Abstract and Applied Analysis

Volume 2014 (2014), Article ID 241650, 9 pages

http://dx.doi.org/10.1155/2014/241650

Research Article

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

Lingling Zhang and Hui Wang

Department of Mathematics, Taiyuan University of Technology, Shanxi 030024, China

Received 26 June 2014; Accepted 28 July 2014; Published 14 August 2014

Academic Editor: Sanling Yuan

Copyright © 2014 Lingling Zhang and Hui Wang. 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.

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).

References

A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475–504, 1998.

J. M. Chadam, A. Peirce, and H. Yin, “The blowup property of solutions to some diffusion equations with localized nonlinear reactions,” Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 313–328, 1992.

J. Wang, Z. J. Wang, and J. X. Yin, “A class of degenerate diffusion equations with mixed boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 589–603, 2004.

N. Wolanski, “Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary,” SIAM Journal on Mathematical Analysis, vol. 24, no. 2, pp. 317–326, 1993.

J. Ding, “Blow-up of solutions for a class of semilinear reaction diffusion equations with mixed boundary conditions,” Applied Mathematics Letters, vol. 15, no. 2, pp. 159–162, 2002.

L. Zhang, “Blow-up of solutions for a class of nonlinear parabolic equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 479–486, 2006.

L. A. Caffarelli and A. Friedman, “Blowup of solutions of nonlinear heat equations,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 409–419, 1988.

V. A. Galaktionov and J. L. Vázquez, “The problem of blow-up in nonlinear parabolic equations,” Discrete and Continuous Dynamical Systems, vol. 8, no. 2, pp. 399–433, 2002.

H. Zhang and X. Guo, “Blow-up for nonlinear heat equations with absorptions,” European Journal of Pure and Applied Mathematics, vol. 1, no. 3, pp. 33–39, 2008.

A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-Up in Problems for Quasilinear Parabolic Equations, Nauka, Moscow, Russia, 1987, (Russian), English Translation: Walter de Gruyter, Berlin, Germany, 1995.

L. E. Payne and P. W. Schaefer, “Lower bounds for blow-up time in parabolic problems under Neumann conditions,” Applicable Analysis, vol. 85, no. 10, pp. 1301–1311, 2006.

H. Amann, “Quasilinear parabolic systems under nonlinear boundary conditions,” Archive for Rational Mechanics and Analysis, vol. 92, no. 2, pp. 153–192, 1986.

J. Ding and S. Li, “Blow-up solutions for a class of nonlinear parabolic equations with mixed boundary conditions,” Journal of Systems Science and Complexity, vol. 18, no. 2, pp. 265–276, 2005.

J. Ding, “Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions,” Computers & Mathematics with Applications, vol. 65, no. 11, pp. 1808–1822, 2013.

C. Enache, “Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition,” Applied Mathematics Letters, vol. 24, no. 3, pp. 288–292, 2011.

H. Zhang, “Blow-up solutions and global solutions for nonlinear parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4567–4574, 2008.

R. P. Sperb, Maximum Principles and Their Applications, vol. 157 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1981.