Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

Abstract

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities.

Keywords:

• variable exponent
• blow-up rate
• blow-up set

AMS Subject Classifications::

• Primary: 35K55, 35B40, 35K15, 35B33

Acknowledgements

This article is partially supported by Shandong Provincial Natural Science Foundation, China (Nos. ZR2009AQ016, ZR2010AQ011), and by the Fundamental Research Funds for the Central Universities. The authors thank the editor and the reviewers for their constructive suggestions to improve the quality of this article.