Blow-up Properties of a Coupled System of Reaction-Diffusion Equations

_____________________________________________ Abstract This paper is concerned with a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions. Firstly, we studied the blow-up set showing that, under some conditions, the blow-up in this problem occurs only at a single point. Secondly, under some restricted assumptions on the reaction terms, we established the upper (lower) blow-up rate estimates. Finally, we considered the Ignition system in general dimensional space as an application to our results.


Introduction
It is well known that many phenomena in the world can be described using partial differential equations. Therefore, since the last decades, the analytical and numerical solutions of partial differential equations have been studied by many authors, see for instance [1,2]. One of the remarkable phenomena in time-dependent problems is the blow-up, which has been considered by many authors (for a single equation and systems), see for instance [3][4][5]. This work is concerned with the blow-up properties of a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions: where are positive and increasing superlinear functions on being integrable at infinity. Moreover, are positive functions in and are nonnegative, smooth, redial non-increasing, and vanishing on That is:

ISSN: 0067-2904
Rasheed Iraqi Journal of Science, 2021, Vol. 62, No. 9, pp: 3052-3060 3503 In addition, the following conditions are assumed to be satisfied: (3) In fact, Problem (1) has been used to describe physical models arising in many fields of sciences [6]; for instance, the chemical concentration, the temperature, and in the chemical reaction process. The coupled reaction-diffusion systems defined in a ball with homogeneous Dirichlet boundary conditions have been studied in [6][7][8][9]. In [7], problem (1) has been considered in one dimensional space: Under some assumptions on and , it has been shown that the blow-up can only occur at a single point. As applications to that result, two special cases of where considered: the power forms and the exponential forms. Later, problem (1) has been studied in a general dimensional space [9], where and are of power type functions: (4) It has been proved that the blow-up can only accour at a single point. In addition, the lower point-wise estimates are as follows: where In [6], it was shown that the upper and lower blow-up rate estimates of this problem are as follows: For another special case of problem (1), where are of exponential type, we have (5) It has been proved that the only blow-up point is and the upper (lower) blow-up rate estimates are as follows [6]: In this paper, under some conditions on the reaction terms, and , we prove that blow-up, in problem (1), occurs only at a single point. Moreover, we established the upper (lower) blow-up rate estimates In addition, the Ignition system [10] will be considered in a general dimensional space as an application to our result.

Preliminaries
By the standard parabolic theory, the local existence and uniqueness of classical solutions to problem (1) are guaranteed [11]. In addition, for many types of the functions, , if the initial functions are suitably large, then [12,13]. Moreover, only simultaneous blow-up can occur and that is because the system in (1) is coupled. In the next lemma, we present some properties to the solutions of problem (1)- (2). For simplicity, we denote Lemma 1, [14]: Let be a classical solution to the problem (1), (2). Then 1.
and are positive and radial. 2.
in Moreover, in 3. 4. If blows-up, then belongs to the blow-up set.

Blow-up Set
Under some assumptions, the next theorem shows that the only possible blow-up point to problem (1)- (2) is . Theorem 1: Let be a blow-up solution to problem (1)- (2). Assume that If there exist two functions such that in In addition, the following conditions are assumed to be satisfied: for some then is the only possible blow-up point Proof Following the technique used in [15] for the scalar problem: As in Lemma 1, and since are radial, for simplicity we denote By the new variables, system (1) can be rewritten as follows: We set that By the parabolic regularity results, we obtain It follows that if , then it cannot be a blow-up point. Therefore, under the assumption of this theorem, the blow-up in problem (1)- (2) can only occur at a single point, which is .

Blow-up Rate Estimates
In this subsection, we consider the lower (upper) blow-up rate estimates for problem (1)-(2) with some restricted assumptions on Theorem 2: Let be a solution to (1), (2), which blows up at only one point ( Similarly, we can find such that

The Ignition System
In this section, we apply Theorem 1 and Theorem 2 to the so called Ignition system [7] which takes the following form: where In order to show that the condition (13) is satisfied for system (23), we need to prove the following lemma. Therefore, (11) and (12) become Thus , or Next, we apply theorem (2) to derive the upper (lower) blow-up rate estimates for problem (23) with (2). Theorem 4: Let be a blow-up solution to problem (23) with (2). Assume that satisfies (6). Then, there exist positive constants , such that

Proof
We define the functions as follows ∫ ∫ It is obviously that Moreover, Therefore, from (14) it follows that Thus, there exist such that By using the same way, we can find such that

Conclusions
In this paper, we studied the blow-up set and the upper (lower) blow-up rate estimates for a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions, under some assumptions. We note that the results of the present work can be applied to many types of systems, including the ignition system. Therefore, by these results, we can easily understand the blowup properties and profiles of such types of systems.