Samuel Tréton

Blow-up vs. global existence for a Fujita-type Heat exchanger system

Keywords: reaction-diffusion system, Fujita blow-up phenomena, critical exponent, global solutions, Heat exchanger system, 35B44, 35B33, 92D25

Blow-up vs. global existence for a Fujita-type Heat exchanger system

Introduction

In this work, we consider the semi-linear Cauchy problem

     ∂ t u = c∆u -µu + νv + u 1+α , t > 0, x ∈ R N , ∂ t v = d∆v + µu -νv + κv 1+β , t > 0, x ∈ R N , (u, v)| t=0 = (u 0 , v 0 ), x ∈ R N , ( 1.1)
in any dimension N ≥ 1. Here the parameters c, d, µ, ν, α, β are positive constants, κ = 0 or κ = 1, and the data u 0 and v 0 are taken non-both-trivial and non-negative in L 1 (R N ) ∩ L ∞ (R N ). This system is motivated by biological issues, particularly the field-road reaction-diffusion system -see (1.8) -for which no Fujita-type result has been established yet.

It is well-known that (1.1) owns a unique maximal solution

(u, v) ∈ (C 1 ((0, T ) , L 1 (R N ) ∩ L ∞ (R N ))) 2 ,
such that (u, v) converges to (u 0 , v 0 ) in (L 1 (R N )) 2 as t → 0. Furthermore, by parabolic regularity, (u, v) also belongs to (C ∞ ((0, T ) × R N )) 2 . Importantly, the cooperative nature of the system ensures the validity of the comparison principle. These results are classical and can be found in [START_REF] Weissler | Local Existence and Nonexistence for Semilinear Parabolic Equations in L p[END_REF], [START_REF] Friedman | Partial Differential Equations of Parabolic Type[END_REF] and [START_REF] Fife | Comparison principles for reaction-diffusion systems: Irregular comparison functions and applications to questions of stability and speed of propagation of disturbances[END_REF] for instance. The main objective of this paper is to determine whether T is finite or not. If T = ∞ we refer to (u, v) as a global solution, while T < ∞ enforces lim

t→T - ∥u(t)∥ L ∞ (R N ) + ∥v(t)∥ L ∞ (R N ) = ∞, (1.2)
and we say that (u, v) blows-up in finite time.

Back in 1966, Fujita conducted the pioneering work [START_REF] Fujita | On the blowing up of solutions of the Cauchy problem for u t = ∆u + u 1+α[END_REF] on the initial value problem

∂ t u = ∆u + u 1+α , t > 0, x ∈ R N , u| t=0 = u 0 , x ∈ R N . ( 1.3)
He showed that for non-negative and non-trivial datum u 0 , there is a critical exponent α F := 2/N (later referred to as the Fujita exponent) that separates systematic blow-up of the solutions from possible global existence. More precisely, if 0 < α < α F , any solution to (1.3) blows-up in finite time. Conversely, if α > α F , problem (1.3) admits global solutions as long as u 0 is chosen "sufficiently small". Additionally, the global solutions provided by this case asymptotically vanish in L ∞ (R N ). The critical case α = α F has subsequently been investigated in various works, including [START_REF] Hayakawa | On nonexistence of global solutions of some semilinear parabolic differential equations[END_REF], [START_REF] Kobayashi | On the growing up problem for semilinear heat equations[END_REF], [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF], [START_REF] Weissler | Existence and non-existence of global solutions for a semilinear heat equation[END_REF], and falls within the systematic blow-up regime. The regime transition that occurs at 1/α = N/2 can intuitively be explained by observing that the equation in (1.3) is a combination of the growth phenomenon resulting from the non-linearity and of the dispersive effect arising from the Laplacian. The growth of the solutions to (1.3) is related to the underlying ODE U ′ = U 1+α whose solutions have the form U (t) = C/(T -t) 1/α and blow-up with algebraic blow-up rate 1/α. On the other hand, solutions to the Heat equation ∂ t u = ∆u vanish for integrable initial datum, and the sharp uniform control ∥u(t)∥ L ∞ (R N ) ≤ C/t N/2 indicates that this vanishing occurs with algebraic decay rate N/2. By formally combining the growth and diffusive actions into (1.3), Fujita's result states that global solutions can only exist if the diffusion decay rate exceeds the growth blow-up rate, which is N/2 > 1/α. Otherwise the dispersive effect of the diffusion is never strong enough to prevent the blowing-up driven by the reaction term.

Since the critical case was classified, many works have shown significant interest in (1.3), either by deepening the established results, or by exploring variations of the problem or additional effects. A wide range of references can be found in the survey articles by Levine [START_REF] Levine | The Role of Critical Exponents in Blowup Theorems[END_REF] and Deng and Levine [START_REF] Deng | The Role of Critical Exponents in Blow-Up Theorems: The Sequel[END_REF], as well as in the books by Hu [START_REF] Hu | Blow-up Theories for Semilinear Parabolic Equations[END_REF] and Quittner and Souplet [START_REF] Quittner | Superlinear Parabolic Problems[END_REF].

The first variation of (1.3) that takes the form of a system appeared in 1991 in the work of Escobedo and Herrero [START_REF] Escobedo | Boundedness and Blow Up for a Semilinear Reaction-Diffusion System[END_REF]. In their study, they considered

∂ t u = ∆u + v 1+α , t > 0, x ∈ R N , ∂ t v = ∆v + u 1+β , t > 0, x ∈ R N . (1.4)
Similarly to the Fujita's problem (1.3), the system (1.4) may be interpreted as the result of the interplay between a growth phenomenon described by the coupled ODE system

U ′ = V 1+α , t > 0, V ′ = U 1+β , t > 0, (1.5)
and an uncoupled diffusion process governed by

∂ t u = ∆u, t > 0, x ∈ R N , ∂ t v = ∆v, t > 0, x ∈ R N .
The results presented in [START_REF] Escobedo | Boundedness and Blow Up for a Semilinear Reaction-Diffusion System[END_REF] highlight that the transition from systematic blow-up from possible global existence is once again determined by the balance between the growth blowup rate and the diffusion decay rate. Specifically, for non-negative and non-both-trivial initial values U 0 , V 0 , both components of the solution (U, V ) to (1.5) blows-up in finite time with the blow-up rates

a := 2 + α α + β + αβ for U, and b := 2 + β α + β + αβ for V.
Consequently, for system (1.4) to have global solutions, N/2 must be larger than both a and b, while systematic blow-up occurs as soon as a or b exceeds N/2. The work of Escobedo and Herrero have been extended in several ways. First, the introduction of different diffusion coefficients has been investigated in [START_REF] Fila | A Fujita-type global existence-global nonexistence theorem for a system of reaction diffusion equations with differing diffusivities[END_REF] by Fila, Levine and Uda. Although they reached the same conclusions as in the case of identical diffusions, it should be noted that considering different diffusion rates significantly increases the complexity of the problem. This draws attention to one of the key challenges involved in working with system (1.1) which is discussed in more details in Section 2. More recently, in [START_REF] Yang | Fujita-type phenomenon of nonlinear coupled nonlocal diffusion system[END_REF], problem (1.4) has also been studied by taking some compactly supported non-local diffusion operators instead of the Laplacians. As for the non-linear effects, the case of time-weighted reactions (f (t)v 1+α , g(t)u 1+β ) was tackled in [START_REF] Uda | The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations[END_REF], [START_REF] Cao | Critical Fujita curve for a semilinear parabolic system with time-weighted sources[END_REF], [START_REF] Castillo | On the critical exponent for some semilinear reaction-diffusion systems on general domains[END_REF], and that of space-weighted reactions (|x| σ 1 v 1+α , |x| σ 2 u 1+β ) was discussed in [START_REF] Mochizuki | Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations[END_REF]. An expansion of (1.4) with a "chain coupling" of more than two unknowns was also handled in [START_REF] Rencławowicz | Global existence and blow-up for a completely coupled Fujita type system[END_REF].

Various other types of systems have been developed based on Fujita's problem (1.3). We mention for instance systems for which the reaction terms take the form (u α v β , u γ v δ ) studied in [START_REF] Qi | Global existence and uniqueness of a reaction-diffusion system via invariant solutions[END_REF], [START_REF] Escobedo | Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations[END_REF], [START_REF] Lu | Global existence and blow-up for a class of semilinear parabolic systems: A Cauchy Problem[END_REF], [START_REF] Snoussi | Global existence, asymptotic behavior and self-similar solutions for a class of semilinear parabolic systems[END_REF], [START_REF] Dickstein | Life span of solutions of a strongly coupled parabolic system[END_REF] or (u α + v β , u γ + v δ ) considered in [START_REF] Cui | Global behavior of solutions to a reaction-diffusion system[END_REF], [START_REF] Souplet | Optimal condition for non-simultaneous blow-up in a reaction-diffusion system[END_REF], [START_REF] Castillo | On the critical exponent for some semilinear reaction-diffusion systems on general domains[END_REF].

We would like to stress that in all the semi-linear systems mentioned above, the interaction between the unknowns occurs through the non-linear growth terms. However, this is not the case for problem (1.1) investigated in this paper. Indeed, as for (1.3) and (1.4), we can extract from (1.1) a "diffusive component":

∂ t u = c∆u -µu + ν v, t > 0, x ∈ R N , ∂ t v = d∆v + µu -ν v, t > 0, x ∈ R N , ( 1.6)
that we call Heat exchanger and which contains all the coupling parts of (1.1). We interpret system (1.6) as a population dynamics model for a single species dispersing on two parallel environments and switching from one to the other. Observe that (1.6) preserves the mass of its initial datum over time, that is

∥u(t) + v(t)∥ L 1 (R N ) ≡ ∥u(0) + v(0)∥ L 1 (R N )
, for all t > 0.

(1.7)

A detailed description of solutions to (1.6) is given in Theorem 2.1, Corollary 2.2 and their surrounding comments. These results are necessary preliminaries to address the systematic blow-up versus possible global existence for the problem (1.1). Such a model which exchanges individuals between two domains is not new in the literature. In particular, the linear Heat exchanger system (1.6) is influenced by the field-road diffusive model

     ∂ t v = d∆v, t > 0, x ∈ R N -1 , y > 0, -d∂ y v| y=0 = µu -ν v| y=0 , t > 0, x ∈ R N -1 , ∂ t u = D∆u -µu + ν v| y=0 , t > 0, x ∈ R N -1 , (1.8)
where (v, u) = (v(t, x, y), u(t, x)), introduced in 2013 by Berestycki, Roquejoffre and Rossi in [START_REF] Berestycki | The influence of a line with fast diffusion on Fisher-KPP propagation[END_REF] and whom both fundamental solutions and asymptotic decay rate were recently studied in [START_REF] Alfaro | The field-road diffusion model: Fundamental solution and asymptotic behavior[END_REF]. Traditionally, the field-road diffusive model is supplemented with a reaction term f in the field which is represented by the first line of (1.8). As far as we know, no Fujita-type reaction f such as f (v) = v 1+α has been considered in the literature. In this case (as with system (1.1)), it is worth noting that the equations are coupled through the diffusion process. In this paper, we therefore propose to study problem (1.1) as a first step towards investigating these "coupled by diffusion" Fujita-type systems.

This work is structured as follows. In Section 2, we start with an introduction of the notations used throughout. Subsequently, we present our results on the diffusive system (1.6) and the semi-linear problem (1.1) in Subsection 2.1 and Subsection 2.2 respectively. The remaining sections are devoted to providing the proofs for the statements outlined in Section 2. Specifically, in Section 3 we prove the results related to linear system (1.6). In Section 4 we address problem (1.1) when global existence is achievable. Lastly, we discuss the systematic blow-up case in Section 5.

Notations and main results

The conventions and notations used in this paper are gathered below.

To avoid confusion, we use bold font (e.g. u, v, , ) to denote the solutions to diffusive problems, while we use regular font (e.g. u, v, σ, δ) for the solutions to non-linear problems. Additionally, if w = w(t, x), the abbreviation w(t) obviously stands for the function w(t, ). We also indicate functions that depend solely on time -such as ODE solutions -in capital letters (e.g. U , V , S, D).

In the whole document the function G = G(t, x) := (4πt) -N/2 e -|x| 2 /4t denotes the Heat kernel in R N with unit diffusion rate, and * is the classical convolution product. For any subset Ω ⊂ R N , we denote ∂Ω its topological frontier. In addition, we use B(0, R) to denote the ball {x ∈ R N such that |x| < R}, and 1 B(0,R) to represent its indicator function.

For any f ∈ L 1 (R N ), the Fourier transform we employ is defined by

F [f ] (ξ) = f (ξ) := R N f (x)e -iξ•x dx,
for which holds the inversion formula f = (2π

) -N F • F [f (-)] as soon as f ∈ L 1 (R N ) and the Hausdorff-Young inequalities ∥f ∥ L ∞ (R N ) ≤ (2π) -N ∥f ∥ L 1 (R N ) , and ∥f ∥ L ∞ (R N ) ≤ ∥f ∥ L 1 (R N ) . (2.1)
We adopt the notation S(R N ) to represent the Schwarz space of rapidly decreasing functions that consists of smooth functions φ : R N → R that, along with all their derivatives, exhibit faster decay than any polynomial as |x| → ∞.

The linear problem (1.6)

To start with, let us consider the Heat exchanger system (1.6) in the case where all the parameters c, d, µ, ν are equal to 1. Then the linear transformation ( , ) := (u + v, u -v) enables the uncoupling of the unknowns, and thus,

u(t) = 1 + e -2t 2 (G(t) * u 0 ) + 1 -e -2t 2 (G(t) * v 0 ) , v(t) = 1 -e -2t 2 (G(t) * u 0 ) + 1 + e -2t 2 (G(t) * v 0 ) . (2.2)
From (2.2), observe that (u, v) separates into an evanescent part (u e , v e ) which decays expo- nentially fast, and a persistent part (u ∞ , v ∞ ), which is the solution to the uncoupled Heat equations

∂ t u ∞ = ∆u ∞ , t > 0, x ∈ R N , ∂ t v ∞ = ∆v ∞ , t > 0, x ∈ R N .
both starting from the averaged datum (u 0 + v 0 )/2. In particular, it becomes clear that both u and v converge to zero with algebraic decay rate N/2. Our first primary contribution is to demonstrate that a similar phenomenon occurs in the general case where c, d, µ, ν may differ.

Theorem 2.1 (Asymptotic behavior of the solutions to the diffusive Heat exchanger) Let (u, v) be the solution to system (1.6) supplemented with non-negative initial datum

(u 0 , v 0 ), such that u 0 , v 0 , u 0 , v 0 are all in L 1 (R N ). Define, for ξ ∈ R N , the radial functions r(ξ) := c -d 2 |ξ| 2 + µ -ν 2 , s(ξ) := µν + [r(ξ)] 2 , (2.3) L(ξ) := s(ξ) - c + d 2 |ξ| 2 + µ + ν 2 , (2.4)
and the dispersal operator L : S(R N ) → S(R N ) through the relation

F [L f ] = L × F [f ] , for all f ∈ S(R N ). (2.5)
Finally, let (u ∞ , v ∞ ) be the solution to the uncoupled diffusion equations

∂ t u ∞ = L u ∞ , t > 0, x ∈ R N , ∂ t v ∞ = L v ∞ , t > 0, x ∈ R N , (2.6)
respectively starting from the data u ∞,0 and v ∞,0 defined via their Fourier transforms:

u ∞,0 := 1 2 1 - r(ξ) s(ξ) u 0 + ν s(ξ) v 0 , v ∞,0 := 1 2 µ s(ξ) u 0 + 1 + r(ξ) s(ξ) v 0 .
(2.7)

Then there are two positive constants k and k ′ which depend on N, c, d, µ, ν such that

∥u(t) -u ∞ (t)∥ L ∞ (R N ) ≤ k ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) e -t ( √ µ + √ ν ) 2 2 , ∥v(t) -v ∞ (t)∥ L ∞ (R N ) ≤ k ′ ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) e -t ( √ µ + √ ν ) 2 2 , ( 2.8
)

for all t > 1.
Remarks on Theorem 2.1.

• The operator L -(2.4)- (2.5). By observing the behavior of L at high and low frequencies, we can discern similarities between L and the Laplacian. Specifically, recalling the identity

F [c∆f ] = -c |ξ| 2 × F [f ] , for all f ∈ S(R N ),
an analysis of L near the null frequency and as |ξ| approaches infinity reveals that L behaves like (cν + dµ)(µ + ν) -1 ∆ at low frequencies and min(c, d)∆ at high frequencies. Note also that L becomes independent on the exchange parameters µ and ν when the diffusion rates c and d are equal. In this case, L ≡ c∆ ≡ d∆.

Figure I below presents radial profiles of L for various combinations of the parameters c, d, µ, ν. The first line is the simple case

c = d = µ = ν = 1, for which L(ξ) = -|ξ| 2 .
Upon examining the different profiles of L in Figure I, we observe two distinct types of shapes that depend on how L changes its behavior from zero to infinity. More precisely, we see "soft transitions" in the first and second lines of the table, and "sharp transitions" in the third and last lines. These differences in the shapes of the decay may indicate varying degrees of homogeneity/heterogeneity in how u and v interact with individuals in their respective environments. Indeed, recalling that µ (resp. ν) represents the outgoing exchange rate of u (resp. v), we can observe that in the sharp transitions cases only, one of the two unknowns has both a large diffusion rate and an outgoing exchange rate less than or equal to the ingoing one. In this way, the strongest diffuser that primarily conserves its individuals mostly affects the low frequencies of L, its contribution quickly vanishing at high frequencies. • The persistent data u ∞,0 and v ∞,0 -(2.7). By analyzing (2.3), observe that |r/ √ s | and 1/ √ s stay bounded between 0 and 1, ensuring the integrability of u ∞,0 and v ∞,0 . Therefore, u ∞,0 and v ∞,0 can be defined correctly by (2.7) and written using the inversion formula given at the beginning of Section 2.

When the diffusion rates c and d are equal, expressions of u ∞,0 and v ∞,0 become straightforward linear combinations of u 0 and v 0 , and the Fourier transform is no longer required to explain them. When the diffusion rates are different -let us say c < d for simplicity -the positive functions (1 + r/ √ s )/2, µ/2 √ s and ν/2 √ s collapse algebraically fast in the high frequencies, acting as low-pass filters. Similarly, the positive function (1 -r/ √ s )/2 acts as a high-pass filter by remaining below the value 1 towards which it converges as |ξ| goes to infinity. Therefore, we can understand the persistent data u ∞,0 and v ∞,0 as linear combinations of u 0 and v 0 that have been altered by high-and low-pass frequency filters.

We now examine the decay rate of (u, v) which is expected to be algebraic. To reach this, it suffices to know how u ∞ (t) and v ∞ (t) decay in L ∞ (R N ), given the exponential convergence towards the persistent part (u ∞ , v ∞ ) ensured by Theorem 2.1. The corollary that follows reveals that this decaying occurs with magnitude N/2, which is tightly related to how the operator L affects the low frequencies of its argument.

∥u(t)∥ L ∞ (R N ) ≤ ℓ ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) (1 + t) N/2 , ∥v(t)∥ L ∞ (R N ) ≤ ℓ ′ ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) (1 + t) N/2 , (2.9)
for all t ≥ 0.

The semi-linear problem (1.1)

Having examined the behavior of the pure diffusive Heat exchanger (1.6) in the previous subsection, we are now prepared to address the question of global existence versus blow-up for our main problem (1.1). Before proceeding with the analysis, it is worth recalling that the comparison principle ensures that, given the same initial datum (u 0 , v 0 ), the solution to (1.

U ′ = U 1+α , t > 0, V ′ = κV 1+β , t > 0, (2.10)
it is apparent that, given any positive initial data U 0 , V 0 ,

• U blows-up at the rate 1/α for κ ∈ {0, 1}, and

• V blows-up at the rate 1/β only when κ = 1.

As observed in introduction, we anticipate distinguishing between systematic blow-up and possible global existence by comparing the decay rate of (1.6) with the blow-up rates of (2.10). To be more specific, the regime transition is expected at N/2 = 1/α when κ = 0 and N/2 = max(1/α, 1/β) when κ = 1. These critical exponents can simply be confirmed when the rates c, d, µ, ν are identical. To see this, let us assume c = d = µ = ν = 1, and define (σ, δ) := (u + v, u -v). Observe that the blowing-up of σ is equivalent to that of (u, v). We first examine the solutions to (1.1)| κ=1 to explore the possible global existence. In an attempt to show that σ may not blow-up, we have

∂ t σ = ∆σ + u 1+α + v 1+β ≤ ∆σ + σ 1+α + σ 1+β =: Lσ.
Thus σ serves as a sub-solution to ∂ t = L. Under the assumption min(α, β) > 2/N , we can find small enough positive values η and R to allow the solution σ to the Cauchy problem

∂ t σ = ∆σ + 2σ 1+min(α,β) , t > 0, x ∈ R N , σ | t=0 = η1 B(0,R) , x ∈ R N ,
to be global and such that σ (t) ≤ 1 for any t > 0. This can be seen from the classical construction of global super-solutions as the product of a time-dependent function with the solution to the Heat equation -see [24 page 130] for instance. Using this estimation on σ , we write

∂ t σ ≥ ∆σ + σ 1+min(α,β) + σ 1+max(α,β) = Lσ,
which indicates that σ is a global super-solution to ∂ t = L. Now, σ ≤ σ as long as u 0 and v 0 are chosen small enough to satisfy u 0 + v 0 ≤ η1 B(0,R) , and with that condition met, this case is resolved.

For the systematic blow-up (still with c = d = µ = ν = 1) we examine (1.1)| κ=0 , and observe that

∂ t δ = ∆δ -2δ + u 1+α ≥ ∆δ -2δ,
which implies that δ remains greater than the sub-solution e -2t (G(t) * (u 0 -v 0 )), that is positive as long as u 0 > v 0 . It is important to note that the assumption u 0 > v 0 is not a strict requirement for proving systematic blow-up. Indeed, we may suppose u 0 > 0 (even if we need to shift the time for this to hold) and then work with the solution arising from the initial datum (u 0 , 0). With the established order relation u > v which holds for all times, we can determine now that σ systematically blows-up when α < 2/N since

∂ t σ = ∆σ + u 1+α ≥ ∆σ + (u/2 + v/2) 1+α = ∆σ + σ 1+α 2 1+α .
In those cases of identical rates, the key argument that enables progress concisely stands in c∆u

± d∆v = c∆(u ± v) = d∆(u ± v).
However, these equalities break down when the diffusion rates differ, making c ̸ = d one of the main challenges of the problem. In the general case where c, d, µ, ν are arbitrary, we cannot rely on our knowledge about Fujita's original problem (1.3) to draw conclusions on the lifetime of solutions to (1.1). Theorem 2.3 and Theorem 2.4 fill this gap and are the main contribution of the present paper.

Theorem 2.3 (Possible global existence) Let N 2 >    1 α if κ = 0, max 1 α , 1 β if κ = 1. (2.11)
Then there is m 0 > 0 that depends on N, c, d, µ, ν, α, β, κ such that the solution (u, v) to problem (1.1) with non-negative (u 0 , v 0 ) exists for all times as soon as

m := ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) < m 0 .
(2.12)

Furthermore, under hypothesis (2.12),

∥u(t)∥ L ∞ (R N ) ≤ M (1 + t) N/2 and ∥v(t)∥ L ∞ (R N ) ≤ M ′ (1 + t) N/2 , (2.13)
for all t > 0 and some positive constants M and M ′ depending on N, c, d, µ, ν, α, β, κ, and m.

Theorem 2.4 (Systematic blow-up) Let N 2 <    1 α if κ = 0, max 1 α , 1 β if κ = 1. (2.

14)

Then any solution (u, v) to problem (1.1) with non-both-trivial and non-negative initial datum

(u 0 , v 0 ) ∈ (L 1 (R N ) ∩ L ∞ (R N )) 2 blows-up in finite time.
Remark on Theorem 2.4.

• Simultaneous versus non-simultaneous blow-up. Based on (1.2), it is clear that at least one of the two components of (u, v) becomes unbounded at the blowing time. Nevertheless, it remains uncertain whether only one component or both components tend to infinity as the blowing-up occurs. The first scenario is called non-simultaneous blow-up, whereas the second is referred to as simultaneous blow-up. Souplet and Tayachi have explored similar issues in [START_REF] Souplet | Optimal condition for non-simultaneous blow-up in a reaction-diffusion system[END_REF] for a Fujita-type system with reactions taking the form of (u α + v β , u γ + v δ ). In their study, they succeeded to separate systematic simultaneous from possible non-simultaneous blow-up with conditions between α and γ and between β and δ. As for problem (1.1), we leave this issue as an open question, but we nevertheless emphasize that the integrability in time of the L ∞ (R N )-norm of the component that goes to infinity could be a decisive factor in determining the behavior of the other component. To illustrate this idea, let us consider the simple case (1.1)| κ=0 with c = d = µ = ν = 1 supplemented with non-negative and identically constant datum (U 0 , V 0 ). In this case the solution (U, V ) does not depend on space and thus solves the ODE system

U ′ = -U + V + U 1+α , t > 0, V ′ = U -V, t > 0.
Observe that if we suppose U 0 > V 0 , then (by subtracting the unknowns) U > V , so that

(U + V ) ′ = U 1+α ≥ (U/2 + V /2) 1+α = 1 2 1+α (U + V ) 1+α ,
and U + V blows-up at some finite time T with at least U becoming unbounded at this time. Now assume the blow-up rate of U to be of magnitude a > 0, namely,

C (T -t) a ≤ U (t) ≤ C (T -t) a ,
for any t ∈ (0, T ) and some positive constants C ≤ C. Turning to V , we have

e -t V 0 + C t 0 e -(t-s) (T -s) a ds ≤ V (t) ≤ e -t V 0 + C t 0 e -(t-s) (T -s) a ds.
Thus, the boundedness of V at the blowing time is equivalent to the convergence of the integral T 0 (T -s) -a ds, which occurs as soon as a < 1. This conditions express that U must diverge slowly in some sense if we want V to be able to explode with it. Obviously, this is a formal argument and a rigorous proof would require a more detailed analysis of the system. Nevertheless, this example illustrates the importance of the blow-up rate in determining the behavior of solutions near the blowing-up.

Asymptotic behavior of the Heat exchanger system

This section focuses on the large time behavior of the solution (u, v) to the Cauchy problem associated with the linear system (1.6).

We begin by providing the proof of Theorem 2.1, which relies on Fourier analysis of the solution. By doing so, we are able to separate (u, v) into two parts, (u ∞ , v ∞ ) and (u e , v e ), where an L ∞

x (R N )/L 1 ξ (R N )-analysis shows that the second part decays exponentially fast. Finally we evaluate the time derivatives of u ∞ and v ∞ which brings us to the formulation of problem (2.6)-(2.7).

Proof of Theorem 2.1. Applying the Fourier transform to the equations in (1.6), namely

∂ t u = c∆u -µu + ν v and ∂ t v = d∆v + µu -ν v,
we are led to the ODE system

∂ t   u v   = -c |ξ| 2 -µ ν µ -d |ξ| 2 -ν =:A(ξ)   u v   . (3.1)
It may be checked that the two eigenpairs, (λ + (ξ), e + (ξ)) and (λ -(ξ), e -(ξ)), of A are given by

λ ± = - c + d 2 |ξ| 2 + µ + ν 2 ± √ s and e± = ν r ± √ s , ( 3.2)
where r and s are defined in (2.3). As a result, we obtain the following expressions for u and v:

   u(t, ξ) v(t, ξ)    = 1 2    1 -r √ s e tλ + + 1 + r √ s e tλ - ν √ s e tλ + -e tλ - µ √ s e tλ + -e tλ - 1 + r √ s e tλ + + 1 -r √ s e tλ -       u 0 (ξ) v 0 (ξ)   .
We split u and v into

   u ∞ (t, ξ) v ∞ (t, ξ)    := e tλ + 2   1 -r √ s ν √ s µ √ s 1 + r √ s      u 0 (ξ) v 0 (ξ)    (3.3) and    u e (t, ξ) v e (t, ξ)    := e tλ - 2   1 + r √ s -ν √ s -µ √ s 1 -r √ s      u 0 (ξ) v 0 (ξ)    , (3.4) so that u = u ∞ + u e and v = v ∞ + v e ,
u = u ∞ + u e and v = v ∞ + v e .
The remainder of the proof consists in demonstrating that the evanescent parts u e (t) and v e (t) decay exponentially fast in L ∞ (R N ). Following this, we establish that (u ∞ , v ∞ ) indeed serves as the solution to the Cauchy problem (2.6)-(2.7).

• Vanishing of (u e (t), v e (t)). Thanks to the Hausdorff-Young inequalities (2.1), controlling u e (t) and v e (t) in L 1 (R N ) enables us to obtain estimates on u e (t) and v e (t) in L ∞ (R N ).

To accomplish this, we first analyze the components of (3.4) which result in the following estimates for all ξ ∈ R N :

λ -≤ - c + d 2 |ξ| 2 + √ µ + √ ν 2 2 , (3.5) 1 ± r √ s ≤ 2, ν √ s ≤ ν µ , µ √ s ≤ µ ν , (3.6)
and using (2.1),

u 0 (ξ) ≤ ∥u 0 ∥ L 1 (R N ) and v 0 (ξ) ≤ ∥v 0 ∥ L 1 (R N ) . (3.7)
Then, by considering the expression for u e (t) in (3.4), the estimation of its L 1 -norm gives, for t > 1,

∥ u e (t)∥ L 1 (R N ) = 1 2 R N 1 + r √ s u 0 (ξ) - ν √ s v 0 (ξ) e tλ -dξ ≤ 1 2 2∥u 0 ∥ L 1 (R N ) + ν µ ∥v 0 ∥ L 1 (R N ) e -t ( √ µ + √ ν ) 2 2 R N e -t c+d 2 |ξ| 2 dξ ≤ max 1, ν 4µ 2π c + d N/2 ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) e -t ( √ µ + √ ν ) 2 2 , (3.8)
where (3.5), (3.6) and (3.7) have been used to go from the first to the second line. Performing a similar calculation for v e (t) leads to

∥ v e (t)∥ L 1 (R N ) ≤ max 1, µ 4ν 2π c + d N/2 ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) e -t ( √ µ + √ ν ) 2 2
, (3.9) so that, using again (2.1), we can retrieve (2.8) from (3.8) and (3.9) with

k := max 1, ν 4µ (2π(c + d)) N/2 and k ′ := max 1, µ 4ν (2π(c + d)) N/2 .
• Problem satisfied by (u ∞ , v ∞ ). Taking first t = 0 in (3.3) straightly recovers the initial datum ( u ∞,0 , v ∞,0 ) (2.7). Then, notice that λ + defined in (3.2) and L defined in (2.4) are actually the same functions. Hence, by working on ∂ t u ∞ from the frequency domain, we have

F [∂ t u ∞ ] = ∂ t u ∞ = λ + × u ∞ = L × u ∞ = F [L u ∞ ] ,
where the last equality precisely comes from the definition of the operator L in (2.5). The same calculation for

∂ t v ∞ gives F [∂ t v ∞ ] = F [L v ∞ ] ,
and thus, (u ∞ , v ∞ ) solves (2.6).

We now proceed with the proof of Corollary 2.2 regarding the decay rate. We start by obtaining uniform controls on u(t) and v(t) for t > 1, specifically,

∥u(t)∥ L ∞ (R N ) ≤ ℓ ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) t N/2 , ∥v(t)∥ L ∞ (R N ) ≤ ℓ ′ ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) t N/2 .
(3.10)

The approach used to derive (3.10) involves splitting high versus low frequencies of the solution and recognizing that only the low frequencies contribute algebraically at large times -refer to [START_REF] Chasseigne | Asymptotic behavior for nonlocal diffusion equations[END_REF] and [START_REF] Alfaro | Fujita blow up phenomena and hair trigger effect: The role of dispersal tails[END_REF] for related phenomena. Given this observation, it is sufficient to note that u(t) and v(t) are bounded for t ≤ 1 to establish (2.9) for all times.

Proof of Corollary 2.2. Recall that u e = u -u ∞ and v e = v -v ∞ . Referring to (2.8) we can write the following for t > 1:

∥u e (t)∥ L ∞ (R N ) ≤ ℓ e ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) t N/2 , ∥v e (t)∥ L ∞ (R N ) ≤ ℓ ′ e ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) t N/2 , ( 3.11)
where ℓ e and ℓ ′ e are positive constants depending on µ, ν and respectively k and k ′ -hence N, c, d, µ, ν.

We then shift to the consistent part of this proof that concerns the estimations of u ∞ v ∞ . Based on the Taylor expansion of L near the origin, specifically,

L(ξ) = - cν + dµ µ + ν |ξ| 2 + o |ξ| 2 , as |ξ| → 0,
we can chose a small enough positive constant a (depending on L, hence c, d, µ, ν) so that

L(ξ) ≤ - cν + dµ 2(µ + ν) |ξ| 2 , as soon as |ξ| ≤ a. ( 3.12)
Basic calculus may be employed to show that L is radially decreasing. This enables to find a positive constant η (depending on L and a, and consequently on c, d, µ, ν) that guarantees L(ξ) ≤ as soon as |ξ| ≥ a.

(3.13)

In the same manner as we demonstrated the vanishing of (u e , v e ), we use the Hausdorff- Young inequalities (2.1) to control ∥u

∞ (t)∥ L ∞ (R N ) and ∥v ∞ (t)∥ L ∞ (R N ) through estimates on ∥ u ∞ (t)∥ L 1 (R N ) and ∥ v ∞ (t)∥ L 1 (R N ) . Starting with u ∞ , we have ∥ u ∞ (t)∥ L 1 (R N ) ≤ |ξ|≤a u ∞ (t, ξ) dξ + |ξ|≥a u ∞ (t, ξ) dξ (3.14) =: ∥ (t)∥ L 1 (R N ) + ∥ (t)∥ L (R N ) .
For the high frequencies, we express u ∞ from (3.3) with λ + = L. This yields

∥ (t)∥ L 1 (R N ) = 1 2 |ξ|≥a 1 - r √ s u 0 (ξ) + ν √ s v 0 (ξ) e tL(ξ) dξ.
Next, using controls on L (3.13) and on 1 -r/ √ s and ν/ √ s (3.6), we eventually get

∥ (t)∥ L 1 (R N ) ≤ e -tη ∥ u 0 ∥ L 1 (R N ) + ν 4µ ∥ v 0 ∥ L 1 (R N ) (3.15)
which collapses exponentially fast. For the low frequencies, we similarly have

∥ (t)∥ L 1 (R N ) = 1 2 |ξ|≤a 1 - r √ s u 0 (ξ) + ν √ s v 0 (ξ) e tL(ξ) dξ.
Using controls on L (3.12), on 1 -r/ √ s and ν/ √ s (3.6), and on u 0 and v 0 (3.7), we find:

∥ (t)∥ L 1 (R N ) ≤ ∥u 0 ∥ L 1 (R N ) + ν 4µ ∥v 0 ∥ L 1 (R N ) |ξ|≤a e -t cν+dµ 2(µ+ν) |ξ| 2 dξ ≤ 2π(µ + ν) cν + dµ N/2 ∥u 0 ∥ L 1 (R N ) + ν 4µ ∥v 0 ∥ L 1 (R N ) t N/2 . ( 3.16)
Finally, we combine (3.15) and (3.16) and use Hausdorff-Young inequalities (2.1) to control u ∞ (t) in L ∞ (R N ). By returning to (3.14), we can follow the same approach to estimate v ∞ (t) as well. As a consequence, we can identify two positive constants ℓ ∞ and ℓ ′ ∞ , which depend on N, c, d, µ, ν, and satisfy the following inequalities for t > 1:

∥u ∞ (t)∥ L ∞ (R N ) ≤ ℓ ∞ ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) t N/2 , ∥v ∞ (t)∥ L ∞ (R ) ≤ ℓ ′ ∞ ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) t N/2 .
(3.17)

To complete the control of (u, v) for t > 1, we combine the estimations on (u e , v e ) (3.11) and (u ∞ , v ∞ ) (3.17) to obtain (3.10) with

ℓ := ℓ e + ℓ ∞ and ℓ ′ := ℓ ′ e + ℓ ′ ∞ .
Now, for t ≤ 1, the comparison principle ensures that (u, v) stays below the solution (u, v) to (1.6) with initial condition (u

0 , v 0 ) ≡ (∥u 0 ∥ L ∞ (R N ) , ∥v 0 ∥ L ∞ (R N )
). Notably, (u, v ) actually does not depend on the space variable and therefore solves the ODE system

∂ t u = -µu + ν v, t > 0, ∂ t v = µu -ν v, t > 0.
Letting := u + v, we clearly have ∂ t ≡ 0, so that,

max(u, v) ≤ max(u, v) ≤ ≡ ∥u 0 ∥ L ∞ (R N ) + ∥v 0 ∥ L ∞ (R N ) ≤ (2π) -N ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) ,
where we used the Hausdorff-Young inequalities (2.1) to obtain the last line. As a consequence, we have,

∥u(t)∥ L ∞ (R N ) ≤ (2π) -N ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) , ∥v(t)∥ L ∞ (R N ) ≤ (2π) -N ∥u 0 ∥ L 1 (R N ) + ∥v 0 ∥ L 1 (R N ) + ∥ u 0 ∥ L 1 (R N ) + ∥ v 0 ∥ L 1 (R N ) , (3.18)
for all t ≤ 1.

To complete the proof, we need to find suitable values for ℓ and ℓ ′ such that (2.9) satisfies both estimates (3.18) (for t ≤ 1) and (3.10) (for t > 1). It can be shown that choosing

ℓ := 2 N/2 max (2π) -N , ℓ and ℓ ′ := 2 N/2 max (2π) -N , ℓ ′
is the optimal solution, so that these values inherit the parameters dependency specified in Corollary 2.2.

Possible global existence

In this section we show the possible existence of global solutions for problem (1.1) when N/2 is larger than both 1/α and κ/β, as stated in Theorem 2.3.

The proof involves constructing a global super-solution (u, v ) for problem (1.1) that drives u and v to 0. More precisely, we try

u(t, x) := F (t) × u(t, x) and v (t, x) := F (t) × v(t, x), ( 4.1)
where (u, v) is the solution to the pure diffusive Heat exchanger (1.6) with initial condi- tion (u 0 , v 0 ), and F is a positive, bounded and continuously differentiable function to be determined. We then search for sufficient conditions on F that guarantee (u, v ) is a global super-solution. This leads to (2.12) which requires the datum (u 0 , v 0 ) to be small in some sense. Finally, the upper control (u, v) ≤ sup(F ) × (u, v) retrieves (2.13).

Observe that our primary focus is on problem (1.1)| κ=1 to prove Theorem 2.3. However, we also investigate the simpler case (1.1)| κ=0 to obtain more accurate values for the constants m 0 , M and M ′ when κ = 0.

Proof of Theorem 2.3. Consider (u, v ) as defined in (4.1). First, we set F (0) = 1 ensuring that (u, v ) and (u, v) have the same initial data. Next, we require the following expressions to be positive for all t > 0 and x ∈ R N :

∂ t u -c∆u + µu -ν v -u 1+α and ∂ t v -d∆ v -µu + ν v -κ v 1+β .
This leads to

F ′ ≥ F 1+α × u α and F ′ ≥ κ × F 1+β × v β . ( 4.2)
Note that u and v can be replaced in (4.2) by their L ∞ (R N )-norms and any control from above of them. Therefore, with Corollary 2.2 providing uniform controls on u and v, we can say that satisfying the following inequalities is sufficient to recover (4.2):

F ′ ≥ F 1+α × ℓm (1 + t) N/2 α and F ′ ≥ κ × F 1+β × ℓ ′ m (1 + t) N/2 β . ( 4.3)
Moving forward, we split the discussion into two parts based on whether κ is 0 or 1.

• The case κ = 0. This case is the simplest of the two since it is sufficient to ask

F ′ = F 1+α × (ℓm) α (1 + t) N α/2 (4.4)
if we require (4.3). By solving the ODE (4.4) with F (0) = 1, we obtain

F (t) = 1 - 2α(ℓm) α N α -2 1 - 1 (1 + t) (N α/2)-1 =:G 0 (t). -1/α (4.5)
It remains to ensure that F exists for all times which is permitted if and only if G 0 does not collide 0. To achieve this, we first need to assume that we are in the regime (2.11) to guarantee the vanishing of 1/(1 + t) (N α/2)-1 in (4.5). Then, because

inf t≥0 [G 0 (t)] = 1 - 2α(ℓm) α N α -2 , ( 4.6)
it suffices to chose

m < m 0 := N α -2 2αℓ α 1/α
to make G 0 positive and so (u, v ) global.

To eventually retrieve the controls in (2.13), we combine (4.5)-(4.6) and (2.9) in Corollary 2.2, which lead us to set

M := (N α -2)(ℓm) α N α -2 -2α(ℓm) α 1/α and M ′ := (N α -2)(ℓ ′ m) α N α -2 -2α(ℓm) α 1/α
, completing the proof in this case.

• The case κ = 1. Similar to the previous case, we aim to set an ODE on F , like (4.4), that would satisfy both ODIs in (4.3). With the loose assumption m < 1, it is clear that

max m (1 + t) N/2 α , m (1 + t) N/2 β ≤ m (1 + t) N/2 min(α,β) . Moreover, (4.2) requires F ′ to be non-negative, so F (t) ≥ F (0) = 1 implies max F 1+α , F 1+β ≤ F 1+max(α,β) .
As a result, it is sufficient to ask

F ′ = F 1+max(α,β) × max(ℓ α , ℓ ′β ) × m (1 + t) N/2 min(α,β) (4.7)
for F to satisfy (4.3). The remainder of the proof is nearly identical to the case κ = 0. Solving (4.7) yields

F (t) = 1 - 2 max(α, β) max(ℓ α , ℓ ′β )m min(α,β) N min(α, β) -2 1 - 1 (1 + t) (N min(α,β)/2)-1 =:G 1 (t). -1/ max(α,β) (4.8)
To ensure the global existence of F , we first need to be in the regime (2.11) to guarantee the vanishing of 1/(1 + t) (N min(α,β)/2)-1 in (4.8). Then, because

inf t≥0 [G 1 (t)] = 1 - 2 max(α, β) max(ℓ α , ℓ ′β )m min(α,β) N min(α, β) -2 , ( 4.9)
it suffices to chose

m < m 0 := min 1, N min(α, β) -2 2 max(α, β) max(ℓ α , ℓ ′β ) 1/ min(α,β)
to make G 1 positive and thus (u, v ) global.

It eventually remains to recover the controls in (2.13). To do this, we combine (4.8)-(4.9) and (2.9) in Corollary 2.2 which leads us to take

M := (N min(α, β) -2)(ℓm) max(α,β) N min(α, β) -2 -2 max(α, β) max(ℓ α , ℓ ′β )m min(α,β) 1/ max(α,β) Since Φ ε is chosen with unit mass, we have, while (u, v) exists in (L ∞ (R N )) 2 , U (t) + V(t) ≤ ∥u(t)∥ L ∞ (R N ) + ∥v(t)∥ L ∞ (R N ) .
(5.5)

As a consequence of this inequality, the blowing-up of (U , V) yields that of (u, v) -possibly at an earlier time.

The objective is now to make (U , V) blow-up. Differentiating U with respect to time yields (the parameters dependency is locally dropped for better visibility)

U ′ = Φ ε * c∆u -µu + νv + u 1+α = c [∆Φ ε * u] -µ [Φ ε * u] + ν [Φ ε * v] + [Φ ε * u 1+α ] ≥ -cλ [Φ ε * u] -µ [Φ ε * u] + ν [Φ ε * v] + [Φ ε * u] 1+α = -(µ + cλ)U + νV + U 1+α ,
where we use an integration by part to go from the first to the second line and (5.3) along Jensen inequality to go from the second to the third line. Applying the same approach to V leads us the following ODI system

U ′ ≥ -(µ + cλ)U + νV + U 1+α , t > 0, V ′ ≥ µU -(ν + dλ)V + κV 1+β , t > 0. (5.6)
Due to its cooperative structure, the system (5.6) enjoys the comparison principle. As a result, U and V are respectively above U and V which solve the associated ODE system U ′ = -(µ + cλ)U + νV + U 1+α , t > 0, V ′ = µU -(ν + dλ)V, t > 0, (5.7)

as long as (U , V) and (U, V ) start from the same initial datum, that is

(U (0), V (0)) = R N Φ ε (z)u 0 (z)dz, R N
Φ ε (z)v 0 (z)dz =: (U 0 , V 0 ). (5.8) From this point, the rest of the demonstration consists in showing that (U, V ) blows-up if the blur parameter ε (= λ/2N ) is chosen sufficiently small. In Figure II below, we present the phase plane associated with ODE system (5.7), and we take this opportunity to introduce some notations that can be understood by looking at Figure II. We begin by finding the isocline curves for system (5.7) that are

{U ′ = 0} = V = U ν (µ + cλ -U α ) , {V ′ = 0} = V = µ ν+dλ U .
(5.9)

These curves intersect each other at the equilibria O := (0, 0) and E 1 := (χ, µ ν+dλ χ), where χ := µ + cλ -µν ν + dλ (5.10)

Next, we define E 0 := (µ 1/α , 0), and for any θ ∈ [0, 1],

E θ := E 0 + θ ---→ E 0 E 1 = µ 1/α (1 -θ) + θχ, µ ν + dλ θχ .
• Step (i). To demonstrate that Ω is stable under the ODE system (5.7), we must establish that the vector field (P, Q)| ∂Ω points inwards Ω. We decompose ∂Ω into Γ 1 ∪ Γ 2 ∪ Γ 3 , as illustrated in Figure II:

∂Ω = {U > µ 1/α , V = 0} Γ 1 ∪ {U > χ, V ′ = 0} Γ 2 ∪ [E 0 E 1 ] Γ 3
.

We observe that (P, Q)| Γ 1 as well as (P, Q)| Γ 2 both point in the correct direction since

• Q > 0 along Γ 1 , and

• Q = 0 < P along Γ 2 .
To address the last portion of ∂Ω, which is Γ 3 = [E 0 E 1 ], we define the matrices

M θ := - → γ θ ---→ E 0 E 1 ,
and our goal is to prove det(M θ ) ≥ 0 for all θ ∈ [0, 1], indicating that -→ γ θ points towards the right-hand side of ---→ E 0 E 1 . Without going into the details of algebraic computations, we find

det(M θ ) = µ ν + dλ χ -µ + cλ µ 1/α (1 -θ) + θχ + θµν ν + dλ χ + µ 1/α (1 -θ) + θχ 1+α + µ 1/α -χ µ 1+1/α 1 -θ ,
where we can verify that det(M 1 ) = 0, what is consistent with -→ γ 1 = (0, 0) at the equilibrium ∂ θ det(M θ ) + µ 1+2/α = 0.

From this we deduce that there exists a positive λ 0 such that ∂ θ det(M θ ) < 0 for all λ ∈ (0, λ 0 ) and all θ ∈ [0, 1]. This implies det(M θ ) > det(M 1 ) = 0, which concludes the step.

• Step (ii). Suppose we have (U 0 , V 0 ) ∈ Ω. Then, thanks to step (i), we know that (U, V ) remains in Ω as long as it exists. Observe that Q > 0 in Ω which causes V to increases. Since the set Ω ∩ {U ′ < 0} does not contain any asymptotically stable trajectories, there exists a time t 0 > 0 for which (U, V ) crosses the curve {U ′ = 0}. After this point, (U, V ) ∈ Ω ∩ {U ′ > 0}, for all t ∈ (t 0 , T ) ,

Figure I -

 I Figure I -Representation of L(ξ) for different combinations of c, d, µ, ν. The curves are shown at different scales for readability reasons. The "soft transition cases" (resp. "sharp transition cases") -see the remarks on L above -are displayed in the first and second (resp. third and fourth) lines. The final column provides a visual representation of how u and v exchange and spread the individuals.

Corollary 2 . 2 (

 22 Decay rate of the solutions to the diffusive Heat exchanger) Under the hypothesis of Theorem 2.1, there are two positive constants ℓ and ℓ ′ depending on N, c, d, µ, ν, such that

  and the boundedness of the matrices in (3.3) and (3.4) provides the necessary integrability to go back into the spatial domain. As a result, we can write

E 1 .

 1 Now, differentiating det(M θ ) with respect to θ yields∂ θ det(M θ ) = µ ν + dλ χ µ+cλ µ 1/α -χ + µν ν + dλ χ-1+α µ 1+α -χ µ 1+α -θ(µ 1+α -χ) α -µ 1+1/α µ 1/α -χ ,and by considering that χ, defined in (5.10), vanishes as λ approaches 0, we have lim λ→0 sup θ∈[0,[START_REF] Alfaro | Fujita blow up phenomena and hair trigger effect: The role of dispersal tails[END_REF]

  1)| κ=0 always remains lower than the solution to (1.1)| κ=1 . Consequently, to demonstrate blow-up, it is sufficient to consider (1.1)| κ=0 , while to establish global existence, we only need to examine (1.1)| κ=1 . Corollary 2.2 states that the solutions to (1.6) decay uniformly to zero at the algebraic rate N/2. With regard to the uncoupled reaction component of (1.1), namely,

Acknowledgement. The author would like to acknowledge his supervisor Matthieu Alfaro for his advices and his reviews, Philippe Souplet for pointing out the references [8], [27], [6], the Laboratoire de Mathématiques Raphaël Salem for the Maple™ licence which helped to investigate around linear system (3.1), and the Région Normandie for the financial support of his PhD.

and M ′ := (N min(α, β) -2)(ℓ ′ m) max(α,β) N min(α, β) -2 -2 max(α, β) max(ℓ α , ℓ ′β )m min(α,β) 1/ max(α,β) which concludes the proof of this second case.

Systematic blow-up

In this section, we tackle the proof of Theorem 2.4 which states the blowing-up of the nonnegative solutions to (1.1) when at least one of the two 1/α and κ/β is greater than N/2.

We cover both cases of κ = 0 and κ = 1 even though the first one would suffice. To do this we need the slight hypothesis α < 2/N which is transparent for the theorem assumptions, up to put κ before u 1+α rather than v 1+β whenever β < 2/N ≤ α. Notice that u and v exchange their roles in this case.

Our method involves passing the solution (u, v) through a Gaussian blur whose intensity is adjusted via a positive parameter ε. As ε decreases, the blur increases. We denote the resulting blurred solution observed at point x = 0 as (U , V). We first observe that the blowing-up of (U , V) implies that of (u, v). Then, we show that (U , V) satisfies an ODI system which ensures its blowing-up if ε is chosen sufficiently small. Proof of Theorem 2.4. For technical reasons we start by adjusting the datum (u 0 , v 0 ) to give it a specific shape. This step is non-limiting, thanks to the comparison principle. First, note that (by shifting time if necessary) we can assume that u 0 and v 0 are positive. As a consequence, there exists small enough η, R > 0 such that

almost everywhere in R N . Therefore, we can assume without loss of generality that the datum (u 0 , v 0 ) takes the form

For ε > 0, we define the family of Gaussian kernels (Φ ε ) ε>0 for all x ∈ R N as

where

if we choose λ = 2N ε, an equality that is maintained throughout the proof. Now we use Φ ε to blur the solution (u, v) by convolution, and we denote the result evaluated at time t and point x = 0 as (U , V). More precisely,

We consider then the open set Ω which is the region above {V = 0}, below {V ′ = 0}, and on the right-hand side of the line (E 0 E 1 ). More precisely,

(5.11)

Finally, we name P and Q the two components of vector field associated with ODE system (5.7), namely,

and we denote -→ γ θ the evaluation of the field (P, Q) at point E θ .

Figure II -Phase plane associated with ODE system (5.7). The equilibria are located at O = (0, 0) = and E 1 = (χ, µ ν+dλ χ) = . The plane splits into two zones depending on whether the trajectories converge to O or blow-up in finite time. As λ approaches zero, the equilibrium E 1 moves towards O until they merge at the limit, and the blowup zone "fills" the area below the isocline {V ′ = 0}, which is defined by {V ′ > 0}. To characterize the blow-up zone, we define E 0 = (µ 1/α , 0) = , and we show that all the points in the region Ω, defined as the intersection of the right-hand side of (E 0 E 1 ), the upper side of {V = 0} and the lower side of {V ′ = 0} are indeed in the blow-up zone provided that λ has been sufficiently reduced.

The continuation of our discussion is divided into three steps: (i) We show that by choosing λ (= 2N ε) smaller, the region Ω remains stable under ODE system (5.7).

(ii) We deduce that any solution to (5.7) starting at initial time in Ω blows-up in finite time.

(iii) Finally, we find an ε > 0 that places (U 0 , V 0 ) in Ω, leading to the blowing-up of (U, V ).

where T denotes the lifetime of (U, V ). From the moment t 0 , U is therefore increasing. If U were bounded, it would converge, so would V in view of the phase plane, which is impossible. Consequently, U is unbounded and we can find a time t 1 > t 0 that is large enough to perform

Finally, by plugging the latter inequality in first line of (5.7), we find

from which the blowing-up of U is evident.

•

Step (iii). The aim of this step is to show that (U 0 , V 0 ) is in Ω if we reduce ε (= λ/2N ).

To achieve this, we must show that (U 0 , V 0 ) belongs to each set constituting the intersection Ω in (5.11).

At the beginning of the proof, we assume that the datum (u 0 , v 0 ) takes the form given in (5.1), where we have v 0 = µ 2ν u 0 . Observing the definition of (U 0 , V 0 ) in (5.8) reveals that V 0 = µ 2ν U 0 as well. Therefore, on the phase plane of Figure II, the datum (U 0 , V 0 ) is located on the line {V = µ 2ν U } which lies below the isocline {V ′ = 0} if λ is chosen sufficiently small -check (5.9) for confirmation. As a result, there exists λ 1 > 0 such that

(

Next, to place (U 0 , V 0 ) on the right-hand side of (E 0 E 1 ), we must ensure that

(5.13)

Since χ vanishes as λ approaches to 0, we may check that the quantity µχ (ν+dλ)(χ-µ 1/α ) is negative if λ is sufficiently small. Therefore, (5.13) is satisfied if

We can also remove dλ and simply require

By further reducing λ (and consequently χ), it is thus sufficient to have

1/α which we can simplified in

Returning to the definition of V 0 in (5.8) with v 0 in (5.1) and Φ ε in (5.2), the latter inequality (5.14) can be expressed as

or with an adequate constant h that depends on N, c, d, µ, ν, α, η,

(5.15)

Now as ε approaches 0, the left-hand side of (5.15) converges towards the Lebesgue measure of the ball B(0, R) that is positive. Meanwhile, on the right-hand side, ε (1/α)-(N/2) collapses since α has been chosen smaller than 2/N . As a result, by taking ε (= λ/2N ) sufficiently small, we can achieve (5.15) and thus (5.13), meaning there exists λ 2 > 0 such that (U 0 , V 0 ) ∈ V > µχ (ν + dλ)(χ -µ 1/α ) (U -µ 1/α ) , for all λ ∈ (0, λ 2 ) .

(5.16)

Lastly, noting the positivity of V 0 and considering (5.11), (5.12) and (5.16), we can conclude that there exists a positive λ (= 2N ε) for which (U 0 , V 0 ) ∈ Ω.

Gathering the elements of the proof from steps (i), (ii) and (iii), we demonstrated that we can find a positive ε for which (U, V ) blows-up in finite time, consequently inducing the blowing-up of (U , V) and (u, v). This concludes the proof.


Ferda Tartanoglu^

Valerie Issarny-^

Alexander Romanovsky^

Nicole Levy^

Dependability in the Web Services Architecture

The Web services architecture is expected to play a prominent role in developing next generation distributed systems. This chapter discusses how to build dependable systems based on the Web services architecture. More specifically, it surveys base fault tolerance mechanisms, considering both backward and forward error recovery mechanisms, and shows how they are adapted to deal with the specifics of the Web in the light of ongoing work in the area. Existing solutions, targeting the development of dependable composite Web services, may be subdivided into two categories that are respectively related to the specification of Web services composition and to the design of dedicated distributed protocols.

Introduction

Systems that build upon the Web services architecture are expected to become a major class of wide-area distributed systems in the near future. The Web services architecture targets the development of applications based on XML-related standards, hence easing the development of distributed systems through the dynamic integration of applications distributed over the Internet, independently of their underlying platforms.

A Web service is a software entity deployed on the Web whose public interface is described in XML. A Web service can interact with other systems by exchanging XML-based messages, using standard Internet transport protocols. The Web service's definition and location (given by a URI) can be discovered by querying common Web service registries. Web services can be implemented using any programming language and executed on heterogeneous platforms; as long as they provide the above features. This allows Web services owned by distinct entities to interoperate through message exchange.

Although the modularity and interoperability of the Web services architecture enable complex distributed systems to be easily built by assembling several component services into one composite service, there clearly is a number of research challenges in supporting the thorough development of distributed systems based on Web services. One such challenge relates to the effective usage of Web services in developing business processes, which requires support for composing Web services in a way that guarantees dependability of the resulting composite services. This calls for developing new architectural principles of building such composed systems, in general, and for studying specialized connectors "glueing" Web services, in particular, so that the resulting composition can deal with failures occurring at the level of the individual component services.

Several properties of the Web services architecture must be taken into account while addressing the above issues. Web services are decentralized in architecture and in administration. Therefore, individual Web services can have different characteristics (e.g., transactional supports, concurrency policies, access rights), which may not be compliant with each other. Moreover, Web services use Internet transport protocols (e.g., HTTP, SMTP) and interacting with them requires dealing with limitations of the Internet such access latency, timeouts, lost requests and security issues. These specifics of Web services require special care for supporting dependability of complex distributed systems in integrating them. The provision of effective support for the dependable integration of Web services is still an open issue, which has led to tremendous research effort over the last couple of years, both in industry and academia, as surveyed in the following.

This chapter is organized as follows. Section 2 introduces base Web services architecture, and discusses Web services composition and related dependability requirements. Then, proposed fault tolerance mechanisms for composite Web services are surveyed: Section 3 overviews transaction protocols for the Web based on backward error recovery, and Section 4 presents how forward error recovery can be applied to Web services composition processes. Finally, Section 5 summarizes our analysis, and sketches our current and future work.

The Web Services Architecture

Although the definition of the overall Web services architecture is still incomplete, the base standards have already emerged from the W3C^, which define a core middleware for Web services, partly building upon results from objectbased and component-based middleware technologies. These standards relate to the specification of Web services and of supporting interaction protocols. Furthermore, there already exist various platforms that are compliant with the Web services architecture, including .NET^, J2EE^ and AXIS'*. Figure 1 -SOAP (Simple Object Access Protocol) defines a lightweight protocol for information exchange that sets the rules of how to encode data in XML, and also includes conventions for partly describing the invocation semantics (either synchronous or asynchronous) as well as the SOAP mapping to an Internet transport protocol (e.g., HTTP) [START_REF]W3C, Simple Object Access Protocol (SOAP) 1.2, W3C Candidate Recommendation[END_REF]. -WSDL (Web Services Description Language) is an XML-based language used to specify: (i) the service's abstract interface that describes the messages exchanged with the service, and (ii) the concrete binding information that contains specific protocol-dependent details including the network end-point address of the service [START_REF]W3C. Web Services Description Language (WSDL) 1.1, W3C Note[END_REF]. -UDDI (Universal Description, Discovery and Integration) specifies a registry for dynamically locating and advertising Web services [START_REF]Universal Description, Discovery and Integration (UDDI), Version 3 Specification[END_REF].

(^Discovery

][ UDDI J Description ) [ WSDL j
Packaging J[ SOAP J ( Transport )( HTTP )

Fig. 1. Web services architecture

Composing Web services then relates to dealing with the assembly of autonomous components so as to deliver a new service out of the components' primitive services, given the corresponding published interfaces. We use the travel agent case study to illustrate a composite Web service. The travel agent service assists the user in booking complete trips according to his/her requests and is built by composing several existing Web services (e.g., accommodation and flight booking, and car rental Web services) located through a public Web services registry (see Figure 2). Each Web service is an autonomous component, which is not aware of its participation into a composition process. A typical scenario for the travel agent service is as follows:

-The user interacts only with the travel agent whose internal computations are hidden from the user. -The user sends a query to the travel agent with the date for the travel and the itinerary. -The travel agent proposes to the user complete trips satisfying his/her request, after querying appropriate accommodation, flight and car rental Web services. -The travel agent makes all the respective bookings and reservations on Web services, according to the user's choices, and returns him/her a confirmation. -If a full trip cannot be found, the travel agent tries to offer the user some replacement trips but no action can be taken without his/her approval. In the current Web services architecture, interfaces are described in WSDL and published through UDDI. However, supporting composition requires further addressing: (i) the specification of the composition, and (ii) ensuring that the services are composed in a way that guarantees the consistency of both the individual services and the overall composition. This calls for the abstract specification of Web services behaviors (see Section 2.1) and of their composition (see Section 2.2) that allows reasoning about the correctness of interactions with individual Web services. In addition, the composition process must not only define the functional behavior of the composite service in terms of interactions with the composed services, but also its non functional properties, possibly exploiting middleware-related services (e.g., services relating to WS-Security [START_REF]Web Services Security (WS-Security)[END_REF] for enforcing secure interactions). Various non-functional properties (e.g., availability, extendibility, rehability, openness, performance, security, scalability) should be accounted for in the context of Web services. However, enforcing dependability of composite Web services is one of the most challenging issues due to the concern for supporting business processes, combined with the fact that the composition process deals with the assembly of loosely-coupled autonomous components (see Section 2.3).

Specifying Conversations

To enable a Web service to be correctly integrated in a composition process, Web services conversations (also referred as to choreography) are to be provided in addition to WSDL interfaces for describing the observable behavior of a Web service by specifying the protocol in terms of message exchanges that should be implemented for correct interaction with the service. As an illustration, a flight booking Web service can publish its behavior as shown in Figure 3: the booking process starts with the call of the Login operation. If the login succeeds, the user can call the FlightRequest operation. Then, if the result returns a valid list of flights, the BookFlight operation is called by sending the FUghtId number of the corresponding flight. The conversation ends if (i) the Login operation fails, (ii) the user calls Logout during the booking process or (iii), the BookFlight operation terminates successfully by sending a confirmation message to the user. The W3C Choreography Working Group^ aims at extending the Web services architecture with a standard for specifying Web services conversations. Existing proposals to the W3C for specifying conversations include WSCL (Web Services Conversation Language) [START_REF]W3C. Web Services Conversation Language (WSCL) 1.0, W3C Note[END_REF] and WSCI (Web Service Choreography Interface) [START_REF] W3c | Web Service Choreography Interface (WSCI) 1.0, W3C Note[END_REF]. In addition, WSCI can be used to describe additional properties related to the service behavior (e.g., transactional properties and exceptional behavior). Note that the specification of conversations, which enriches the definition of Web services interfaces, should be accounted for, for the definition of Web services registries like UDDI. Instances of Web services should be retrieved with respect to the definition of both the service's interfaces, observable behaviors and non-functional properties (e.g., transactional behavior). WSDL extensibility elements allow the Web service interface definition to be extended and can be, for example, used to add information describing conversations and other properties. In addition, WSCL can further be published in UDDI registries for retrieving Web services with respect to the conversations that the services support [START_REF] Beringer | Using WSCL in a UDDI Registry 1.02[END_REF].

Specifying Composition Processes

Web services are integrated according to the specification of a composition process. Such a process may be specified as a graph (or process schema) over the set of composed Web services (see Figure 4). It is worth noting that if some Web services used in a composition process have associated conversation descriptions (e.g., WSCL or WSCI) for the operations they support, the overall composition must conform to all these descriptions. The specification of a composition graph may be:

1. Automatically inferred from the specification of individual services as addressed in [START_REF] Narayanan | Simulation, verification and automated composition of Web services[END_REF]. 2. Distributed over the specification of the component Web services as in the XL language [START_REF] Florescu | XL: An XML language for Web service specification and composition[END_REF]. 3. Given separately with XML-based declarative languages as BPEL, BPML, CSDL and SCSL [START_REF]Business Process Execution Language for Web Services (BPEL4WS), Version 1.0[END_REF][START_REF] Bpmi | Business Process Modeling Language (BPML), Version 1.0[END_REF][START_REF] Casati | Developing e-services for composing e-services[END_REF][START_REF] Yang | Web component: A substrate for Web service reuse and composition[END_REF], or in the form of state-charts as undertaken in [START_REF] Fauvet | Peer-to-peer traced execution of composite services[END_REF].

The first approach is quite attractive but restricts the composition patterns that may be applied, and cannot thus be used in general. The second approach is the most general, introducing an XML-based programming language. However, this limits the re-usability and evolution of (possibly composite) Web services due to the strong coupling of the specification of the composition process with that of the composed services. The third approach directly supports reuse, openness, and evolution of Web services by clearly distinguishing the specification of component Web services (comprising primitive components that are considered as black-box components and/or inner composite components) from the specification of composition. Hence, although there is not yet a consensus about the best approach for specifying composite Web services, it may be anticipated that this will most likely rely on the XML-based specification of a graph over Web services that is decoupled from the specification of the composed Web services. The main reasons that lead to this conclusion include compliance and complementarity with established W3C standards (i.e., WSDL and SOAP), thus providing reusability, openness and extensibility, but also the fact that it is the approach undertaken by most industrial consortia.

StartO- 4. Composition graph for the travel agent service

( ^ T.Agent Request ^ / \^ \ \ \ Hotel Search Flight Search Car Search V J \ \ U^ ^ T.Agent Select / / / / / \ \ \ Hotel Reserve Flight Login Car Rent _ Flight Search h Flight Book \ \ OEnd Fig.

Dependability Requirements

Composite Web services have high dependability requirements that call for dedicated fault tolerance mechanisms due to both the specifics of the Web services architecture and limitations of the Internet, which is not a reliable media [START_REF] Kalyanakrishnan | Rehability of Internet hosts: A case study from the end user's perspective[END_REF]. The autonomy of component Web services raises challenging issues in specifying composition processes and in particular exceptional behaviors of composite services when dealing with faults. These faults include but are not limited to (i) faults occurring at the level of the Web services, which may be notified by error messages, (ii) faults at the underlying platform (e.g., hardware faults, timeouts), and (iii) faults due to online upgrades of service components and/or of their interfaces.

In general, the choice of fault tolerance techniques to be exploited for the development of dependable systems depends very much on the fault assumptions and on the system's characteristics and requirements. There are two main classes of error recovery [START_REF] Lee | Fault Tolerance Principles and Practice, volume 3 of Dependable Computing and Fault-Tolerant Systems[END_REF]: backward (based on rolling system components back to the previous correct state) and forward error recovery (which involves transforming the system components into any correct state). The former uses either diverselyimplemented software or simple retry; the latter is usually application-specific and relies on an exception handling mechanism [START_REF] Cristian | Dependability of Resilient Computers, chapter Exception Handling[END_REF]. It is a widely-accepted fact that the most beneficial way of applying fault tolerance is by associating its measures with system structuring units as this decreases system complexity and makes it easier for developers to apply fault tolerance [START_REF] Randell | Recursive structured distributed computing systems[END_REF]. Structuring units applied for both building distributed systems and providing their fault tolerance are well-known: they are distributed transactions and atomic action^.

Distributed transactions [START_REF] Gray | Transaction Processing: Concepts and Techniques[END_REF] use backward error recovery as the main fault tolerance measure in order to satisfy completely or partially the AciD (atomicity, consistency. also referred to as conversations, but we will not use this term to avoid confusion with Web services conversations. isolation, durability) properties. Atomic actions [START_REF] Campbell | Error recovery in asynchronous systems[END_REF] allow programmers to apply both backward and forward error recovery. The latter relies on coordinated handling of action exceptions that involves all action participants. Backward error recovery has a limited applicability, and in spite of all its advantages, modern systems are increasingly relying on forward error recovery, which uses appropriate exception handling techniques as a means [START_REF] Cristian | Dependability of Resilient Computers, chapter Exception Handling[END_REF]. Examples of such applications are complex systems involving human beings, COTS components, external devices, several organizations, movement of goods, operations on the environment, real-time systems that do not have time to go back. Integrated Web services clearly fall into this category. A typical example that shows the need for a specialized fault tolerance mechanism for the Web services architecture is the issue of running distributed transactions over several autonomous Web services. As an illustration, Figure 5 depicts a sequence diagram example associated with the travel agent service. We consider that for a given trip request (expressed by giving the journey dates and the location), the travel agent finds a list of available hotel rooms and flights to the destination. Then, the user selects a complete trip by choosing a hotel and a flight from the list of trips. Once the user has confirmed the booking, the travel agent makes the hotel reservation, which completes successfully and then, attempts for the flight booking. The flight booking Web service returns an error, informing that the server is busy, and after several attempts, the server returns a NoFlightAvailable error message. Several solutions can be applied to handle this type of faults:

Hotel Flight

Request

1. The hotel reservation can be canceled -if possible. The user is then informed that this trip is no longer available and he/she can retry booking by selecting another trip (e.g., by changing the dates of the journey). 2. Alternative transport means instead of flight can be proposed to the user (e.g., a train ticket to a close city and renting a car).

These examples reflect two different fault tolerance mechanisms; (i) backward error recovery with cancellation/compensation and retry, and (ii) forward error recovery with an application-specific exception handler that handles the error without necessarily trying to restore the system state back.

Developing fault tolerant mechanisms for composite Web services has been an active area of research over the last couple of years. Existing proposals mainly exploit backward error recovery, and more specifically, transactions. However, the autonomy of Web services and the Web latency have led to exploit more flexible transactional models and forward error recovery techniques, as discussed in the next two sections.

Backward Error Recovery for the Web

Transactions have been proven successful in enforcing dependability in closed distributed systems and are extensively exploited for the implementation of primitive (non-composite) Web services. However, transactions are not suited for making the composition of Web services fault tolerant in general, for at least two reasons: ~ The management of transactions that are distributed over Web services requires cooperation among the transactional supports of individual Web services, which may not be compliant with each other and may not be willing to do so given their intrinsic autonomy and the fact that they span different administrative domains. -Locking resources (i.e., the Web service itself in the most general case) until the termination of the embedding transaction is in general not appropriate for Web services, still due to their autonomy, and also to the fact that they potentially have a large number of concurrent clients that will not stand extensive delays.

Enhanced transactional models have been considered to alleviate the latter shortcoming. In particular, the split model (also referred to as open-nested transactions) where transactions may split into a number of concurrent subtransactions that can commit independently allows reduction of the latency due to locking [START_REF] Pu | Split-transactions for open-ended activities[END_REF]. Typically, sub-transactions are matched to the transactions already supported by Web services (e.g., transactional booking offered by a service). Hence, transactions over composite services do not increase the access latency as offered by the individual services. Enforcing the atomicity property over a transaction that has been split into a number of sub-transactions then requires using compensation over committed sub-transactions in the case of transaction abortion. However, to support this, Web services should provide compensating operations for all the operations they offer. Such an issue is in particular addressed by the BPEL [START_REF]Business Process Execution Language for Web Services (BPEL4WS), Version 1.0[END_REF] and WSCI [START_REF]W3C. Web Services Conversation Language (WSCL) 1.0, W3C Note[END_REF] languages for specifying composite services, which allow compensating operations associated with the services operations to be defined. It is worth noting that using compensation for aborting distributed transactions must extend to all the participating Web services (i.e., cascading compensation by analogy with cascading abort). Such a concern is addressed in [START_REF] Mikalsen | Transactional attitudes: Reliable composition of autonomous Web services[END_REF]. This paper introduces a middleware whose API may be exploited by clients of a composite service for specifying and executing a (open-nested) transaction over a set of Web services whose termination is dictated by the outcomes of the transactional operations invoked on the individual services. In addition to client-side solutions to the coordination of distributed open-nested transactions, work is undertaken in the area of distributed transaction protocols supporting the deployment of transactions over the Web, while not imposing long-lived locks over Web resources. We discuss here the two main proposals aimed at the Web services architecture: (i) the Business Transaction Protocol (BTP) [START_REF]Business Transaction Protocol (BTP)[END_REF], and (ii) Web Services Transaction (WS-Transaction) [START_REF]Web Services Transaction (WS-Transaction)[END_REF].

BTP introduces two different transaction models for the Web: (i) the atomic business transactions (or atoms), and (ii) the cohesive business transactions (or cohesions). A composite application can be built from both atoms and cohesions that can be nested. In the atomic business transaction model, several processes are executed within a transaction and either all complete or all fail. This is similar to distributed AciD transactions on tightly coupled systems. However, the isolation property is relaxed and intermediate committed values can be seen by external systems (i.e., systems not enrolled in the transaction). Figure 6 illustrates the atomic business transaction model using the travel agent service involving a flight booking Web service {Flight) and an accommodation booking Web service {Hotel). In this scenario, the hotel room booking fails while the flight booking succeeds, which leads to cancellation of the booked flight before the end of the transaction: A travel agent service scenario example for illustrating cohesive business transactions is given in Figure 7, where the flight booking is performed on two distinct Web services. In this example, the transaction, which was originally initiated with three participants, ends with two commits and one abortion:

1. Travel Agent sends the request messages to the two flight booking Web services. Air France and British Airways and to the Hotel Web service. 2. Web services return response messages to the Travel Agent. 3. Travel Agent selects Air France for the flight booking, and therefore sends a Cancel message to British Airways Web service and a Prepare message to the two other Web services. 4. Air France and Hotel Web services acknowledge with the Prepared message and British Airways confirms the cancellation with the Canceled message. 5. Travel Agent confirms commits {Confirm messages). 6. Web services acknowledge (Con/irrned messages). [START_REF]Web Services Transaction (WS-Transaction)[END_REF] defines a specialization of WS-Coordination [START_REF]Web Services Coordination (WS-Coordination)[END_REF], which is an extensible framework for specifying distributed protocols that coordinate the execution of Web services, and that can be used in conjunction with BPEL. Like BTP, it offers two different transaction models: (i) atomic transactions (AT) and (ii) business activity (BA). An atomic transaction adheres to the traditional ACID properties with a two-phase commit protocol. Note that as opposed to the BTP atomic business transactions, the isolation property is not relaxed in WS-Transactions, which as we mentioned before, is not suitable for the majority of Web service applications. The business activity protocol specifically serves

WS-Transaction

Travel Agent

Air France British Airways Hotel If there is a need for a coordinated activity to be compensated, the coordinator sends compensate messages to all the participants involved in the activity. Then, each participant replies by sending back either a compensated or a faulted message, depending on whether the required compensation operation was successfully completed or not. However, there is no requirement for an agreement on the outcome, and any participant can leave the coordinated activity in which it is engaged, prior to the termination of peer participants. A WS-Transaction business activity example is shown in Figure 8, with an Airline Web service and an Hotel Web service: Although there is not yet a consensus on a standard protocol for managing transactions on the Web, various implementations of these protocols are already available: the JOTM transaction manager^ and Cohesions ^° implement ® http://www.objectweb.org/jotin/. ^' ^ http://www.choreology.com/.

the BTP protocol, and the CoUaxa Orchestration Server^^ permits the use of the WS-Transaction protocol when composing Web services with the BPEL language. However, solutions to the dependable composition of Web services that use primarily transactions do not cope with all the specifics of Web services. A major source of penalty lies in the use of backward error recovery in an open system such as the Internet, which is mainly oriented towards tolerating hardware faults but poorly suited to the deployment of cooperation-based mechanisms over autonomous component systems that often require cooperative application-level exception handling among component systems. Even with the use of cohesions, which does not necessarily roll back all participant states in case of abortion of one of them, they do not specify any specific way of providing fault tolerance, so everything is left to programmers. Moreover, cancellation or compensation does not always work in many real-life situations, which involve documents, goods, money as well as humans (clients, operators, managers, etc.) and which require application-specific error handling. [START_REF] Casati | Developing e-services for composing e-services[END_REF] Forward Error Recovery for the Web Forward error recovery, using an exception handling mechanism is extensively exploited in the specifications of composite Web services in order to handle error occurrences (e.g., [START_REF]Business Process Execution Language for Web Services (BPEL4WS), Version 1.0[END_REF], [START_REF] Bpmi | Business Process Modeling Language (BPML), Version 1.0[END_REF], [START_REF] W3c | Web Service Choreography Interface (WSCI) 1.0, W3C Note[END_REF]). For instance, in BPEL, exception handlers (referred to as fault handlers) can be associated to a (possibly nested) activity so that when an error occurs inside an activity, its execution terminates, and the corresponding exception handler is executed. However, when an activity is defined as a concurrent process and at least one embedded activity signals an exception, all the embedded activities are terminated as soon as one signaled exception is caught, and only the handler for this specific exception is executed. Hence, error recovery actually accounts for a single exception and thus cannot ensure recovery of a correct state. The only case where correct state recovery may be ensured is when the effect of all the aborted activities are rolled back to a previous state, which may not be supported in general, in the context of Web services, as discussed previously. The shortcoming of BPEL actually applies to all XML-based languages for Web services composition that integrate support for specifying concurrent activities and exception handling.

A solution to the above issue lies in structuring the composition of Web services in terms of coordinated atomic actions. The Coordinated Atomic Action (or CA action) concept [START_REF] Xu | Fault tolerance in concurrent object-oriented software through coordinated error recovery[END_REF] is a unified scheme for coordinating complex concurrent activities and supporting error recovery between multiple interacting components. Atomic actions are used to control cooperative concurrency and to implement coordinated error recovery whilst AciD transactions are used to maintain the consistency of shared resources. A CA action is designed as a set of participants cooperating inside it and a set of resources accessed by them. In the course of the action, participants can access resources that have AciD properties. Action participants either reach the end of the action and produce a http: //www. collELxa.com/. normal outcome or, if one or more exceptions are raised, they all are involved in their coordinated handling. If several exceptions have been raised concurrently, they are resolved [START_REF] Campbell | Error recovery in asynchronous systems[END_REF] using a resolution tree imposing a partial order on all action exceptions, and the participants handle the resolved exception. If this handling is successful the action completes normally, but if handling is not possible then all responsibility for recovery is passed to the containing action where an external action exception is propagated. CA actions provide a base structuring mechanism for developing fault tolerant composite Web services: a CA action specifies the collaborative realization of a given function by composed services, and Web services correspond to external resources. However, as for transactions, AciD properties over external resources are not suited in the case of Web services. We have therefore introduced the notion of Web Services Composition Action (WSCA) that differs from CA actions in relaxing the transactional requirements over external resources (which are not suitable for wide-ajea open systems) and the introduction of dynamic nesting of WSCAs (i.e., nested calls of WSCAs for the sake of modularity) [START_REF] Tartanoglu | Dependability in the web service architecture[END_REF].

The travel agent service is used to illustrate how WSCAs can be applied for specifying the composition of Web services. We consider joint booking of accommodation and flights using separate hotel and airline Web services. Then, the composed Web service's operation is specified using WSCAs as follows. The top-level TraweM^eni WSCA comprises the t/ser and the TraweZ participants; the former interacts with the user while the latter achieves joint booking according to the user's request through call to the WSCA that composes the Flight and the Hotel participants^^. Figure 9 depicts the software architecture of the travel agent service where rectangles represents components which can be either WSCA participants or Web services, and elliptical nodes represents connectors such as WSCAs connectors and SOAP connectors.

A diagrammatic specification of the WSCAs is shown in Figure 10. In Trav-elAgent, the User participant requests the Travel participant to book a flight ticket and a hotel room for the duration of the given stay. This leads the Travel participant to invoke the JointBooking WSCA that composes the Hotel Web service and the Airline Web service. The participants of the JointBooking WSCA respectively requests for a hotel room and a flight ticket, given the destination and departure and return dates provided by the user. Each request is subdivided into reservation for the given period and subsequent booking if the reservation succeeds^^. In the case where either the reservation or the booking fails, the participant raises the unavailable exception that is cooperatively handled at the ^ Such a workflow process is certainly not the most common since the user is in general requested for confirmation prior to booking. However, this scenario that applies most certainly to in-hurry-not-bother users enables concise illustration of the various recovery schemes that are supported. ' •^ We assume here that all Web services understand and use term unavailable in the same way. To deal with the problem of meanings in a general fashion one could apply techniques that use ontology and standard ways of representing semantic information. Such an ontology for travel business partners can be found on http://opentravel.org. Compared to the solutions that introduce transactional supports for composed Web services, using WSCAs mainly differs in that it exploits forward error recovery at the composition level, while enabling exploitation of transactional supports offered by individual Web services -if available. Hence, the underlying protocol for interaction among Web services remains the one of the Web services architecture (i.e., SOAP) and does not need to be complemented with a distributed transaction protocol. Similarly to this solution, the one of [START_REF] Mikalsen | Transactional attitudes: Reliable composition of autonomous Web services[END_REF] does not require any new protocol to support distributed open-nested transactions. An open-nested transaction is declared on the client side by grouping transactions of the individual Web services, through call to a dedicated function of the middleware running on the client. The transaction then gets aborted by the middleware using compensation operations offered by the individual Web services, according to conditions set by the client over the outcomes of the grouped transactions. The solution offered by WSCA is then more general since it allows forward error recovery involving several services to be specified at the composition level, enabling in particular to integrate non-transactional Web services while still enforcing dependability of the composite service and partial results of a nested action to be reported to the higher level action. There is a number of Java and Ada implementations of CA actions developed for different platforms, environments and applications. A complete RMI Java framework was developed several years ago [START_REF] Zorzo | An object-oriented framework for dependable multiparty interactions[END_REF] and since then it has been applied in a number of industry-oriented case studies. It offers a number of classes (for defining actions, action participants, exception handlers) and a runtime support in a form of the action manager object. Recently it has been used for a preliminary experimental work on implementing a prototype Travel Agent system [START_REF] Romanovsky | Structuring integrated Web applications for fault tolerance[END_REF]. A Java-based local runtime support for WSCA is under development now. It is built as an adaptation of this extended CA action Java framework.

Conclusion

The Web services architecture is expected to play a major role in developing next generation distributed systems. However, the architecture needs to evolve to support all the requirements appertained to distributed systems. Addressing such requirements relates, in particular, in reusing solutions from the distributed system community. However, most solutions will not be reusable as is, mainly because of the openness of the Internet. Hence, making evolve the Web services architecture to support the thorough development of distributed systems raises a number of challenges.

This paper has addressed one of the issues raised in this context, which is the dependable composition of Web services, i.e., understanding how fault tolerance should be addressed in the Web services architecture. While dependability in closed distributed systems is conveniently addressed by transactions when concerned with both concurrency control and failure occurrences, it can hardly rely on such a mechanism in an open environment. A major source of difficulty lies in the use of backward error recovery which is mainly oriented towards tolerating hardware faults but poorly suited to the deployment of cooperation-based mechanisms over autonomous component systems that often require cooperative application-level exception handling among component systems. One solution to this concern lies in forward error recovery that enables accounting for the specific of Web services and that leads to structure Web services-based systems in terms of co-operative actions. In particular, we are able to address dependable service composition in a way that neither undermines the Web service's autonomy nor increases their individual access latency.

We are currently working on the formal specification of WSCAs for enabling rigorous reasoning about the behavior of composite Web services regarding both the correctness of the composition and offered dependability properties. The specification of composite Web services allows carrying out a number of analyzes with respect to the correctness and the dependable behavior of composite services. Except classical static type checking, the correctness of the composite service may be checked statically with respect to the usage of individual services. In addition, the same specification can be used for implementing executable assertions to check the composite service behavior online. Reasoning about the dependable behavior of composite Web services lies in the precise characterization of the dependability properties that hold over the states of the individual Web services after the execution of WSCAs. We are in particular interested in the specification of properties relating to the relaxed form of atomicity that is introduced by the exploitation of open-nested transactions within WSCA.

We are further implementing a base middleware support for WSCAs. The middleware includes the generation of composite Web services from WSCA specifications and a service for locating Web services that implements behavioral specification matching. In addition, we target development of related middleware support for improving the overall quality of composite Web services. We are in particular interested in developing a specialized caching support intended for reducing response time.

R

  . de Lemos et al. (Eds.): Architecting Dependable Systems, LNCS 2677, pp. 90-109, 2003. (c) Springer-Verlag Berlin Heidelberg 2003

  depicts the technology stack of the base Web services architecture, each layer being defined with common protocol choices. Main standards for the Web services architecture ^ World Wide Web Consortium, http: //www. w3. org. ^ Microsoft .NET, http://www.microsoft.com/net/. ^ Java 2 Platform, Enterprise Edition, http://java.sun.com/j2ee. ^ Apache AXIS, http://xml.apache.org/jixis. Ferda Taxtanoglu et al. being defined by the W3C Web Service Activity^ and the Oasis Consortium^ are the following:

^

  http://www.w3c.org/2002/ws/. ^ http://www.oasis-open.org.

Fig. 2 .

 2 Fig. 2. A composite Web service example: The travel agent service

Fig. 3 .

 3 Fig. 3. A conversation for the flight booking Web service

Fig. 5 .

 5 Fig. 5. Sequence diagram for the travel agent service

1 . 2 .Fig. 6 .

 126 Fig. 6. BTP Atomic Business Transaction

Fig. 7 .

 7 Fig. 7. BTP Cohesive Business Transaction

  1. The Travel Agent initiates the Business Activity with the Flight and Hotel participants by sending the Request messages. 4. The Flight Web service returns Completed to confirm commitment, while the Hotel Web service returns an error message Faulted and can no longer commit the transaction. 5. In order to abort the whole transaction and restore the state, the Travel Agent sends a Compensate message to the Flight Web service which has already completed the (sub)-transaction and a Forget message to the Hotel Web service. 6. The Flight Web service compensates the committed transaction by canceling the booking order and confirms with the Compensated message sent back to the Travel Agent. 7. If the Flight Web service cannot compensate the booked operation, it returns an error message Faulted back to the Travel Agent 8. In this case, the Travel Agent sends a Forget message to the Flight Web service. The iiight has been booked and cannot be canceled.

	Travel Agent	Flight	Hotel
	Request
	Request
		Register
			Register
	Complete
	Completed
			Faulted
	Compensate
	Forget
	Compensated
		Faulted
	Forget
		'	'
	Fig. 8. WS-Transaction Business Activity

  JointBooking WSCA denoted by the greyed box in the figure. If both participants signal the unavailable exception, then Travel signals the abort exception so that the exception gets handled by TravelAgent in a cooperation with the User (e.g., by choosing an alternative date). If only one participant raises the unavailable exception, cooperative exception handling includes an attempt by the other participant to find an alternative booking. If this retry fails, the booking that has succeeded is canceled and the abort exception is signaled to the calling TravelAgent WSCA for recovery with user intervention.

	TravelAgent WSCA ( WSCA
	(T SOAP^	(T SOAP "}
	Air France	British Airways	Hilton
	Fig. 9. WSCA architecture
	level of the

The Flight and Hotel Web services enroll in the transaction {Register messages) by returning list of respective availabilities.

3.Travel Agent initiate booking on the Flight Web service {Complete message).

Acknowledgments

We would like to thank Marie-Claude Gaudel and Brian Randell for their helpful comments and discussions. This research is partially supported by the European 1ST DSoS (Dependable Systems of Systems) project (IST-1999-11585)i'*.