Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

Abstract

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: , in , , on , , in , where is a bounded domain with smooth boundary . Under some appropriate assumption on the functions , , , , and and initial value , we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

1. Introduction

The study of global and blow-up solutions for nonlinear parabolic equations has received a lot of attention in the past several decades (see [1–4]). In most works, so far, a variety of approaches have been developed in dealing with different nonlinear parabolic problems, such as the existence of global solution, blow-up solution, an upper bound for “blow-up time,” an upper estimate of “blow-up rate,” or global solution. So far, some applications in physics, chemistry, and biology are relevant to blow-up phenomena which can be found in [5–11]. In this paper, we consider the global and blow-up solutions of the following nonlinear parabolic equation with Robin boundary condition: where ,    is a bounded domain with smooth boundary , represents the outward normal derivative on , is positive constant, is the initial value, is the maximal existence time of , and is the closure of . Set . We assume, throughout the paper, that is a positive function, for any , is a positive function, is a positive function, is a positive function, is a nonnegative function, and is a positive function. Under these assumptions, the classical parabolic equation theory [12] ensures that there exists a unique classical solution with some for the problem (1), and the solution is positive over . Moreover, by the regularity theorem [13], .

The problems of the global and blow-up solutions for nonlinear parabolic equations have been investigated extensively by many authors and have got a lot of meaningful results. Some special cases of problem (1) have been treated already. Ding [14] deals with the following problem: where is a bounded domain of    with smooth boundary . By constructing auxiliary functions and using a first-order differential inequality technique, Ding derives conditions on the data, which guarantee the existence of blow-up or global solution. The following problem is investigated by Enache in [15]: where is a bounded domain of    with smooth boundary . By constructing auxiliary functions and first-order differential inequality technique, Enache establishes some conditions on nonlinearities and the initial date to guarantee that exists for all times or blows up at some finite time . Besides, the following problem is investigated by Zhang in [16]: where is a bounded domain in    with smooth boundary. Under appropriate assumptions on the functions , , and , Zhang obtains the conditions under which the solutions may exist globally or blow up in a finite time. Moreover, upper estimates of the “blow-up time,” blow-up rate, and global solutions are obtained also.

In this paper, we obtain the existence theorem of global and blow-up solution by constructing completely different auxiliary functions and technically using maximum principles. As a result, the sufficient conditions for the existence of a global solution and an upper estimate of the global solution and the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate” are specified under some appropriate assumption on the functions , , , , and and initial value . Our results extend and supplement those obtained in [14–16].

The content of this paper is organized as follows. In Section 2, we study the existence of the global solution of (1). In Section 3, we investigate the blow-up solution of (1). In Section 4, we will give a few examples to explain our results.

2. Global Solution

Our main result for the global solution is the following Theorem 1.

Theorem 1. Let be a solution of (1). Suppose that the following conditions are satisfied.(i)For any , (ii)For any , (iii)Consider the integration (iv)Consider
Then the solution to problem (1) must be a global solution and where and is the inverse function of .

Proof. Consider the auxiliary function Then, we have By (1), We have Then By (13) and (16), it follows that By (14), we have Substitute (18) into (17) to get By (12), we have Next, we substitute (20) into (19) to obtain So we have According to (11), we have Substituting (23) into (22), we have Namely, The assumptions (5) and (6) guarantee that the right-hand side of (25) is nonnegative; that is, By applying maximum principle (see [17]), it follows from (26) that can attain its nonnegative maximum only for or .
For , by (8), we have
For , we claim that cannot take a positive maximum at any point . In fact, suppose that can take a positive maximum at one point ; then Combine (1) and (11) with (23); we have Next, by using a part condition of (5) ,   for any , we can obtain which contradicts with inequality (28). Thus, we know that the maximum of in is zero; that is, With (11), we know For each fixed , we integrate (32) from to : which implies that must be a global solution of (1). In fact, suppose that blows up at finite time ; then Passing to the limit as in (33) yields which contradicts with the condition (iii). This shows that is global solution. Moreover, it follows from (33) that Since is an increasing function, we have The proof is completed.

3. Blow-Up Solution

The following theorem is the main result for the blow-up solution of (1).

Theorem 2. Let be a solution of problem (1). Assume that the following conditions are satisfied.(i)For any , (ii)For any , (iii)Consider the integration (iv)Consider
Then the solution of problem (1) must blow up in finite time , and where and is the inverse function of .

Proof. Construct the following auxiliary function: So we have As the previous derivation from (14) to (25), we can obtain It is seen from (38) and (39) that the right-hand side of (46) is nonpositive; that is, By applying maximum principle (see [17]), it follows from (47) that can attain its nonpositive minimum only for or .
For , with (41), we have
For , substituting and with and in (29), respectively, we have Combining (47)–(49) with condition (i), we can apply the maximum principles again to obtain that the minimum of in is zero. Thus, At the point , where , we can integrate (51) from to to get which implies that must blow up in finite time. Actually, if is a global solution of (1), then, for any , (52) shows Letting in (53), we have which contradicts with assumption (40). This shows that must blow up in finite time . Moreover, letting in (52), we have By integrating inequality (51) over , for each fixed , we obtain Hence, by letting , we have Since is a decreasing function, we obtain The proof is completed.

4. Applications

When , , or , , , , or , , , , the conclusions of Theorems 1 and 2 still hold true. In this sense, our results extend and supplement the results in [14–16]. In what follows, we present several examples to demonstrate the applications of the abstract results.

Example 1. Let be a solution of the following problem: where , is the unit ball of . Now we have In order to determine the constant , we assume and then and It is easy to check that (5)–(7) hold. By Theorem 1, must be a global solution, and

Example 2. Let be a solution of the following problem: where , is the unit ball of . Now we have In order to determine the constant , we assume and then and It is easy to check that (38)–(40) hold. By Theorem 2, must blow up in finite time , and where and is the inverse function of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors especially thank the reviewers for their very useful suggestions. The research was partially supported by the National Natural Science Foundation of China (no. 61250011) and the Science Foundation of Shanxi Province (no. 2012011004-4) and Postdoctoral Science Foundation (no. 2012M510786).