Discrete Dynamics in Nature and Society
Volume 2020 (2020), Article ID 7480676, 7 pages
https://doi.org/10.1155/2020/7480676

Research Article

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

Baiping Ouyang,¹ Wei Fan,² and Yiwu Lin³

¹College of Data Science, Huashang College Guangdong University of Finance & Economics, Guangzhou 511300, China
²The School of Gifted Young, University of Science and Technology of China, Hefei, China
³Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521, China

Correspondence should be addressed to Yiwu Lin; 26-062@gduf.edu.cn

Received 6 March 2020; Accepted 12 May 2020; Published 5 June 2020

Academic Editor: Rodica Luca

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.

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 [3–12]. There are also some works under nonlinear boundary conditions [13–15]. 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 [16–18]). Recently, some scholars have started to investigate the blow-up problems with time-dependent coefficients [19–21]. 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 satisfy

where 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 Hölder 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 is

where , , and are positive constants, and satisfies .

2. Lower Bound for the Blow-Up Time

In this part, we define an auxiliary function of the form

where , , and .

We establish the following theorem:

Theorem 1. Let be the weak solution of problems (1)–(3) in a bounded convex domain . Then, the quantity defined in (8) satisfies the integral inequality

which follows that the blow-up time is bounded below. We have

where , , , , and will be defined later.

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

where .

For the second term on the right side of (11), we apply the divergence theorem, the trace embedding, and (3) to get

where , is the outward normal vector of and .

For the second term on the right side of (12), using (11), we obtain

where is a positive constant which will be defined later. Using (4), we have

where .

Choosing and and using (4), (5), and (7), we have

where and is a positive constant which will be defined later.

For the first term on the right side of (15), using (4) and Young’s inequality, we have

where , , and is a positive constant which will be defined later.

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

where , , and .

Combining (12), (13), (14), and (17), we have

where 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 obtain

where is a positive constant which will be defined later. Similarly, we have

where and .

Choosing and and using (4), (6), and (7), we have

where and is a positive constant which will be defined later.

For the first term on the right side of (22), using (4), we have

where , , and is a positive constant which will be defined later.

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

where .

Combining (19)–(21) and (24), we have

where and

For the third term on the right side of (11), using Hölder inequality and Young’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 have

where , , and .

For the first term on the right side of (27), using (4) and Young’s inequality, we get

where , , and is a positive constant which will be defined later.

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

where , , 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 have

where , , , and is a positive constant which will be defined later.

For the first term on the right side of (32), using (4), we get

where , , and is a positive constant which will be defined later.

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

where , , and .

Combining (31) and (34), we obtain

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

where , , , and

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

where , , , and .

Let

where .

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 get

where 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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