Universal bound for global solution of nonlinear heat equation

The aim of this paper is to study some properties of positive solutions to the nonlinear diffusion equation ∂u(x,t)∂t=Δpu(x,t)+c(x)f(u(x,t)),(x,t)∈Ω×(0,∞).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial u(x,t)}{\partial t} = \Delta _p u(x,t) + c(x)f(u(x,t)), \;\; (x,t) \in \Omega \times (0,\infty ). \end{aligned}$$\end{document}Assuming that f is of a bistable type with stable constant steady states 0 and c0>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0 >0$$\end{document}, 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 c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0$$\end{document} as t tends to +∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+\,\infty $$\end{document}. Some examples where our results can be applied are provided.