Blow-Up Analysis for a Reaction-Diffusion Model with Nonlocal and Gradient Terms

In this paper, we investigate the blow-up phenomena for the following reaction-diﬀusion model with nonlocal and gradient terms: ≥ 0 in Ω ⎧⎪⎨⎪⎩ . Here Ω ⊂ R N ( N ≥ 3 ) is a bounded and convex domain with smooth boundary, and constants m, p, q, α are supposed to be positive. Utilizing the Sobolev inequality and the diﬀerential inequality technique, lower bound for blow-up time is derived when blow-up occurs. In addition, we give an example as application to illustrate the abstract results


Introduction
As we all know, reaction-diffusion models can be used to illustrate many natural phenomena such as heat flow, combustion, and gravitational potentials, so they have received extensive attention from many scholars [1,2]. Since early sixties, lots of papers concerning the problem of blow-up or global existence of solutions to reaction-diffusion models have been published. After that, qualitative properties of reactiondiffusion models were investigated, such as the blow-up set, blow-up rate, blow-up profile, and boundedness of global solutions (see the papers [3][4][5][6] and books [7,8]).
Especially, when the solution of reaction-diffusion models blows up, one would like to know in what time the solution blows up. Weissler in [9] firstly studied the blow-up time of reaction-diffusion models, and then much attention has been paid to finding the blow-up time of solutions. However, much work has been done in deriving the upper bound for the blow-up time [10]. In practical situations, lower bound for the blow-up time is more useful than upper bound for the blowup time in predicting the critical state of the systems. is makes the study of the lower bound for the blow-up time more meaningful. In 2006, Payne and Schaefer introduced the first differential inequality technique to give the lower bound for the blow-up time (see [11]). Based on Payne's methods, lots of works are devoted to giving the lower bound for the blow-up time when blow-up occurs (refer to [12][13][14][15][16][17]).
In this paper, we investigate the blow-up time of the following reaction-diffusion model with nonlocal and gradient terms: where Ω ⊂ R N (N ≥ 3) is a bounded and convex domain with smooth boundary, m, p, q, α are positive constants, and t * is the maximal existence time of the solution u. Also, we assume that h is nonnegative C 1 (R + ) function, where R + � (0, +∞). u 0 (x) is assumed to be nonnegative C 1 (R + ) function, which is compatible with the boundary conditions. Problem (1) can describe many physical phenomena and biological species theories. For example, in the density of some biological species for population dynamics, nonlocal source term represents the births of the species, and gradient terms can illustrate the natural or the accidental deaths.
To complete our research, we focus our attention on the following blow-up phenomena of the reaction-diffusion models (see [18][19][20][21]). Marras, et al. in [20] studied where Ω ⊂ R N (N ≥ 1) is a bounded and convex domain with smooth boundary. When p > q, they determined lower bounds for t * when blow-up occurs in Ω ⊂ R 2 and Ω ⊂ R 3 .
In addition, when p < q, a global existence criterion for u was derived on Ω ⊂ R N (N ≥ 1). Ding and Shen in [21] investigated the following nonlocal reaction-diffusion model: where Ω ⊂ R N (N ≥ 2) is a bounded and convex domain whose boundary is sufficiently smooth. Assuming that they obtained a lower bound for the blow-up time when Ω ⊂ R 3 . Moreover, an upper bound for the blow-up time and global solution were also discussed. Inspired by the research studies mentioned above, we deal with the blow-up phenomena of problem (1). e highlight of this paper is to investigate the model with both nonlocal terms and gradient terms, making the research closer to reality. In addition, there is little research on the blow-up phenomenon of the solution of problem (1) and even less research on the lower bound for the blow-up time.
e key to achieving our work is to build suitable auxiliary functions. Since auxiliary functions given in problems (2) and (3) are no longer applicable, we need to establish new auxiliary functions and use the Sobolev inequality to accomplish our research. e paper is organized as follows. In Section 2, when Ω ⊂ R N (N ≥ 3), we derive a lower bound for the blow-up time when blow-up occurs. In Section 3, an example is given to illustrate the application of the abstract results obtained in this paper.

Lower Bound for Blow-Up Time
When In this section, we seek the lower bound for the blow-up time when Ω ⊂ R N (N ≥ 3). For this aim, we firstly assume with constants b > 0, β > 1. Moreover, we suppose constants m > 0, p > 1, q > 2. Let the auxiliary function be defined as follows: where Owing to Corollary 9.14 in [22], we know that Here w ∈ W 1,2 (Ω) and C � C(N, Ω) is a Sobolev embedding constant depending on N and Ω. Inequality (9) will be used in our proof. e main results are stated next. Theorem 1. Let u be a nonnegative classical solution of problem (1). Suppose (5)-(7) hold. If the solution u blows up in the measure A(t) at some finite time t * , then t * is bounded by where , where |Ω| is the measure of the bounded and convex domain Proof. Using (5)-(7) and the divergence theorem, we have It follows from the Hölder inequality that Mathematical Problems in Engineering 3 which is equivalent to Inserting (18) into (16), we derive Recalling inequality (2) in [20], we have We apply the Hölder inequality and the Young inequality to the term where ε 1 is given in (15). e substitution of (21) into (20) yields We insert (22) into (19) to get From (7), the Hölder inequality, and the Young inequality, we can deduce that (25) Combining (24) and (25) with (23), we obtain where B 1 and B 2 are defined in (13) and (14), respectively. Applying (7) and Sobolev inequality (9), we have Mathematical Problems in Engineering 5 anks to the basic inequality we rewrite (27) and (28) as It follows from the Hölder inequality and the Young inequality that 6 Mathematical Problems in Engineering where ε 2 , ε 3 are given in (15). Moreover, combining (33) and (31) and (32) and (30), respectively, we have Inserting (34) and (35) into (26), we obtain where D, D 1 , . . . , D 4 are defined in (11) and (12). Integrating (36) between 0 and t * , we arrive at Mathematical Problems in Engineering

Application
In what follows, an example is given as application to illustrate the abstract results in eorem 1.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.