Qualitative Study in a Parabolic Equation with Nonstandard Growth Conditions and Singular Medium Void

Abstract

This paper deals with qualitative properties of solutions in a parabolic equation with nonstandard growth conditions and singular medium void. For the sublinear source, we prove that all solutions are global. For the linear source, the solutions are global provided that the measure of domain is small. For the superlinear source, the solutions blow up at infinite time given the initial data belonging to the unstable set, and blow up in finite time with positive or negative energy, respectively. The exponential decay rates of global solutions are demonstrated as well and the bounds of blow-up time are determined for all dimensions of space domain. At last, we discuss the blow-up rates of solutions.