Universal bound for global solution of nonlinear heat equation

Abstract

The aim of this paper is to study some properties of positive solutions to the nonlinear diffusion equation $$\begin{aligned} \frac{\partial u(x,t)}{\partial t} = \Delta _p u(x,t) + c(x)f(u(x,t)), \;\; (x,t) \in \Omega \times (0,\infty ). \end{aligned}$$ Assuming that f is of a bistable type with stable constant steady states 0 and $$c_0 >0$$ , we show, that there exist a universal, a priori upper bound for all positive solutions of the previous equation. Moreover, we prove the convergence of these solutions to the constant $$c_0$$ as t tends to $$+\,\infty $$ . Some examples where our results can be applied are provided.