Necessary Conditions for Local and Global Existence to a Reaction-diffusion System with Fractional Derivatives

We give some necessary conditions for local and global existence of a solution to reaction-diffusion system of type (FDS) with temporal and spacial fractional derivatives. As in the case of single equation of type (STFE) studied by M. Kirane et al. (2005), we prove that these conditions depend on the behavior of initial conditions for large |x|.


Introduction
This paper deals with the following reaction-diffusion system of type (FDS): where N ≥ 1, p and q are two positive reals.
The system (FDS) is completed by the following initial conditions: and we will assume that both u 0 and v 0 are nonnegative continuous functions.
The system (FDS) was considered in the case α = δ = 1, β = γ = 2, by many authors in several contexts, see [2,3,5] (with ν = μ = 1).Moreover, concerning nonexistence result and basing one's argument on [4], Escobedo and Herero proved in [2] that if pq ≥ 1, then the only global solution of the system (FDS), reduced to the following reaction-diffusion problem: is the trivial one, that is, u ≡ v ≡ 0, while, in a recent paper (see [7]), the authors study the system (FDS) and they find a bound on N leading to the absence of global nonnegative solutions.More precisely, recovering the case studied in [2] (when , they proved the following.
Theorem 1.1.If p > 1 and q > 1 and supposing then the system (FDS) does not admit nontrivial global weak nonnegative solution.
Therefore, they also establish some necessary conditions for the existence of local and global solutions to the following problem: and these conditions depend on the behavior of the initial data u 0 and on the function h for large |x|.Similar results can be found in [6] and [1].
Our results can be viewed as analogous to those obtained in [7], since the system (FDS) can be considered as a system of two equations of type (STFE).Therefore, we can extend these results to the more general system, under some suitable conditions on h, g, k, and l.Of course, taking such form for reaction terms, all results presented here will depend also on the functions h, g, k, and l.
For the convenience of the reader, we recall here some definitions and properties of fractional derivatives in the sense of Caputo and of Riemann-Liouville.
Definition 1.2.The left-handed derivative and the right-handed derivative in the sense of Caputo for ψ ∈ L 1 (0,T) are defined, respectively, by where Γ denotes, as usual, the Euler gamma function.

K. Haouam and M. Sfaxi 3
Up to replace ψ by ψ and to keep the derivative operator before the integral in expressions (1.8), we obtain the definitions of the left-handed derivative and the right-handed derivative in the sense of Riemann-Liouville denoted, respectively, by D α 0|t and D α t|T .See [8] for more details.
Recall that the Caputo derivative is related to the Riemann-Liouville one by the following formula: Finally, taking into account the following integration by parts formula: we adopt the following.
and satisfies

Statement of the results
Our main result is the following.Theorem 2.1.Assume that p, q > 1 and let (u,v) be a local solution (T < +∞) of problem (FDS).Then, the following estimates hold: where C and C are some positive constants.
Proof.Thanks to variational formulation (1.12), we have And thus where As above, using once again Hölder's inequality, we obtain where Furthermore, keeping the first terms in the left-hand sides of (1.12) and recalling that both u 0 and v 0 are nonnegative functions, we obtain as above ( Now, we consider some test functions in (2.12) and (2.13), introduced in [7], of the form where ) is nonnegative, with support in {R < |x| < 2R} and satisfies (−Δ) β/2 Φ + ≤ kΦ for some constant k > 0, (−Δ) γ/2 Φ + ≤ hΦ for some constant h > 0. (2.15) The exponent l, introduced in (2.14), is any positive real number if where p and q are, respectively, the conjugate exponents of p and q.Moreover, note that where Similarly, where 6 Necessary conditions for local and global existence Next, as in [7], consider the change of variables Taking into account (2.15), we obtain Analogously, using once again (2.15), we get Hence, (2.21), (2.23), and (2.24) with inequality (2.12) lead to where C 1 , C 2 , C 3 , and C 4 are positive constants independent on R and T. Consequently, (2.26) Finally, letting R → +∞ in (2.26), we conclude that (2.27) Estimate (2.1) is then proved.Using estimate (2.13) and applying the change of variables t = Tτ and x = Ry in the expressions of Ꮽ and Ꮾ, we obtain estimate (2.2).
In the sequel, we also assume that p, q > 1. Concerning existence of global and local solution, we give the following necessary condition results.