Blow-up phenomena for p-Laplacian parabolic equations under nonlocal boundary conditions

Abstract

The purpose of this paper is to deal with the blow-up problems of the following p-Laplacian parabolic equations with nonlocal boundary conditions: $\begin{array}{rl}& {\left(h\left(u\right)\right)}_t=\mathrm{\nabla }\cdot \left(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\right)+k_1\left(t\right)f\left(u\right)\text{in}\mathrm{\Omega }\times (0,t^{\ast }),\\ & |\mathrm{\nabla }u{|}^{p-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=k_2\left(t\right){\int }_{\mathrm{\Omega }}g\left(u\right)\mathrm{d}x\text{on}\mathrm{\partial }\mathrm{\Omega }\times (0,t^{\ast }),\\ & u(x,0)=u_0\left(x\right)\ge 0\text{in}\overline{\mathrm{\Omega }},\end{array}$ where p>2, $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$ is a bounded convex region, and the boundary $\mathrm{\partial }\mathrm{\Omega }$ is smooth. With the help of differential inequality techniques and Sobolev inequalities, we prove that the blow-up does occur on some certain conditions of the data. In addition, we obtain upper bounds and lower bounds of the blow-up time in $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$.

Keywords:

• p-Laplacian parabolic equations
• nonlocal boundary conditions
• blow-up

AMS CLASSIFICATIONS:

• 35K65
• 35B44

Acknowledgments

The authors are greatly indebted to the reviewers for many helpful suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (No. 61473180).