Blow-Up Solutions and Global Existence for Quasilinear Parabolic Problems with Robin Boundary Conditions

Abstract

We study the blow-up and global solutions for a class of quasilinear parabolic problems with Robin boundary conditions. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of blow-up solution, an upper bound for the “blow-up time,” an upper estimate of the “blow-up rate,” the sufficient conditions for the existence of global solution, and an upper estimate of the global solution are specified.

1. Introduction

In this paper, we are going to investigate the blow-up and global solutions of the following quasilinear parabolic problem with Robin boundary conditions: where () is a bounded domain with smooth boundary , represents the outward normal derivative on , is a 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 function, for any , is a positive function, is a positive function, is a positive function, is a positive function, and is a positive function. Under the above assumptions, the classical parabolic equation theory [1] assures that there exists a unique classical solution with some for problem (1) and the solution is positive over . Moreover, by regularity theorem [2], .

Many authors have studied the blow-up and global solutions of nonlinear parabolic problems (see, for instance, [3–14]). Some special cases of the problem (1) have been treated already. Enache [15] investigated the following problem: where () is a bounded domain with smooth boundary . Some conditions on nonlinearities and the initial data were established to guarantee that is global existence or blows up at some finite . In addition, an upper bound and a lower bound for were derived. Zhang [16] dealt with the following problem: where () is a bounded domain with smooth boundary . By constructing auxiliary functions and using maximum principles, the sufficient conditions characterized by functions , and were given for the existence of blow-up solution. Ding [17] considered the following problem: where () is a bounded domain with smooth boundary . By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, the sufficient conditions were specified for the existence of blow-up and global solutions. For the blow-up solution, a lower bound on blow-up time is also obtained. Some authors also discussed blow-up phenomena for parabolic problems with Robin boundary conditions and obtained a lot of interesting results (see [18–24] and the references cited therein).

As everyone knows, parabolic equation describes the process of heat conduction. Blow-up and global solutions for parabolic equations reflect the unsteady state and steady state of heat conduction process, respectively. In the problems (2) and (4), the heat conduction coefficient depends only on the temperature variable . In the problem (3), the heat conduction coefficient depends on the temperature variable and space variable . However, in a lot of processes of heat conduction, heat conduction coefficient depends not only on the temperature variable but also on the space variable and the time variable . Therefore, in this paper, we study the problem (1). It seems that the method of [15–17] is not applicable for the problem (1). In this paper, by constructing completely different auxiliary functions with those in [15–17] and technically using maximum principles, we obtain some existence theorems of blow-up solution, an upper bound of “blow-up time,” an upper estimates of “blow-up rate,” the existence theorems of global solution, and an upper estimate of the global solution. Our results extend and supplement those obtained in [15–17].

We proceed as follows. In Section 2, we study the blow-up solution of (1). Section 3 is devoted to the global solution of (1). A few examples are presented in Section 4 to illustrate the applications of the abstract results.

2. Blow-Up Solution

The main results for the blow-up solution are Theorems 1–3. For simplicity, we define the constant In Theorems 1–3, the three cases , and are considered, respectively. In the first case, , we have the following conclusions.

Theorem 1. Let be a solution of the problem (1). Suppose the following.(i)Consider  (ii)For ,  (iii)For , (iv)Consider Then, the solution of the problem (1) must blow up in a finite time , and where and and are the inverse functions of and , respectively.

Proof. In order to discuss the blow-up solution by using maximum principles, we construct an auxiliary function from which we have By (14) and (15), we have It follows from (1) that Next, we substitute (17) into (16) to obtain With (13), it has Substitute (19) into (18) to get In view of (12), we have Substituting (21) into (20), we get The assumptions (6) and (7) imply that the right-hand side of (22) is nonpositive; that is, Applying the maximum principle [25], it follows from (23) that can attain its nonpositive minimum only for or . For , (5) implies
We claim that cannot take a negative minimum at any point . Indeed, if take a negative minimum at point , then It follows from (1) and (21) that Next, by using (8) and the fact , it follows from (26) that which contradicts inequality (25). Thus, we know that the minimum of in is zero. Thus, that is, At the point , where , integrate (29) over to get which shows that must blow up in finite time. In fact, suppose is a global solution of (1), then, for any , it follows from (30) that Passing to the limit as in (31) yields which contradicts assumption (9). This shows that must blow up in a finite time . Furthermore, letting in (30), we have that is, which implies that By integrating inequality (29) over , for each fixed , one gets Hence, by letting , we obtain Since is a decreasing function, we have The proof is complete.

In the second case, , the following two assumptions and can guarantee that inequality (23) holds.Consider   For ,    Hence, by repeating the proof of Theorem 1, we have the following results.

Theorem 2. Let be a solution of the problem (1). Suppose that and hold and assumptions (iii) and (iv) of Theorem 1 hold. Then, the conclusions of Theorem 1 are valid.

In the third case, , the following two assumptions and imply that inequality (23) holds.  Consider   For ,

Theorem 3. Let be a solution of the problem (1). Suppose that and hold and assumptions (iii) and (iv) of Theorem 1 hold. Then, the results stated in Theorem 1 still hold.

Remark 4. When (9) implies that When (9) implies that

3. Global Solution

We define the constant The following Theorems 5–7 are the main results for the global solution. In Theorems 5–7, we study the three cases , and , respectively. In the first case, , we have the following results.

Theorem 5. Let be a solution of the problem (1). Suppose the following.(i)Consider (ii)For , (iii)For , (iv)Consider Then, the solution to the problem (1) must be a global solution and where and is the inverse function of .

Proof. In order to study the global solution by using maximum principles, we construct an auxiliary function Substituting and with and in (22), respectively, gives Assumptions (48) and (49) guarantee that the right side in equality (55) is nonnegative; that is, It follows from (47) that Replacing and with and in (26), respectively, we have Combining (56)–(58) with (50) and applying the maximum principles again, it follows that the maximum of in is zero. Thus, For each fixed , integration of (60) from to yields which implies that must be a global solution. Actually, if blows up at a finite time , then Letting in (61), we have which contradicts with assumption (51). This shows that is global. Moreover, it follows from (61) that Since is an increasing function, we have The proof is complete.

In the second case and the third case , we have the following results.

Theorem 6. Let be a solution of the problem (1). Suppose that assumptions and hold.  Consider   For , And assumptions (iii) and (iv) of Theorem 5 hold. Then, the results of Theorem 5 are valid.

Theorem 7. Let be a solution of the problem (1). Suppose that assumptions and hold.Consider For , And assumptions (iii) and (iv) of Theorem 5 hold. Then, the conclusions stated in Theorem 5 still hold.

Remark 8. When (51) implies that When (51) implies that or

4. Applications

When and or and or and , the conclusions of Theorems 1–3 and 5–7 still hold true. In this sense, our results extend and supplement the results of [15–17].

In what follows, we present several examples to demonstrate the applications of the abstract results.

Example 9. Let be a solution of the following problem: where is a bounded domain with smooth boundary , . Here,
Assume that and one of the following three assumptions holds.(i)In the case or .(ii)In the case or .(iii)In the case .

It follows from Theorems 1–3 that blows up in a finite time , and Assume that and one of the following three assumptions holds.(i)In the case or .(ii)In the case .(iii)In the case .

By Theorems 5–7, must be a global solution and

Example 10. 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 Then, and It is easy to check that (6)-(9) hold. By Theorem 1, must blow up in a finite time , and

Example 11. Let be a solution of the following problem: where is the unit ball of . Now,

By setting we have and It is easy to check that (68)-(69) and (50)-(51) hold. By Theorem 7, must be a global solution, and

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 61074048 and 61174082), the Research Project Supported by Shanxi Scholarship Council of China (Nos. 2011-011 and 2012-011), and the Higher School “131" Leading Talent Project of Shanxi Province.