Blowing-up solutions for a nonlinear time-fractional system

A nonlinear system with different fractional derivative terms is considered. The existence of positive blowing-up solutions is proved.


Introduction
We are concerned with blowing-up solutions of the nonlinear fractional system: (1.1) for u > 0, v > 0, where D σ t for 0 < σ < 1 (σ = α, β) stands for the Riemann-Liouville fractional derivative defined for an integrable function f by ( (t−τ ) σ dσ , p, q, r, s are positive real numbers to be fixed later. There are a couple of physical motivations for considering the system (1.1). Firstly, the type of nonlinearities in the system (1.1) appears in the systems describing processes of heat diffusion and combustion in two component continua with nonlinear heat conduction and volumetric release (u t − a u = u p v q , v t − b u = u r v s , the subscript t stands for the time derivative, while stands for the Laplacian operator) [6]. Secondly, as suggested recently [2] one may take (0)) instead of u and v if the process takes place in a porous medium. To simplify the analysis one may start by replacing (u − u(0)) and respectively. Before, we state our results, let us dwell a while on the existing literature. In [3], Furati and Kirane considered blowing-up solutions to the system for u > 0, v > 0 and 0 < α, β < 1. Then Kirane and Malik in [4] studied the profile of the blowing-up solutions of system (1.2). The study of the reduced system: for u > 0, v > 0 is well documented in the book [6]; in fact, it admits the first integral where a 1 = r + 1 − p, a 2 = q + 1 − s. it can then be decoupled in From here, the occurrence of finite time blow-up in each component can be derived. If, for example, a 1 > 0, a 2 > 0, then u blows-up whenever p + a 1 q/a 2 > 1; this inequality is equivalent to the condition −rq + ( p − 1)(s − 1) > 0, which is satisfied in this case (since p < 1 + r, s < 1 + q). Similarly, it can be checked that if a 1 > 0, a 2 > 0, the second component v also blows-up in finite time. From the identity (1.4) it follows that the blow-up times of u(t) and v(t) are the same. A different situation arises when a 1 a 2 < 0, for example, if a 1 > 0, a 2 < 0. In this case C 0 > 0, and since s > 1 In the case a 1 < 0, a 2 < 0, the constant C 0 can be either sign. For C 0 = 0, both components of the solutions of equations (1.5) and (1.6) lead to finite time blow-up.
However, such a decoupling and analysis is not directly possible for system (1.1).

Results
As argued in [5], For the sequel, we need the following lemma.
Let us recall the Mittag-Leffler function The solutions of (1.1) satisfies the system of integral equations Now, we prove the positivity of u and v .
we write the first equation of system (1.1) as: where I is the identity operator; so, formally, where * denotes convolution. We clearly have u > 0. Similarly, v > 0 can be proved. As a first consequence, the solution (u, v) of system (1.1) satisfies the system of inequalities so it is an upper solution of the system Our first result is the following blow-up result concerning solutions to system (1.1).
Proof The proof is based on the results above concerning system (1.3).

Theorem 2.4 Let (u, v) be the solution of (1.1) associated to the initial condition
is satisfied, then (u, v) blows-up in a finite time.

Estimate of the blow-up time and profile of the solution
We consider only solutions under conditions of Theorem 2.4. Without loss of generality, we assume that θ > λ ⇔ s > p. Then we have: Solutions of (3.1) and (3.2) are explicitly given by Whereupon,