Research Article | Open Access

Baiping Ouyang, Wei Fan, Yiwu Lin, "Lower Bound for the Blow-Up Time for the Nonlinear Reaction-Diffusion System in High Dimensions", *Discrete Dynamics in Nature and Society*, vol. 2020, Article ID 7480676, 7 pages, 2020. https://doi.org/10.1155/2020/7480676

# Lower Bound for the Blow-Up Time for the Nonlinear Reaction-Diffusion System in High Dimensions

**Academic Editor:**Rodica Luca

#### Abstract

In this paper, we study the blow-up phenomenon for a nonlinear reaction-diffusion system with time-dependent coefficients under nonlinear boundary conditions. Using the technique of a first-order differential inequality and the Sobolev inequalities, we can get the energy expression which satisfies the differential inequality. The lower bound for the blow-up time could be obtained if blow-up does really occur in high dimensions.

#### 1. Introduction

During the past decades, the blow-up phenomena for the solutions to the parabolic problems have been widely concerned. It is important in practice that how to determine the bound of the blow-up time of the solutions about the parabolic equations and systems. Their applications are included in physics, chemistry, astronomy, biology, and population dynamics [1, 2]. Actually, when the blow-up occurs at , it is difficult to get the exact value of . We mainly focus on estimating its bounds. At present, the studies on the blow-up phenomena of parabolic problems mainly focus on homogeneous Dirichlet boundary condition and homogeneous Neumann and Robin boundary conditions [3â€“12]. There are also some works under nonlinear boundary conditions [13â€“15]. Most of these articles are focused on . There are only a few papers dealing with a lower bound for the blow-up time in high dimensions (see [16â€“18]). Recently, some scholars have started to investigate the blow-up problems with time-dependent coefficients [19â€“21]. In paper [21], the authors considered the following nonlinear reaction-diffusion system with time-dependent coefficients:

The authors obtained the lower and upper bounds for the blow-up time when the blow-up occurred. In this paper, we further consider the blow-up phenomena for the following system with time-dependent coefficients under nonlinear boundary conditions in high dimensions:

We assume that are continuous, and , and satisfywhere is a positive constant to be defined later.

Our goal in this paper is to obtain a lower bound for the blow-up time of the solutions to systems (2) and (3) in for any . The nonlinear terms and and the boundary conditions are difficult to tackle. We cannot get the result by following the method proposed in [21], so we must use a new method to overcome these difficulties. To the best of our knowledge, no results exist in that direction, and we think our result is new and interesting.

In the further discussions, we will use the following HĂ¶lder inequality:where is a nonnegative function and , , , and are positive constants satisfying .

We also need the following Sobolev inequality [22]:with which is a Sobolev embedding constant depending on and .

And the classical (or elementary) inequality iswhere , , and are positive constants, and satisfies .

#### 2. Lower Bound for the Blow-Up Time

In this part, we define an auxiliary function of the formwhere , , and .

We establish the following theorem:

Theorem 1. *Let be the weak solution of problems (1)â€“(3) in a bounded convex domain . Then, the quantity defined in (8) satisfies the integral inequalitywhich follows that the blow-up time is bounded below. We havewhere , , , , and will be defined later.*

Now, we prove Theorem 1. For simplicity, assume that the solution is classical of problems (1)â€“(3). The general case can be done by approximation. Differentiating , we havewhere .

For the second term on the right side of (11), we apply the divergence theorem, the trace embedding, and (3) to getwhere , is the outward normal vector of and .

For the second term on the right side of (12), using (11), we obtainwhere is a positive constant which will be defined later. Using (4), we havewhere .

Choosing and and using (4), (5), and (7), we havewhere and is a positive constant which will be defined later.

For the first term on the right side of (15), using (4) and Youngâ€™s inequality, we havewhere , , and is a positive constant which will be defined later.

Combining (15) and (16), if we choose suitable such that , we havewhere , , and .

Combining (12), (13), (14), and (17), we havewhere and .

Similarly, for the fourth term on the right side of (11), using the divergence theorem and (3), we have

For the second term on the right side of (19), using (4), we obtainwhere is a positive constant which will be defined later. Similarly, we havewhere and .

Choosing and and using (4), (6), and (7), we havewhere and is a positive constant which will be defined later.

For the first term on the right side of (22), using (4), we havewhere , , and is a positive constant which will be defined later.

Combining (22) and (23), if we choose suitable such that , we havewhere .

Combining (19)â€“(21) and (24), we havewhere and

For the third term on the right side of (11), using HĂ¶lder inequality and Youngâ€™s inequality, we have

For the first term on the right side of (26), using (4), (5), and (7) and taking care of the given condition , we havewhere , , and .

For the first term on the right side of (27), using (4) and Youngâ€™s inequality, we getwhere , , and is a positive constant which will be defined later.

Combining (27) and (28), if we choose suitable such that , we havewhere , , and .

Combining (26) and (29), we obtain

By the same way, for the fifth term on the right side of (11), using (4), we have

For the first term on the right side of (31), using (4), (6), and (7) and taking care of the given condition , we havewhere , , , and is a positive constant which will be defined later.

For the first term on the right side of (32), using (4), we getwhere , , and is a positive constant which will be defined later.

Combining (32) and (33), if we choose suitable such that , we havewhere , , and .

Combining (31) and (34), we obtain

Combining (11), (18), (25), (30), and (35), we havewhere , , , and

If we choose suitable , , , and such that , we can rewrite (36) aswhere , , , and .

Letwhere .

Integrating (37) from 0 to , we have

Considering , the integration of the right side of (39) exists. It is clear that is an increasing function. So, we can getwhere is the inverse function of .

The proof of Theorem 1 is complete.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61907010), the Foundation for Natural Science in Higher Education of Guangdong, China (Grant no. 2018KZDXM048), the General Project of Science Research of Guangzhou (Grant no. 201707010126), and Huashang College Guangdong University of Finance & Economics (Grant no. 2019HSDS26).

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#### Copyright

Copyright © 2020 Baiping Ouyang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.