Nonlinear degenerate prabolic equations with singular lower-order term

Abstract

We use variational methods to study the nonexistence of positive solutions for the following nonlinear parabolic partial differential equations: \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta( u^m)+V(x)u^m & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T), \end{cases} \] and \[ \begin{cases} \frac{\partial u}{\partial t}=\Delta_p u+V(x)u^{p-1} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega ,\\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T ), \end{cases} \] where $0 < m < 1$, $1 < p < 2$, $V\in L_{loc}^1(\Omega)$ and $\Omega$ is a bounded domain with smooth boundary in $ \mathbb{R}^N$.