4.6). //// and thus

1 Λ ∂ r Φ(r 0 ) ≤ -m δ < 0 which implies |∂ r Φ(r 0 )| ≥ Λm δ . Consequently, (4.15) becomes f (r 0 , ∂ r Φ(r 0 ), ∂ 2 r Φ(r 0 )) ≥ Λ p-1-q -1 Λ q m q δ =: ε δ,Λ > 0,
which ends the proof. ////

We are now able to prove that the half-relaxed limits u * and u * coincide. Proof. We fix Λ > 1 > λ > 0 such that λ > r * and

δ := M ∞ ∇u * L ∞ (B)
1 -λ (p-q)/(p-1-q) ∈ (0, λ -r * ).

Defining now U (r) := Λu * (r), r ∈ [0, 1], and V (r) := λ (p-q)/(p-1-q) u * r λ , r ∈ [0, λ],

we obtain due to (4.13) ≥ λ (p-q)/(p-1-q) M ∞ ≥ V (r).

U (r) ≥ u * (r) = M ∞ ≥ V (
Recalling (4.16), we have thus shown that U (r) ≥ V (r) for r ∈ [0, r * + δ] . (4.17)

Next, we define I λ := (r * +δ, λ). On the one hand, V is a viscosity subsolution to f (r, ∂ r z, ∂ 2 r z) = 0 in I λ . Indeed, take Φ ∈ C 2 (I λ ) and assume that V -Φ has a local maximum at r 1 ∈ I λ . Then u * -Ψ has a local maximum at r 1 /λ, where Ψ(r) := λ -(p-q)/(p-1-q) Φ(λr) for r ∈ ((r * + δ)/λ, 1). Owing to (4.1), we obtain

f r 1 λ , ∂ r Ψ r 1 λ , ∂ 2 r Ψ r 1 λ ≤ 0.
Consequently, 0 ≥ λ q/(p-1-q) f r 1 λ , λ -1/(p-1-q) ∂ r Φ(r 1 ), λ 1-1/(p-1-q) ∂ 2 r Φ(r and V is a viscosity subsolution to f (r, ∂ r z, ∂ 2 r z) = 0 in I λ . On the other hand, it follows from Lemma 4.2 that U is a viscosity supersolution to f (r, ∂ r z, ∂ 2 r z) = ε δ,Λ in I λ with some ε δ,Λ > 0. As furthermore V (r) = 0 ≤ U (r) for r = λ and U (r) ≥ V (r) for r = r * + δ due to (4.17

  , (3.8),(3.10), and(3.11), (u ε ) ε is bounded in, say, C 0,1/2 (B × (0, ∞)) because the uniform Lipschitz continuity in r implies a uniform C 0,1/2bound in r; thus (u ε ) ε is relatively compact in C( B × [0, T ])for all T > 0. It follows from the stability theorem [10, Section 6] and the comparison principle for (1.1) [11, Theorem 2.1] that (u ε ) ε converges uniformly towards the unique viscosity solution u to (1.1) on compact subsets of B × [0, ∞). The properties of u and the bounds listed in Proposition 3.1 then readily follow from this convergence, the properties of u ε , (3.4), (3.8), and (3.10), the function W being the solution to the ordinary differential equation

Lemma 4 . 3

 43 We have u * = u * on B.

  r) for r ∈ [0, r * ]. (4.16)Furthermore, we infer from the Lipschitz continuity of u * that, for r ∈ (r * , r * + δ],U (r) ≥ u * (r) ≥ u * (r * ) -∇u * L ∞ (B) |r -r * | = M ∞ -∇u * L ∞ (B) |r -r * | ≥ M ∞ -δ ∇u * L ∞ (B)

1 )=

 1 -(p -1)|∂ r Φ(r 1 )| p-2 ∂ 2 r Φ(r 1 ) -N -1 r 1 |∂ r Φ(r 1 )| p-2 ∂ r Φ(r 1 ) -|∂ r Φ(r 1 )| q = f (r 1 , ∂ r Φ(r 1 ), ∂ 2 r Φ(r 1 ))

  ), we conclude that U (r) ≥ V (r) for r ∈ [r * + δ, λ] by[START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] Section 5C]. Using (4.17), we end up withΛu * (r) ≥ λ (p-q)/(p-1-q) u * r λ for r ∈ [0, λ].Letting now Λ ց 1 and λ ր 1, we conclude u * ≥ u * in [0, 1] which, together with (4.4), implies u * = u * . //// Finally, we prove Theorem 1.3. Proof of Theorem 1.3. Defining u ∞ := u * = u * by Lemma 4.3, (4.1), (4.2), and Lemma 4.1 imply that u ∞ is a radially symmetric, non-increasing, and Lipschitz continuous viscosity solution to F (∇z,D 2 z) = 0 in B satisfying u ∞ = 0 on ∂B. Moreover, u ∞ L ∞ (B) = M ∞ > 0 due to (4.6). Hence, owing to Theorem 1.1, there is a unique ϑ ∈ [0, 1) such that u ∞ = w ϑ . In particular, the equality u * = u * and the definition of u * and u * provide the uniform convergence of u(t) towards u * = w ϑ in every compact subset of B as t → ∞, see [3, Lemme 4.1] or [2, Lemma V.1.9]. Combining this local convergence with (4.3) and (4.7) gives lim t→∞ u(t) -w ϑ C( B) = 0 and the claim is proved. ////

  0} of the solution u to (1.1) is bounded in L ∞ (B). More precise information are gathered in the next lemma. Assume that u 0 , p, and q fulfil (1.2) and(1.3). Let u be the viscosity solution to (1.1) described in Proposition 3.1. Then t -→ u(t) L ∞ (B) is a non-increasing function and

	2-p + (p -2)(N -1)t creasing and converges to zero as t → ∞.	-1/(p-2) for t ≥ 0 and W is obviously positive, de-////
	By (3.1), the trajectory {u(t) : t ≥ Lemma 3.2

Acknowledgements

The authors would like to thank Olivier Ley for helpful discussions and comments. Part of this work was done during visits of Ph. Laurençot to the Fachbereich Mathematik of the Universität Duisburg-Essen and of C. Stinner to the Institut de Mathématiques de Toulouse, Université Paul Sabatier -Toulouse III. We would like to express our gratitude for the invitation, support, and hospitality.

* Partially supported by the ANR projects "Hamilton-Jacobi et théorie KAM faible" (ANR-07-BLAN-3-187245) and EVOL (ANR-08-0242)

Now, owing to the monotonicity and radial symmetry of u * and u * , there are r * ∈ [0, 1] and r * ∈ [0, 1] such that

Due to (4.4), (4.6), and (4.7), we have

Next, we show that Λu * is a strict supersolution to the stationary equation in a subset of B for Λ > 1.

Lemma 4.2 Fix Λ > 1 and δ ∈ (0, 1 -r * ). Then there are

Proof.

Fix δ ∈ (0, 1 -r * ). Then, due to (4.3), (4.5), and (4.11), there is

, it is also a viscosity supersolution to f 0 (r, ∂ r z, ∂ 2 r z) = 0 in (0, 1) and it follows from Lemma 2.1 that

for a.e. r ∈ (r δ , 1). Integrating and using the continuity of u * we conclude that

)) and assume that Λu * -Φ has a local minimum at some r 0 ∈ (r δ , 1). Then u * -(Φ/Λ) has a local minimum at r 0 and (4.2) implies

Thus, we have


José Ignacio Manzano

Florent Cochet

Benjamin Boucherle

Verónica Gómez- Pérez

Ahcène Boumendjel

Francisco Gamarro

Marine Peuchmaur
email: marine.peuchmaur@univ-grenoble-alpes.fr

Arylthiosemicarbazones as Antileishmanial agents

Keywords:

Based on a screening process, we targeted substituted thiosemicarbazone as potential antileishmanial agents. Our objective was to identify the key structural elements contributing to the anti-parasite activity that might be used for development of effective drugs. A series of 32 compounds was synthesized and their efficacy was evaluated against the clinically relevant intracellular amastigotes of Leishmania donovani. From these, 22 compounds showed EC50 values below 10 µM with the most active derivative (compound 14) showing an EC50 of 0.8 μM with very low toxicity on two different mammalian cell lines. The most relevant structural elements required for higher activity indicate that the presence of a fused bicyclic aromatic ring such as a naphthalene bearing an alkyl or an alkoxy group substituent are prerequisites. Owing to the easy synthesis, high activity and low toxicity, the most active compounds could be considered as a lead for further development.

Introduction

Leishmaniasis, a neglected disease caused by protozoan parasites of the genus Leishmania, is a major health problem, mainly in developing countries, since it is endemic in 98 countries and affects more than 12 million people with more than 2 million new cases reported per year. [1,2] The World Health Organization classified leishmaniasis as a major tropical disease, ranking second only after malaria. This pathogenic disease presents four major syndromes, including cutaneous, mucocutaneous, visceral, post-kala azar dermal leishmaniasis and is caused by at least 20 different Leishmania species. [3] The life cycle of Leishmania parasite is characterized by two distinct morphologies: the promastigote form, flagellated and found in the alimentary tract of the female sand fly vector and the amastigote intracellular form, as the most relevant clinical form found inside the host mammalian cells. [4] Since no effective vaccines are currently available against leishmaniasis, chemotherapy remains the only mean to treat the disease. However, existing treatments are inadequate due to route of administration, side effects, cost, duration of treatment, and they are threatened by resistance issues and therapeutic failure. [5,6] Therefore, the demand for new, effective, affordable and better tolerated drugs is needed.

The first-line treatments consist of pentavalent anti-monials, amphotericin B and its liposomal formulation AmBisome, paromomycin and miltefosine. Current research interest is mainly directed towards the discovery and the optimization of novel chemical entities or reworking existing drugs. Among the recently identified scaffolds, acridines, imidazoles, benzodiazepines, chromenes, coumarins, chalcones and steroids were identified as potential scaffolds for the design of new drugs candidate for treating leishmaniasis. [7][8][9][START_REF] Nagle | Recent developments in drug discovery for leishmaniasis and human African trypanosomiasis[END_REF] Despite the poor understanding of relevant targets in these parasites, some of them are reported, including DNA topoisomerases, protein kinases, folate biosynthesis and cysteine protease. [8,[START_REF] Nagle | Recent developments in drug discovery for leishmaniasis and human African trypanosomiasis[END_REF] In this regard, thiosemicarbazones were previously described as inhibitors of cysteine proteases which could potentially be used to treat protozoan infections such as leishmaniasis. [START_REF] De Melos | Novel 3,4-methylenedioxyde-6-Xbenzaldehyde-thiosemicarbazones: synthesis and antileishmanial effects against Leishmania amazonensis[END_REF][START_REF] Britta | Cell death and ultrastructural alterations in Leishmania amazonensis caused by new compound 4-Nitrobenzaldehyde thiosemicarbazone derived from S-limonene[END_REF][13][START_REF] Schroeder | Identification of semicarbazones, thiosemicarbazones and triazine nitriles as inhibitors of Leishmania mexicana cysteine protease CPB[END_REF][START_REF] Dodd | Hybrid molecules: growth inhibition of Leishmania donovani pro-mastigotes by thiosemicarbazones of 3-carboxy-β-carbolines[END_REF][START_REF] Siles | Preparation of aryl (thio)semicarbazones as inhibitors of cysteine proteases for treatment of protozoan infections such as trypanosomiasis, malaria and leishmaniasis[END_REF] So far, the available data needs to be strengthened by investigating large series of compounds and withdrawing pertinent structure-activity relationship.

With the aim to find more active compounds against Leishmania but non-toxic on mammalian cells and to shed more light on the structure-activity relationship aspect, we screened an in house library of thiosemicarbazone derivatives (Figure 1) against intracellular amastigotes of Leishmania donovani (HU3 strain). This first primary screening highlighted that non polar compounds were the most promising. Considering the results obtained with this first series, we decided to synthesize new thiosemicarbazone derivatives to optimize the selectivity index (SI, EC50 MRC-5 / EC50 parasite) of these series.

Results and discussion

In order to refine the SAR study, we mainly focused our synthesis on two sub-families of thiosemicarbazones: the N-alkyl indole derivatives and the non-polar bicyclic compounds. For both sub-families, the last step of the synthesis is a condensation of an arylaldehyde with the thiosemicarbazide in the presence of acetic acid in ethanol (Scheme 1). The non-commercial arylaldehyde were prepared through an alkylation reaction. The N-alkyl indole carboxaldehyde 1a-h were obtained as previously described [START_REF] Meguellati | B-ring modified aurones as promising allosteric inhibitors of hepatitis C virus RNA-dependent RNA polymerase[END_REF] by treatment of an indole carboxaldehyde with a halogenoalkane, in the presence of NaH in DMF. The alkoxy arylaldehydes 2a-o were conventionally synthesized starting from hydroxyl naphthalene carboxaldehyde and halogenoalkane in presence of a base (K2CO3 or NaH depending on the alkyl group introduced).

Following the same synthetic procedure and starting from [1,1'-biphenyl]-4,4'dicarboxaldehyde, [START_REF] Meguellati | New pseudodimeric aurones as palm pocket inhibitors of hepatitis C virus RNA-dependent RNA polymerase[END_REF] one bis-thiosemicarbazide 34 was synthesized to include it in the biological evaluation. Finally, 32 new compounds were obtained and fully characterized previous to the evaluation of their activity and toxicity. Scheme 1. Synthesis of thiosemicarbazone derivatives 3-34. Reagents and conditions: (i) thiosemicarbazide, EtOH, AcOH, reflux, 10 h, 21-88%; (ii) R"X (X = Br or Cl), NaH 60% in oil, dry DMF, rt, 18-72 h, 41-99%; [START_REF] Meguellati | B-ring modified aurones as promising allosteric inhibitors of hepatitis C virus RNA-dependent RNA polymerase[END_REF] (iii) for R = alkyl: K2CO3, RX (X = I or Br), acetone, reflux, 4-24 h, 37-92%; (iv) for R = MOM: MOMBr, NaH 60%, dry DMF, 0°C then rt 1-3 h, 68-76%.

It is well established that the clinically relevant form of Leishmania is the intracellular amastigote form that proliferate inside the mammalian cells. Therefore, the series of compounds was evaluated against intracellular amastigotes of L. donovani (HU3 strain). Additionally, we evaluated the toxicity of the investigated compounds on two mammalian cell lines: MRC-5 (human fetal fibroblast) and THP-1 (acute monocyclic leukemia) cells. The obtained results are shown in Table 1.

As described in the introduction, our first preliminary screening pointed out the importance of polarity that seems to be unfavorable for the antileishmanial activity. In the other hand, available literature reports dealing with semicarbazones as antileishmanial agents or acting on targets involved in the parasite spread revealed that the presence of aromatic moieties is often synonymous of higher activity. Therefore, we focused our SAR study on apolar or compounds with moderate polarity (Table 1). A first global analysis of the results highlighted that anthracene 4 and phenanthrene 3 thiosemicarbazones are weakly active or relatively toxic compounds.

Most of indole thiosemicarbazones are toxic (compounds 8-11), some possess no antileishmanial activity (compounds 5-7), whereas compound 12 is active on intracellular amastigotes of L. donovani (with a EC50 of 9 μM) and nontoxic on THP-1 cells, even if toxicity is noticeable on MRC-5 cells, displaying a low SI of 3.4. If we consider the bis-thiosemicarbazone 34, it is both non active on intra-cellular amastigotes of L. donovani and no toxic on MRC-5 and THP-1 cells.

Among the naphthalene thiosemicarbazone derivatives, it is interesting to point out the great variability of antileishmanial activity and toxicity depending both on the relative position of substituent and thiosemicarbazone moiety and the nature of the substituent on the naphthalene.

For instance, compounds 16, 22 and 25 bearing a methoxy group on the naphthalene moeity present an antileishmanial activity ranging from 4 to more than 10 μM whereas their toxicity vary from 42 to more than 250 μM with compound 25 being the most active, displaying the best SI of this series (>20). Among all O-alkyl substituent evaluated on the naphthalene structure, it was observed that the benzyl and the methylcyclohexyl groups seem to enhance the toxicity on MRC-5 or THP-1 cells (compounds 19, 23, 28 and 31 for the benzyl group and compounds 20 and 24 for methylcyclohexyl group) even if EC50 on intracellular amastigotes of L. donovani is in the low micromolar range and the SI obtained with these compounds are relatively high (3.7-22.6), with the exception of compound 24. Regarding the methoxymethyl (MOM) substituent, we observe that it is not beneficial to the antileishmanial activity: compounds 21, 29 and 32 have an EC50 values ranging from 6.8 to more than 10 μM. Naphthalene thiosemicarbazides bearing a methoxy, an ethoxy, a butoxy, a methyl or a fluoro substituent (compounds 13, 14, 15, 18, 26, 27 or 30) generally present a better activity on intracellular amastigotes (0.8-6.5 µM and a promising SI (19.5>250) due to the weak cytoxicity toward human cells. The most active compound found so far was compound 14 (EC50 = 0.8 µM that displayed a high SI (>250).

Finally, the disubstituted naphthalene compound 33 show a low toxicity (EC50 > 200 μM on MRC-5 and THP-1 cells), but also its antileishmanial activity is lower (9 μM). b SI = Selectivity Index (EC50 MRC-5 / EC50 parasite).

c AmB has been used as an antileishmanial reference compound.

d -CH2Cy : methylcyclohexyl.

Conclusion

In conclusion, we elaborated, evaluated and conducted a relation structure-activity study on thiosemicarbazone bearing either a naphthyl or indolyl moieties. Indeed, we identified compounds with an antileishmanial activity in the low micromolar range. Particularly, we have observed that naphthalene thiosemicarbazides bearing a methoxy, an ethoxy, a butoxy, a methyl or a fluoro substituent compounds presented the best activity against L. donovani, highlighting compound 14, the most active compound of this study with an EC50 of 0.8 μM, low toxicity in MRC-5 cells and the best SI (>250). These results are in perfect line with specification needed for development of novel and costly-effective antileishmanial drugs and highlight the advantage of this series of compounds which will be further evaluated by in vivo studies.

Experimental section

Chemistry

Commercially available reagents and solvents were used without further purification. Reactions were monitored by thin-layer chromatography (plates coated with silica gel 60 F254 from Macherey-Nagel). Products were purified with column chromatography on Silica gel 60 (230-400 mesh from Macherey-Nagel) or by automatic flash chromatography with the Grace device:

Reveleris X2 using Grace Resolv cartridges. Melting points were measured on a Büchi B540 apparatus and are uncorrected. Purity of the reported compounds were ≥ 95%.

General procedure A for the alkylation of hydroxy-naphtalene-carboxaldehyde derivatives (2a-e, g-k, m-n).

The mixture of hydroxy-naphtalene-carboxaldehyde (1 equiv.), K2CO3 (1.5 equiv.) and alkyl halide (1.5 equiv.) in acetone (12 mL/mmol) was stirred under reflux for 4 to 24 h. When TLC showed complete disappearance of the starting materials, solvent was evaporated under reduced pressure. The residue was purified by column chromatography on silica gel (cyclohexane/EtOAc 90:10 to 70:30).

General procedure B for the synthesis of methoxymethyl derivatives from hydroxynaphthaldehydes (2f, 2l, 2o).

To a solution of hydroxy-naphthaldehyde (1 equiv.) in dry DMF (1 mL/mmol) was added, at 0°C, sodium hydride (60% dispersion in mineral oil, 2 equiv.). The reaction mixture was stirred for 15 min. then MOM-Br (2 equiv.) was added and the reaction mixture was stirred for 1 to 3 h at rt. The reaction was quenched with brine and extracted 3 times with EtOAc. The organic layer was washed twice with ice cold brine, dried over MgSO4, filtered and evaporated. The residue was purified by flash chromatography on silica gel (cyclohexane/EtOAc 80:20).

General procedure C for the synthesis of thiosemicarbazide derivatives (3-33 and 35).

To a solution of the appropriate aldehyde (1.1 equiv.) in anhydrous ethanol (10 mL/mmol), was added thiosemicarbazide (1.0 équiv.) and acetic acid (0.2 mL/mmol). The reaction mixture was refluxed for 10 h and then was cooled to rt. The precipitate was filtered, washed with ether and recrystallized from ethanol to obtain the corresponding 1-(1arylethylidene)thiosemicarbazone compounds. [M+Na] + (100).

4-methoxy-1-naphthaldehyde (2a)

2-benzyloxy-1-naphthaldehyde (2g).

This product was prepared according to previously described procedure. [START_REF] Quideau | 2-Alkoxyarenol-derived orthoquinols in carbon-oxygen, carbon-nitrogen and carbon-carbon bond-forming reactions[END_REF] 2-(cyclohexylmethoxy)-1-naphthaldehyde (2h). The crude product was prepared according to general procedure A starting from 2-hydroxy-1-naphthaldehyde (50 mg, 0.29 mmol, 1.0 equiv.)

and cyclohexylmethyl bromide (0.08 mL, 0.58 mmol, 2.0 equiv.). The pure product (

1-benzyloxy-2-naphthaldehyde (2n

). This product was prepared according to previously described procedure. [START_REF] Quideau | 2-Alkoxyarenol-derived orthoquinols in carbon-oxygen, carbon-nitrogen and carbon-carbon bond-forming reactions[END_REF] 1-(methoxymethoxy)-2-naphthaldehyde (2o). The crude product was prepared according to general procedure B starting from 1-hydroxy-2-naphthaldehyde (100 mg, 0.58 mmol, 1.0 equiv.). The pure product (86 mg, 68%) was obtained as a white solid. Rf 0.52 (cyclohexane/EtOAc 80:20). Analyses of the compound are in agreement with the literature data. [START_REF] Harvey | A new coumarin synthesis and its utilization for the synthesis of polycyclic coumarin compounds with anticarcinogenic properties[END_REF] 2-(phenanthren-9-ylmethylene)hydrazinecarbothioamide (3), 2-(anthracen-9ylmethylene)hydrazinecarbothioamide ( 4). These products were prepared according to previously described procedure.

[24]

2-((1-methyl-1H-indol-2-yl)methylene)hydrazinecarbothioamide (5), 2-((1-methyl-1H-indol-5yl)methylene)hydrazinecarbothioamide (6), 2-((1-methyl-1H-indol-3-

yl)methylene)hydrazinecarbothioamide (7). These products were prepared according to previously described procedure. [START_REF] Yi | Refinement of arylthiosemicarbazone pharmacophore in inhibition of mushroom tyrosinase[END_REF] 2-((1-(cyclohexylmethyl)-1H-indol-3-yl)methylene)hydrazinecarbothioamide (8). The crude product was prepared according to general procedure C starting from N-(cyclohexylmethyl)-indole-3-carboxaldehyde [START_REF] Meguellati | B-ring modified aurones as promising allosteric inhibitors of hepatitis C virus RNA-dependent RNA polymerase[END_REF]

2-((1-benzyl-1H

-indol-3-yl)methylene)hydrazinecarbothioamide (9). The crude product was prepared according to general procedure C starting from N-benzyl-indole-3-carboxaldehyde [START_REF] Meguellati | B-ring modified aurones as promising allosteric inhibitors of hepatitis C virus RNA-dependent RNA polymerase[END_REF] 1e (100 mg, 0.45 mmol, 1.1 equiv. [START_REF] Dodd | Hybrid molecules: growth inhibition of Leishmania donovani pro-mastigotes by thiosemicarbazones of 3-carboxy-β-carbolines[END_REF]. This product was prepared according to previously described procedure. [START_REF] Du | Synthesis and structure-activity relationship study of potent trypanocidal thio semicarbazone inhibitors of the trypanosomal cysteine protease cruzain[END_REF] 2-((4-methoxynaphthalen-1-yl)methylene)hydrazinecarbothioamide [START_REF] Siles | Preparation of aryl (thio)semicarbazones as inhibitors of cysteine proteases for treatment of protozoan infections such as trypanosomiasis, malaria and leishmaniasis[END_REF]. The crude product was prepared according to general procedure C starting from 2a (130 mg, 0.70 mmol, 1.1 equiv.).

The pure product (89 mg, 55%) was obtained as a yellow solid. Rf 0. of the intracellular parasite growth, as previously described. [29] Additionally, Selectivity Indices (SI) of compounds were subsequently calculated on the basis of their antiparasitic activities (EC50 parasite) and host cell cytotoxicity (EC50 MRC-5 cells), according to the formula: EC50(MRC-5) / EC50 parasite.

Figure 1 .

 1 Figure 1. General structures of the first series of thiosemicarbazone screened.

a

  Yields are given for the last step of the synthesis (condensation of bis-aldehyde and benzofuranone).

  1 H and13 C NMR spectra were recorded at room temperature in deuterated solvents on a Brüker Avance-400 instrument (at 400 MHz and 100 MHz respectively). Chemical shifts (δ) are reported in parts per million (ppm) relative to TMS as internal standard or relative to the solvent [ 1 H: δ(DMSO-d6) = 2.50 ppm, δ(CDCl3) = 7.24 ppm, δ(CD3OD) = 3.31 ppm, δ(acetone-d6) = 2.05 ppm; 13 C: δ(DMSO-d6) = 39.51 ppm, δ(CDCl3) = 77.23 ppm, δ(CD3OD) = 49.00 ppm, δ(acetone-d6) = 29.84 ppm]. Electrospray ionization ESI mass spectra were acquired by the Analytical Department of Grenoble University on an Esquire 3000 Plus Bruker Daltonis instrument with a nanospray inlet. Accurate mass measurements (HRMS) were carried out on a ESI/QTOF with the Waters Xevo G2-S QTof device. HPLC analyses were performed with an Agilent 1100 series using a diode array detector and a C18 reversed-phase column (Nucleosil C18, Macherey-Nagel, 5 μm particle size, 125 mm × 3 mm) at 45°C, with a mobile phase composed of A = water and TFA 0.1% and B = MeOH and TFA 0.1% with a gradient 85:15 to 0:100 A:B over 14 min, 1 mL/min, 10 μL injection, detection at 254 nm.

7 (

 7 The crude product was prepared according to general procedure A starting from 4-hydroxy-1-naphthaldehyde (150 mg, 0.87 mmol, 1.0 equiv.) and iodomethane. The pure product (133 mg, 82%) was obtained as a colorless oil. Rf 0.39 (cyclohexane/EtOAc 80:20);1 H NMR (400 MHz, CDCl3) δ ppm 4.07 (s, 3H), 6.88 (d, J = 8.1 Hz, 1H), 7.56 (ddd, J = 8.4, 6.9, 1.3 Hz, 1H), 7.69 (ddd, J = 8.4, 6.9, 1.3 Hz, 1H), 7.88 (d, J = 8.1 Hz, 1H), 8.29-8.34 (m, 1H), 9.25-9.37 (m, 1H), 10.19 (s, 1H); 13 C NMR (100 MHz, CDCl3) δ ppm 56.1 (CH3), 103.0 (CH), 122.5 (CH), 124.9 (CH), 125.1 (C), 125.6 (C), 126.5 (CH), 129.6 (CH), 132.0 (C), 139.8 (CH), 160.9 (C), 192.4 (CH); LRMS (ESI+) m/z (%) 209 [M+Na] + (100). 4-ethoxy-1-naphthaldehyde (2b). The crude product was prepared according to general procedure A starting from 4-hydroxy-1-naphthaldehyde (200 mg, 1.16 mmol, 1.0 equiv.) and iodoethane. The pure product (161 mg, 69%) was obtained as a brown solid. Rf 0.60 (cyclohexane/EtOAc 70:30); m.p. 75°C; 1 H NMR (400 MHz, CDCl3) δ ppm 1.60 (t, J = 9.3 Hz, 3H), 4.31 (d, J = 9.3 Hz, 2H), 6.88 (d, J = 10.7 Hz, 1H), 7.57 (ddd, J = 11.1, 9.2, 1.6 Hz, 1H), 7.69 (ddd, J = 11.1, 9.2, 1.9 Hz, 1H), 7.90 (d, J = 10.7 Hz, 1H), 8.37 (m, 1H), 9.29 (m, 1H), 10.20 (s, 1H); 13 C NMR (100 MHz, CDCl3) δ ppm 14.8 (CH3), 64.6 (CH2), 103.7 (CH), 122.CH), 125.0 (CH), 125.1 (C), 125.8 (C), 126.5 (CH), 129.7 (CH), 132.2 (C), 140.0 (CH), 160.5 (C), 192.4 (CH); LRMS (ESI+) m/z (%) 201 [M+H] + (100).

Table 1

 1 Antileishmanial activity and intrinsic toxicity of the different compounds on MRC-5 and THP-1 cells.

					EC50 (μM)
	Compd	Ar	Yield a (%)	Intracellular amastigotes L. donovani	MRC-5 cells	THP-1 cells	SI b
		AmB c		0.09 ± 0.01	23 ± 2	38 ± 5	260
	3		99	> 10	192 ± 10	> 200	-
	4		48	5.3 ± 0.4	45 ± 9	67 ± 11	8.5
	5		92	> 10	> 200	> 200	-
	6		90	> 10	> 200	> 200	-

  1d (100 mg, 0.44 mmol, 1.1 equiv.). The pure product (113 mg, 91%) was obtained as a yellow solid. Rf 0.33 (cyclohexane/EtOAc 60:40); m.p. 207°C; 1 H NMR (400 MHz, DMSO-d6) δ ppm 0.90-1.84 (m, 11H), 4.02 (d, J = 7.2 Hz, 2H), 7.09-7.17 (m, 1H), 7.20-7.26 (m, 1H), 7.40 (s, 1H), 7.52 (d, J = 8.2 Hz, 1H), 7.80 (s, 1H), 8.01 (s, 1H), 8.21 (d, J =

	7.8 Hz, 1H), 8.27 (s, 1H), 11.15 (s, 1H); 13 C NMR (100 MHz, DMSO-d6) δ ppm 49.2 (CH2), 110.4
	(CH), 110.5 (C), 120.9 (CH), 122.4 (CH), 122.7 (CH), 124.7 (C), 127.1 (2xCH2), 127.5 (CH),
	128.5 (2xCH2), 133.9 (CH2), 136.7 (C), 137.4 (CH), 140.1 (CH), 176.4 (C); LRMS (ESI+) m/z
	(%) 315 [M+H] + (100), 337 [M+Na] + (32); HRMS (ESI+) m/z calc. for C17H23N4S 315.1643, found
	315.1632.

  (benzyloxy)naphthalen-1-yl)methylene)hydrazinecarbothioamide[START_REF] Harvey | A new coumarin synthesis and its utilization for the synthesis of polycyclic coumarin compounds with anticarcinogenic properties[END_REF]. The crude product was prepared according to general procedure C starting from 2g (80 mg, 0.31 mmol, 1.1 equiv.). The pure product (67 mg, 72%) was obtained as a yellow solid. Rf 0.26 (cyclohexane/EtOAc 80:20); m.p. 142°C; 1 H NMR (400 MHz, CDCl3) δ ppm 5.25 (s, 2H), 6.39 (s, 1H), 7.11 (s, 1H), 7.22-7.49 (m, 5H), 7.49-7.64 (m, 2H), 7.69-7.98 (m, 3H), 8.64 (s, 1H), 8.83 (d, J = 8.8 Hz, 1H), 9.43 (s, 1H); 13 C NMR (100 MHz, CDCl3) δ ppm 71.8 (CH2), 114.1 (CH), The crude product was prepared according to general procedure C starting from 2h (50 mg, 0.19 mmol, 1.1 equiv.). The pure product (33 mg, 57%) was obtained as a yellow solid. Rf 0.35 (cyclohexane/EtOAc 80:20); m.p. 181°C; 1 H NMR (400 MHz, DMSO-d6) δ ppm 1.06-2.02 (m, 11H), 3.97 (d, J = 6.1 Hz, 2H), 7.00 (d, J = 8.3 Hz, 1H), 7.52-7.58 (m, 1H), 7.60-7.65 (m, 1H), 7.86 (s, 1H), 8.02 (d, J = 8.3 Hz, 1H), 8.17 (s, 1H), 8.21-8.27 (m, 1H), 8.43 (d, J = 8.4 Hz, 1H), /parasite ratio of 1:10 with late-stage promastigotes. Infected cell cultures were incubated with different compound concentrations in RPMI 1640 medium plus 10% iFBS at 37°C with 5% CO2 for 72 h and then lysed. Luminescence intensity was measured as indicative

	(cyclohexane/EtOAc 80:20); m.p. 199°C; 1 H NMR (400 MHz, acetone-d6) δ ppm 3.53 (s, 3H),
	5.50 (s, 2H), 7.20 (d, J = 8.2 Hz, 1H), 7.44 (s, 1H), 7.58 (ddd, J = 8.3, 6.8, 1.3 Hz, 1H), 7.65
	(ddd, J = 8.3, 6.8, 1.3 Hz, 1H), 7.77 (s, 1H), 8.04 (d, J = 8.2 Hz, 1H), 8.35 (d, J = 8.3 Hz, 1H),
	8.63 (d, J = 8.3 Hz, 1H), 8.88 (s, 1H), 10.39 (s, 1H); 13 C NMR (100 MHz, acetone-d6) δ ppm
	56.6 (CH3), 95.5 (CH2), 108.4 (CH), 123.4 (CH), 123.9 (C), 124.4 (CH), 126.6 (CH), 128.8 (C),
	128.5 (CH), 128.7 (CH), 133.0 (C), 143.1 (CH), 155.6 (C), 180.1 (C); LRMS (ESI+) m/z (%) 290
	[M+H] + (100), 312 [M+Na] + (23); HRMS (ESI+) m/z calc. for C14H16N3O2S 290.0963; found
	290.0952.
	2-((2-methoxynaphthalen-1-yl)methylene)hydrazinecarbothioamide (22). This product was
	prepared according to previously described procedure.[27]
	14 (cyclohexane/EtOAc
	90:10); m.p. 237°C; 1 H NMR (400 MHz, acetone-d6) δ ppm 4.08 (s, 3H), 7.05 (d, J = 8.2 Hz, 1H), 7.41 (s, 1H), 7.56 (ddd, J = 8.3, 6.8, 1.2 Hz, 1H), 7.64 (ddd, J = 8.3, 6.8, 1.2 Hz, 1H), 7.74 (s, 1H), 8.04 (d, J = 8.2 Hz, 1H), 8.29-8.34 (m, 1H), 8.64 (d, J = 8.5 Hz, 1H), 8.86 (s, 1H), 10.36 2-((2-114.6 (C), 124.6 (CH), 125.0 (CH), 127.8 (2xCH), 126.5 (CH), 126.6 (CH), 128.8 (CH), 129.0 (2xCH), 129.4 (C), 131.4 (C), 133.3 (CH), 136.3 (C), 142.0 (CH), 157.7 (C), 178.5 (C); LRMS (ESI+) m/z (%) 336 [M+H] + (50), 358 [M+Na] + (100); HRMS (ESI+) m/z calc. for C19H18N3OS (s, 1H); The pure product (120 mg, 97%) was obtained as a yellow solid. Rf 0.41(cyclohexane/EtOAc 336.1171, found 336.1162.
	70:30); m.p. 240°C; 1 H NMR (400 MHz, DMSO-d6) δ ppm 1.47 (t, J = 7.1 Hz, 3H), 4.27 (q, J =
	7.1 Hz, 2H), 7.02 (d, J = 8.1 Hz, 1H), 7.55 (ddd, J = 8.3, 6.9, 1.3 Hz, 1H), 7.66 (ddd, J = 8.3, 6.9,
	1.3 Hz, 1H), 7.82 (s, 1H), 8.05 (d, J = 8.1 Hz, 1H) 8.17 (s, 1H), 8.25 (d, J = 9.1 Hz, 1H), 8.44 (d,
	J = 9.1 Hz, 1H), 8.77 (s, 1H), 11.30 (s, 1H); 13 C NMR (100 MHz, DMSO-d6) δ ppm 14.5 (CH3),
	63.8 (CH2), 105.1 (CH), 121.4 (C), 122.2 (CH), 123.2 (CH), 124.8 (C), 125.5 (CH), 127.8 (CH),
	128.0 (CH), 131.3 (C), 142.1 (CH), 155.8 (C), 177.5 (C); LRMS (ESI+) m/z (%) 274 [M+H] +
	(100); HRMS (ESI+) m/z calc. for C14H16N3OS 274.1014, found 274.1021. 8.75 (s, 1H), 11.30 (s, 1H); 13 C NMR (100 MHz, DMSO-d6) δ ppm 27.0 (2xCH2), 27.6 (CH2),
	30.8 (2xCH2), 39.3 (CH), 76.1 (CH2), 114.8 (CH), 115.7 (C), 125.1 (CH), 126.3 (CH), 129.1
	2-((4-butoxynaphthalen-1-yl)methylene)hydrazinecarbothioamide (18). The crude product was (CH), 129.6 (CH), 130.5 (C), 132.6 (C), 134.0 (CH), 143.9 (CH), 159.4 (C), 178.5 (C); LRMS
	prepared according to general procedure C starting from 2c (60 mg, 0.26 mmol, 1.1 equiv.). The

13

C NMR (100 MHz, acetone-d6) δ ppm 56.3 (CH3), 105.0 (CH), 122.9 (C), 123.3 (CH), 124.5 (CH), 126.4 (C), 126.5 (CH), 128.6 (CH), 129.3 (CH), 132.8 (C), 143.4 (CH), 158.2 (C), 180.2 (C); LRMS (ESI+) m/z (%) 179 (100), 282 [M+H] + (23), 282 [M+Na] + (83); HRMS (ESI+) m/z calc. for C13H14N3OS 260.0858, found 260.0852.

2-((4-ethoxynaphthalen-1-yl)methylene)hydrazinecarbothioamide

[START_REF] Meguellati | B-ring modified aurones as promising allosteric inhibitors of hepatitis C virus RNA-dependent RNA polymerase[END_REF]

. The crude product was prepared according to general procedure C starting from 2b (100 mg, 0.50 mmol, 1.1 equiv.). pure product (43 mg, 60%) was obtained as a yellow solid. Rf 0.45 (cyclohexane/EtOAc 70:30); m.p. 196°C; 1 H NMR (400 MHz, DMSO-d6) δ ppm 1.00 (t, J = 7.4 Hz, 3H), 1.47-1.64 (m, 2H),

2-((2-(cyclohexylmethoxy)naphthalen-1-yl)methylene)hydrazinecarbothioamide (24).

(ESI+) m/z (%) 342 [M+H] + (100); HRMS m/z (%) calc. for C19H24N3OS 342.1640, found 342.1639. macrophage

Acknowledgements

The authors wish to acknowledge the support from the ICMG Nanobio Platform, Grenoble on which the NMR and MS were performed. This work was supported by the Spanish Grants Proyecto de Excelencia, Junta de Andalucía, Ref. CTS-7282 (to F.G.), SAF2012-34267 (to F.G.), FEDER funds from EU to F.G.

The authors are grateful to ANR (Agence Nationale pour la Recherche) for financial support (Labex Arcane (ANR-11-LABX-0003-01)).

Appendix A. Supplementary data

Supplementary data ( 1 H and 13 C NMR spectra) related to this article can be found at http://dx.doi.org/ thiosemicarbazide, EtOH, AcOH, reflux, 10 h, 21-88%; (ii) R"X (X = Br or Cl), NaH 60% in oil, dry DMF, rt, 18-72 h, 41-99%; [START_REF] Meguellati | B-ring modified aurones as promising allosteric inhibitors of hepatitis C virus RNA-dependent RNA polymerase[END_REF] (iii) for R = alkyl: K2CO3, RX (X = I or Br), acetone, reflux, 4-24 h, 37-92%; (iv) for R = MOM: MOMBr, NaH 60%, dry DMF, 0°C then rt 1-3 h, 68-76%.

List of captions

Table 1

Antileishmanial activity and intrinsic toxicity of the different compounds on MRC-5 and THP-1 cells.