Blow-Up Phenomena in Reaction-Diffusion Problems with Nonlocal and Gradient Terms

This paper considers the blow-up phenomena for the following reaction-diﬀusion problem with nonlocal and gradient terms: in Ω . ⎧⎪⎪⎪⎨⎪⎪⎪⎩ Here m > 1, and Ω ⊂ R N ( N ≥ 2 ) is a bounded and convex domain with smooth boundary. Applying a Sobolev inequality and a diﬀerential inequality technique, lower bounds for blow-up time when blow-up occurs are given. Moreover, two examples are given as applications to illustrate the abstract results


Introduction
Since the 1960s, the blow-up phenomena of reaction-diffusion problems have received considerable interest. From then on, questions concerning finite-time blow-up of solutions to reaction-diffusion problems, as well as other classes of problems, have attracted more attention. A number of studies appeared on the topic of blow-up in time (see [1][2][3][4]) or global existence and boundedness of solutions (see [5]). Moreover, qualitative properties were investigated such as the blow-up set, rate, and profile of the blow-up [6].
From a practical point of view, apart from considering the above problems, one would like to know whether the solutions blow up, and if so, at what time blow-up occurs. However, when the solution does blow up at some finite time, the blowup time can seldom be determined explicitly, and much effort has been devoted to the calculations of bounds for the blow-up time. As we know, the studies of many papers are lead to upper bounds for the blow-up time when blow-up does occur (see [7][8][9][10]). For the practical situations, lower bounds for the blowup time are more important, which can be used to predict the unsteady state of the systems more accurately.
In 2006, Payne and Schaefer [11] used the differential inequality technique to derive the lower bound for the blowup time when Ω ⊂ R 3 . Based on their work, many scholars tend to seek lower bounds for quantity of reaction-diffusion problems when Ω ⊂ R 3 (see [12] and the reference therein) and when Ω ⊂ R N (N ≥ 3) (see [13,14]). In order to expand the underpinning theory of the mathematical analysis of problem, we aim to derive the results that have been extended to more general reaction-diffusion problems when Ω ⊂ R N (N ≥ 2). In addition, aiming to be closer to more realistic models, in this paper, we deal with the following reaction-diffusion problem with nonlocal and gradient terms: where m > 1, Ω ⊂ R N (N ≥ 2) is a bounded and convex domain with smooth boundary, zu/z] denotes the outward normal derivative on zΩ, and t * is the blow-up time if blowup occurs. roughout this paper, we suppose g are nonnegative C(R + ) functions and u 0 are nonnegative C 1 (Ω) functions satisfying compatibility conditions. Problem (1) appears in the mathematical models for gas or fluid flow in porous media (see [15]); it can also be used to describe the evolution of some biological population u (cells, bacteria, etc.), which live in a certain domain and whose growth is influenced by the law + Ω u r dx − |∇u| s . Nonlocal terms + Ω u r dx represent the births of the species, and − |∇u| s represents the accidental deaths of the species. For other related references, readers can refer to [16,17]. In order to achieve our goal, we mainly pay our attention to the following works [18,19]. Marras et al. in [18] investigated the following problem: where Ω ⊂ R N (N ≥ 2) is a bounded and convex domain with smooth boundary. ey obtained A lower bound for the blow-up time when Ω ⊂ R 3 . Song in [19] considered the following reaction-diffusion problem: where Ω ⊂ R 3 is a bounded and convex domain with smooth boundary. e authors obtained A lower bound for the blow-up time when blow-up occurs.
Inspired by the aforementioned research studies, we consider the blow-up phenomena of problem (1). e highlight of this paper is considering both gradient term and nonlocal term sources, which make the problem more closer to the reality. In addition, there is little research on the blowup phenomenon of the solution of problem (1) and even less research on the lower bound for the blow-up time. e main difficulty in studying (1) is to build suitable auxiliary functions. Since auxiliary functions defined in problems (2) and (3) are no longer applicable for our study, it is necessary to construct new auxiliary functions and use Sobolev inequalities to accomplish our research. e paper is organized as follows. In Section 2, when Ω ⊂ R N (N ≥ 3), we obtain a criterion for blow-up of the solution of problem (1) and give a lower bound for blow-up time. In Section 3, when Ω ⊂ R 2 , a lower bound for blow-up time is derived. In Section 4, we present two examples to illustrate the applications of the abstract results obtained in this paper.

Lower Bound for Blow-Up Time
When In this section, our aim is to determine a lower bound for blow-up time t * when Ω ⊂ R N (N ≥ 3). We now assume with constants a > 0, l > 1. Moreover, we suppose constants r > 1, s > 2, and Let us define the following auxiliary function: where It is known from Corollary 9.14 in [20] that Here w ∈ W 1,2 (Ω) and C � C(N, Ω) is a Sobolev embedding constant depending on N and Ω. In this section, we need to use Sobolev inequality (9). e main result is formulated next. Theorem 1. Let u be a nonnegative classical solution of problem (1). Suppose (4)- (7) hold. If the solution u blows up in the measure J(t) at some finite time t * , then t * is bounded by Discrete Dynamics in Nature and Society where |Ω| is the measure of the bounded and convex domain Proof. We compute by using (4), (7), and the divergence theorem It follows from the Holder inequality that Discrete Dynamics in Nature and Society which is equivalent to Inserting (18) into (16), we derive In view of the lemma in [21], we have For the second term on the right side of (20), we apply the Ho .. lder inequality and the Young inequality to get 4 Discrete Dynamics in Nature and Society where ε 1 is given in (15). Substituting (20) and (21) into (19), we derive It follows from (5)- (7), the Ho .. lder inequality, and the Young inequality that Discrete Dynamics in Nature and Society 5

(29)
For the first term on the right side of the (29), we make use of the Ho .. lder inequality and the Young inequality to get Discrete Dynamics in Nature and Society Moreover, it follows from the Ho .. lder inequality and the Young inequality that |Ω|.

(31)
Again making use of the Ho .. lder inequality and the Young inequality to the second term on the right side of (29), we obtain Inserting (31)-(33) into (29), we have Discrete Dynamics in Nature and Society 7 Combining (33) and (26), we derive where A 1 , A 2 are given in (11) and (12). Integrating between 0 and t * , we arrive at

Lower Bound for Blow-Up Time
When Ω ⊂ R 2 In this section, we will give a lower bound for the blow-up time when Ω ⊂ R 2 . Here we still suppose that conditions (4) and (5) hold. Since the embedding theorem in (9) is no longer available when N � 2, before proving our main theorem, we note that the following Sobolev embedding: implies that where C � C(Ω) is the embedding constant depending on Ω. In addition, assume that where Our main results are stated next.

Theorem 2.
Let u be a nonnegative classical solution of problem (1). Assume that (4), (5), and (39) hold. If the solution u blows up in the measure H(t) at some finite time t * , a lower bound for t * is where where |Ω| is the measure of the bounded and convex domain Ω, ρ 0 � min zΩ (x · ]) > 0, and d � max Ω |x|.
Proof. Repeating the calculations in (16)-(25), we have Again using inequality (28), we rewrite (38) as For the first term on the right side of inequality (48), we make use of the Ho .. lder inequality and the Young inequality where ε 2 is given in (45). It follows from (48)-(51) that