Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy–Leray inequalities

Abstract

The purpose of this paper is twofold. First is the study of the nonexistence of positive solutions of the parabolic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u}{\partial t}=\varDelta _{p} u+V(x)u^{p-1}+ \lambda u^{q} &{}\quad {\text{in}}\; \varOmega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\ge 0 &{}\quad {\text{in}} \; \varOmega ,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu }=\beta |u|^{p-2} u &{}\quad {\text{on}}\; \partial \varOmega \times (0,T), \end{array}\right. } \end{aligned}$$

where $\varOmega$ is a bounded domain in ${\mathbb{R}}^{N}$ with smooth boundary $\partial \varOmega$, $\varDelta _{p} u= {\text{div}}(|\nabla u|^{p-2} \nabla u)$ is the p-Laplacian of u, $V\in L_{\mathrm{loc}}^{1}(\varOmega )$, $\beta \in L_{\mathrm{loc}}^{1}(\partial \varOmega )$, $\lambda \in {\mathbb{R}}$, the exponents p and q satisfy $1<p<2$, and $q>0$. Then, we present some sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation we are interested in.