Blow-Up Phenomena for Porous Medium Equation with Nonlinear Flux on the Boundary
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.
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 . 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  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 2010, Payne et al.  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
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.
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.
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.
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.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford, UK, 2008.View at: MathSciNet