On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators

Abstract

The purpose of this paper is to study the nonexistence of positivesolutions of the doubly nonlinear equation\[\begin{cases}\frac{\partial u}{\partial t}=\nabla_{\gamma}\cdot(u^{m-1}|\nabla_{\gamma} u|^{p-2}\nabla_{\gamma}u) +Vu^{m+p-2} & \text{in}\quad \Omega \times (0, T ) ,\\u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T),\end{cases}\] where $\nabla_{\gamma}=(\nabla_x, |x|^{2\gamma}\nabla_y)$, $x\in \mathbb{R}^d, y\in \mathbb{R}^k$, $\gamma>0$, $\Omega$ is a metric ballin $\mathbb{R}^{N}$, $V\in L_{\text{loc}}^1(\Omega)$, $m\in\mathbb{R}$, $1 0$. The exponents $q^{*}$ arefound and the nonexistence results are proved for $q^{*} ≤ m+p < 3$.