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 2010, Payne et al. [10] considered a semilinear heat equation with nonlinear boundary condition ( = 1 in (1)) and established conditions on nonlinearities sufficient to guarantee that (, ) exists for all time  > 0 as well as conditions on data forcing the solution (, ) to blow up at some finite time  * .When  = 1, 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][14][15][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 ( ≥ 2).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) blowup 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).

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 Ω be a bounded star-shaped region and assume that  >  satisfy If  and  satisfy the following conditions: where  1 ,  2 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 div(∇  ) = Δ  + ∇ ⋅ ∇  .By the divergence theorem again, we get Let Point out that  0 is positive because Ω is star-shaped by hypothesis.Notice also that div 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 Therefore, the hypotheses that  >  and 2 <  +  imply that So, by Hölder's inequality, we have For  > 0, we obtain from (23) that Thus, inserting (24) in (20), we obtain where and let  be sufficiently small to ensure  2 > 0. By Hölder's inequality again, we have where we assume throughout the paper that |Ω| = ∫ Ω  is the measure of Ω.Using ( 25) and ( 27), we obtain Moreover, using Hölder's inequality once more, we have Finally, from (28) and (30), we obtain We deduced from (31) that Φ() ≤ max{Φ(0), ( 2 / 1 ) 2/(−) |Ω|}.On the other hand, Φ() is nonnegative function by assumption.So that Φ() keeps bounded continuously under the conditions given in Theorem 1, the solutions exsit for all time  > 0. 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.

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.