Journal of Function Spaces
Volume 2016 (2016), Article ID 8107657, 7 pages
http://dx.doi.org/10.1155/2016/8107657

Research Article

Blow-Up Phenomena for Nonlinear Reaction-Diffusion Equations under Nonlinear Boundary Conditions

Juntang Ding

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

Received 20 November 2015; Accepted 6 March 2016

Academic Editor: Leszek Olszowy

Copyright © 2016 Juntang Ding. 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.

This paper deals with blow-up and global solutions of the following nonlinear reaction-diffusion equations under nonlinear boundary conditions: where is a bounded domain with smooth boundary . We obtain the conditions under which the solutions either exist globally or blow up in a finite time by constructing auxiliary functions and using maximum principles. Moreover, the upper estimates of the “blow-up time,” the “blow-up rate,” and the global solutions are also given.

1. Introduction

During the past few decades, the blow-up phenomena for the nonlinear reaction-diffusion equations have been studied by a large number of authors, and the reader is referred to [1–8] and the references therein. In this paper, we consider the following nonlinear reaction-diffusion problem under nonlinear conditions:

where is a bounded domain with smooth boundary , represents the outward normal derivative on , and is the maximal existence time of . Set . We assume, throughout the paper, that is a positive function, is a function, for any , is a nonnegative function, is a nonnegative function, and is a positive function and satisfies the compatibility conditions. Under the above assumptions, the local existence and uniqueness of classical solution of problem (1) were established by Amann [9]. Furthermore, it follows from maximum principle [10] and regularity theorem [11] that the solution is positive and .

Many authors have investigated blow-up and global solutions of nonlinear reaction-diffusion equations under nonlinear boundary conditions and have obtained a lot of interesting results (see, e.g., [12–20]). To my knowledge, some special cases of (1) have been studied. Zhang [21] considered the following problem:

where is a bounded domain with smooth boundary . By constructing auxiliary functions and using maximum principles, the existence of blow-up and global solutions were obtained under appropriate assumptions on the functions , and . Zhang et al. [22] dealt with the following problem:

where is a bounded domain with smooth boundary . Some conditions on nonlinearities and the initial data were given to ensure that exists globally or blows up at some finite time . In addition, the upper estimates of the global solution, the “blow-up time,” and the “blow-up rate” were also established.

In this paper, we study reaction-diffusion problem (1). It is well known that , , , and are nonlinear reaction, nonlinear diffusion, nonlinear convection, and nonlinear boundary flux, respectively. What interactions among the four nonlinear mechanisms result in the blow-up and global solutions of (1) is investigated in this work. We note that the boundary flux function depends not only on the concentration variable but also on the space variable and the time variable . Hence, it seems that the methods of [21, 22] are not applicable for problem (1). In this paper, by constructing completely different auxiliary functions from those in [21, 22] and technically using maximum principles, we obtain the existence theorems of the blow-up and global solution. Moreover, the upper estimates of “blow-up time,” “blow-up rate,” and global solution are also given. Our results can be seen as the extension and supplement of those obtained in [21, 22].

The present work is organized as follows. In Section 2, we deal with the blow-up solution of (1). Section 3 is devoted to the global solution of (1). As applications of the obtained results, some examples are presented in Section 4.

2. Blow-Up Solution

In this section, we discuss what interactions among the four nonlinear mechanisms of (1) result in the blow-up solution. Our main result in this section is the following theorem.

Theorem 1. Let be a solution of problem (1). Assume that the following conditions (i)–(iv) are satisfied.

(i)For ,

(ii)For ,

(iii)Consider the following:

(iv)Consider the following:

Then blows up in a finite time and

where

and is the inverse function of .

Proof. Introduce an auxiliary function

and then we have

It follows from (12) and (13) that

The first equation of (1) implies

Inserting (15) into (14), we obtain

It follows from (11) that

Substituting (17) into (16), we get

By (10), we have

Now, we insert (19) into (18) to deduce

Assumptions (4) ensure that the right side in equality (20) is nonnegative; that is,

Next, it follows from (1) and (10) that

We note that (6) implies

There, (21)–(23), assumption (5) and the maximum principle [10] imply that the maximum of the function in is zero. In fact, if the function takes a positive maximum at point , then we have

Using assumption (5) and the fact that , it follows from (22) that

which contradicts the second inequality in (24). Hence, the maximum of the function in is zero. Now, we have

that is,

At the point , where , integrating inequality (27) from 0 to , we arrive at

Inequality (28) and assumption (7) imply that blows up in finite time . Now, we let in (28) to deduce

For each fixed , integrating inequality (27) from to , we get

In the above inequality, letting , we obtain

We note that is a strictly decreasing function. Hence,

The proof is complete.

3. Global Solution

In this section, we study what interactions among the four nonlinear mechanisms of (1) result in the global solution of (1). The main results of this section are formulated in the following theorem.

Theorem 2. Let be a solution of problem (1). Assume that the following conditions (i)–(iv) are fulfilled.

(i)For ,

(ii)For ,

(iii)Consider the following:

(iv)Consider the following:

Then must be a global solution and

where

and is the inverse function of .

Proof. We consider an auxiliary function

Using the same reasoning process as that of (11)–(20), we obtain

Assumptions (33) imply that the right side of (40) is nonpositive; that is,

By (1) and (39), we get

Assumption (35) implies

It follows from (41)–(43), (34), and the maximum principle that the minimum of in is zero. Hence, we have the following inequality:

that is

For each fixed , we integrate (45) from 0 to to deduce

Inequality (46) and assumption (36) imply that must be a global solution. Furthermore, it follows from (45) that

Since is a strictly increasing function, we obtain

The proof is complete.

4. Applications

When and , the results of Theorems 1–2 still hold. In this sense, our results extend and supplement those obtained in [21, 22].

In the following, we give a few examples to demonstrate the applications of Theorems 1–2.

Example 3. Let be a solution of the following problem:

where . We note

In order to calculate the constant , we set

and then and

We can check that (4), (5), and (7) hold. It follows from Theorem 1 that blows up in a finite time and

Example 4. Let be a solution of the following problem:

where . Now we have

Setting

we get and

We can also check that (33), (34), and (36) hold. Hence, Theorem 2 implies must be a global solution and

Remark 5. When the functions , and are all exponential functions, Theorems 1 and 2 can be used. When not all of them are exponential functions, Theorems 1 and 2 can be used in some special cases.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61473180).

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