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

Shen, Xuhui; Wu, Dandan

doi:https://doi.org/10.1155/2022/8364982

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2022 | Article ID 8364982 | https://doi.org/10.1155/2022/8364982

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

Xuhui Shen¹and Dandan Wu²

Academic Editor: Genni Fragnelli

Received29 Oct 2021

Accepted12 Jan 2022

Published19 Apr 2022

Abstract

This paper considers the blow-up phenomena for the following reaction-diffusion problem with nonlocal and gradient terms: Here , and 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 obtained in this paper.

1. 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 [1–4]) 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 blow-up 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 [7–10]). For the practical situations, lower bounds for the blow-up 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 blow-up time when . Based on their work, many scholars tend to seek lower bounds for quantity of reaction-diffusion problems when (see [12] and the reference therein) and when (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 . 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 , is a bounded and convex domain with smooth boundary, denotes the outward normal derivative on , and is the blow-up time if blow-up occurs. Throughout this paper, we suppose are nonnegative functions and are nonnegative 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 . Nonlocal terms represent the births of the species, and 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 is a bounded and convex domain with smooth boundary. They obtained A lower bound for the blow-up time when . Song in [19] considered the following reaction-diffusion problem:where is a bounded and convex domain with smooth boundary. The 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). The 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 blow-up phenomenon of the solution of problem (1) and even less research on the lower bound for the blow-up time. The 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.

The paper is organized as follows. In Section 2, when , 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 , 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.

2. Lower Bound for Blow-Up Time When

In this section, our aim is to determine a lower bound for blow-up time when . We now assumewith constants . Moreover, we suppose constants , , and

Let us define the following auxiliary function:where

It is known from Corollary 9.14 in [20] thatwhich implies

Here and is a Sobolev embedding constant depending on and . In this section, we need to use Sobolev inequality (9). The main result is formulated next.

Theorem 1. Let be a nonnegative classical solution of problem (1). Suppose (4)–(7) hold. If the solution blows up in the measure at some finite time , then is bounded bywherewhere is the measure of the bounded and convex domain , , and .

Proof. We compute by using (4), (7), and the divergence theoremIt follows from the Holder inequality thatwhich is equivalent toInserting (18) into (16), we deriveIn view of the lemma in [21], we haveFor the second term on the right side of (20), we apply the Hlder inequality and the Young inequality to getwhere is given in (15). Substituting (20) and (21) into (19), we deriveIt follows from (5)–(7), the Hlder inequality, and the Young inequality thatSubstituting (23)–(25) into (22) yieldswhere are defined in (13) and (14), respectively. Applying (7) and the Sobolev inequality (9), we haveUsing the inequalitywe rewrite inequality (27) asFor the first term on the right side of the (29), we make use of the Hlder inequality and the Young inequality to getMoreover, it follows from the Hlder inequality and the Young inequality thatAgain making use of the Hlder inequality and the Young inequality to the second term on the right side of (29), we obtainInserting (31)–(33) into (29), we haveCombining (33) and (26), we derivewhere are given in (11) and (12). Integrating between 0 and , we arrive at

3. Lower Bound for Blow-Up Time When

In this section, we will give a lower bound for the blow-up time when . Here we still suppose that conditions (4) and (5) hold. Since the embedding theorem in (9) is no longer available when , before proving our main theorem, we note that the following Sobolev embedding:implies thatwhere is the embedding constant depending on . In addition, assume thatwhere

Our main results are stated next.

Theorem 2. Let be a nonnegative classical solution of problem (1). Assume that (4), (5), and (39) hold. If the solution blows up in the measure at some finite time , a lower bound for iswherewhere is the measure of the bounded and convex domain , , and .

Proof. Repeating the calculations in (16)–(25), we havewhere are defined in (43) and (44). Now, we pay our attention to the term . It follows from the Hlder inequality and Sobolev inequality (37) thatAgain using inequality (28), we rewrite (38) asFor the first term on the right side of inequality (48), we make use of the Hlder inequality and the Young inequality to obtainMoreover,Similar to the computation process in (50), we getwhere is given in (45). It follows from (48)–(51) thatSubstituting (52) into (46), we havewhere are defined in (41) and (42). Integrating (53) between 0 and , we deduce that

4. Applications

In this section, two examples are given to illustrate the abstract results of Theorems 1 and 2.

Example 1. Let be the nonnegative classical solution of the following problem:where is a ball in . Here we choose , and . It is easy to check that conditions (4)–(6) hold. Moreover, from Theorems 2.1 and 3.2 in [22], we have . By inserting the above constants into (12)–(16), we get . It follows from Theorem 1 thatwith

Example 2. Let be the nonnegative classical solution of following problem:Here is a circular domain in . Select , , , and . Moreover, combining (41)–(45), we can compute . It is easy to verify that (4), (5), and (39) hold. According Theorem 2, we have the lower bound for the blow-up time:where

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 study was supported by the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (no. 2020L025) and the Youth Natural Science Foundation of Shanxi Province (no. 20210302124533).

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Copyright

Copyright © 2022 Xuhui Shen and Dandan Wu. 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.

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