Journal of Applied Mathematics

Volume 2013 (2013), Article ID 952126, 5 pages

http://dx.doi.org/10.1155/2013/952126

Research Article

Blow-Up Phenomena for Porous Medium Equation with Nonlinear Flux on the Boundary

Yan Hu,¹ Jing Li,² and Liangwei Wang¹

¹School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, China

²College of Science, Minzu University of China, Beijing 100081, China

Received 19 July 2013; Accepted 1 November 2013

Academic Editor: Malgorzata Peszynska

Copyright © 2013 Yan Hu et al. 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 investigate the blow-up phenomena for nonnegative solutions of porous medium equation with Neumann boundary conditions. We find that the absorption and the nonlinear flux on the boundary have some competitions in the blow-up phenomena.

1. Introduction

In this paper, we are concerned with the blow-up of solutions of porous medium equations with nonlinear flux on the boundary. Consider

where , the nonnegative initial value , is a bounded region in () with the sufficiently smooth boundary , is the unit normal vector on , is the blow-up time if blow-up occurs, or else .

The blow-up phenomena for the nonnegative solutions of the heat equation with nonlinear sources ( and in (1)) in the whole space was first found by Fujita in 1966, see [1]. He proved the following results:

(a) if , then (1) has no global positive solutions; (b) if , then there exist global positive solutions.

The critical case was proved to belong to the blow-up case in 1970’s by several authors [2–4]. In 1980, Galaktionov and others [5] considered the nonnegative solutions of (1) (with and ) in whole space . They found some results similar to those for the heat equation () as follows

(a) if , then (1) has no global solutions; (b) if , then there exist global positive solutions that decay like .

In [6, 7], Galaktionov, Mochizuki and Suzuki, had also revealed that the critical case belongs to the blow-up case, see also [8, 9].

In 2010, Payne et al. [10] considered a semilinear heat equation with nonlinear boundary condition ( in (1)) and established conditions on nonlinearities sufficient to guarantee that exists for all time as well as conditions on data forcing the solution to blow up at some finite time . When , the blow-up phenomena for the solutions of the porous medium equation with nonlinear flux on the boundary had also been studied by several authors [11, 12]. For other interesting results on the large time behavior on the solutions of the porous medium equation, we refer the reader to papers [13–16].

Inspired by the above papers, we will study the blow-up phenomena for the solutions of the porous medium equation with nonlinear flux on the boundary in higher dimensional space (). In fact, we find that if the absorption is more powerful than the boundary flux, then the solutions of the problem (1)–(3) exist for all time on a bounded star-shaped region. On the other hand, if the boundary flux is more powerful, then the solutions of the problem (1)–(3) blow-up at a finite time. Moreover, we will give the upper-bound estimates for the blow-up time.

The paper is organized as follows. In Section 2, we concentrate our attention on the conditions of the global existence for the solutions of the problem (1)–(3). Section 3 is devoted to the investigation of the blow-up phenomena for the solutions of the problem (1)–(3).

2. Criterion for Global Existence

In this section, we investigate the global solutions of problem (1)–(3). The main result of this section is the following theorem.

Theorem 1. Let be a bounded star-shaped region and assume that satisfy

If and satisfy the following conditions:

where , are nonnegative constants, then the nonnegative solutions of the problem (1)–(3) do not blow up.

Proof. Let

Differentiating (7) and making use of (1), we obtain that

From the hypothesis (5), we get

By (2), (6) and the divergence theorem, we have

Here we used the identity . By the divergence theorem again, we get

Let

Point out that is positive because is star-shaped by hypothesis. Notice also that

We thus have

On the another hand

Therefore, from (10)–(15), we have

We obtain from the Young inequality that

where

This leads to

Combining this with (16), we get

Let

Therefore, the hypotheses that and imply that

So, by Hölder's inequality, we have

For , we obtain from (23) that

Thus, inserting (24) in (20), we obtain

where

and let be sufficiently small to ensure . By Hölder's inequality again, we have

where we assume throughout the paper that is the measure of . Using (25) and (27), we obtain

Moreover, using Hölder's inequality once more, we have

that is,

Finally, from (28) and (30), we obtain

We deduced from (31) that . On the other hand, is nonnegative function by assumption. So that keeps bounded continuously under the conditions given in Theorem 1, the solutions exsit for all time . That is, we find that the global solution exists when the absorption is more powerful than the nonlinear boundary flux and this accomplishes the proof of Theorem 1.

3. Criterion for Blow-Up

In this section, we concentrate on the finite time on which blow-up occurs. We construct two auxiliary functions to redefine and , then the nonlinear boundary-flux is more powerful than the absorption, and we obtain the following result.

Theorem 2. Suppose

Let

If

then the solutions of the problem (1)–(3) blow up at time with

Here is defined in (7). Moreover, if , then .

Proof. Differentiating (7) and using the hypothesis (33), we have

Differentiating (33), we thus obtain from (15) that

Note the identity that

So, from (37), we get

Therefore,

Here, we have used the identities

So, the hypothesis implies that for all , the following inequality holds :

By the Schwarz inequality, we have

Together with (36), we have

That is,

Integrating this from to , we obtain

Substituting (46) in (36) we obtain the differential inequality

If , then

This leads to

If , then

holds for . This implies that and completes the proof of Theorem 2.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of “CQ CSTC” (cstc2012jjA00013), and Scientific and Technological Research Program of Chongqing Municipal Education Commission.

References

H. Fujita, “On the blowing up of solutions of the Cauchy problem for ,” Journal of the Faculty of Science, vol. 13, pp. 109–124, 1966.

K. Hayakawa, “On nonexistence of global solutions of some semilinear parabolic differential equations,” Proceedings of the Japan Academy, vol. 49, pp. 503–505, 1973.

K. Kobayashi, T. Sirao, and H. Tanaka, “On the growing up problem for semilinear heat equations,” Journal of the Mathematical Society of Japan, vol. 29, no. 3, pp. 407–424, 1977.

D. G. Aronson and H. F. Weinberger, “Multidimensional nonlinear diffusion arising in population genetics,” Advances in Mathematics, vol. 30, no. 1, pp. 33–76, 1978.

V. A. Galaktionov, S. P. Kurdjumov, A. P. Mihaĭlov, and A. A. Samarskiĭ, “On unbounded solutions of the Cauchy problem for the parabolic equation ,” Doklady Akademii Nauk SSSR, vol. 252, no. 6, pp. 1362–1364, 1980.

V. A. Galaktionov, “Blow-up for quasilinear heat equations with critical Fujita's exponents,” Proceedings of the Royal Society of Edinburgh A, vol. 124, no. 3, pp. 517–525, 1994.

K. Mochizuki and R. Suzuki, “Critical exponent and critical blow-up for quasilinear parabolic equations,” Israel Journal of Mathematics, vol. 98, pp. 141–156, 1997.

T. Kawanago, “Existence and behaviour of solutions for ,” Advances in Mathematical Sciences and Applications, vol. 7, no. 1, pp. 367–400, 1997.

H. A. Levine, “The role of critical exponents in blowup theorems,” SIAM Review, vol. 32, no. 2, pp. 262–288, 1990.

L. E. Payne, G. A. Philippin, and S. Vernier Piro, “Blow-up phenomena for a semilinear heat equation with nonlinear boundary conditon, I,” Zeitschrift für Angewandte Mathematik und Physik, vol. 61, no. 6, pp. 999–1007, 2010.

Z. Jiang, S. Zheng, and X. Song, “Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions,” Applied Mathematics Letters, vol. 17, no. 2, pp. 193–199, 2004.

F. Li and J. Li, “Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 1005–1014, 2012.

L. Wang, J. Yin, and C. Jin, “-limit sets for porous medium equation with initial data in some weighted spaces,” Discrete and Continuous Dynamical Systems B, vol. 18, no. 1, pp. 223–236, 2013.

S. Kamin and L. A. Peletier, “Large time behaviour of solutions of the porous media equation with absorption,” Israel Journal of Mathematics, vol. 55, no. 2, pp. 129–146, 1986.

L. Wang and J. Yin, “Complicated asymptotic behavior of solutions for heat equation in some weighted space,” Abstract and Applied Analysis, vol. 2012, Article ID 463082, 15 pages, 2012.

J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford, UK, 2008.