On the semistability of the minimal positive steady state for a nonhomogeneous semilinear Cauchy problem

Abstract

This paper is contributed to the study of the Cauchy problem \[ \left\{ \begin{array}{ll} u_{t}=\Delta u+K(|x|)u^{p}+\mu f(|x|) \mbox{in} R^{n}\times (0,T), & u(x,0)=\varphi (x) \mbox{in} R^{n}, & u(x,0)=\varphi (x) \mbox{in} R^{n}, & \end{array}% \right. \]% with non-negative initial function $\varphi \not\equiv 0$. We will study the asymptotic behavior and the semistability of the minimal positive steady state. In addition, we will prove that all slow decay positive steady states are stable and weakly asymptotically stable in some weighted $L^{\infty }$ norms.