Nonlinear degenerate parabolic equations with time-dependent singular potentials for Baouendi-Grushin vector fields

Abstract

In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials:

$$\begin{array}{*{20}c} {\frac{{\partial u^q }} {{\partial t}} = \nabla _\alpha \cdot \left( {\left\| z \right\|^{ - p\gamma } \left| {\nabla _\alpha u} \right|^{p - 2} \nabla _\alpha u} \right) + V(z,t)u^{p - 1} ,} \\ {\frac{{\partial u^q }} {{\partial t}} = \nabla _\alpha \cdot \left( {\left\| z \right\|^{ - 2\gamma } \nabla _\alpha u^m } \right) + V(z,t)u^m ,} \\ {\frac{{\partial u^q }} {{\partial t}} = u^\mu \nabla _\alpha \cdot \left( {u^\tau \left| {\nabla _\alpha u} \right|^{p - 2} \nabla _\alpha u} \right) + V(z,t)u^{p - 1 + \mu + \tau } } \\ \end{array}$$

in a cylinder Ω × (0, T) with initial condition u (z, 0) = u ₀ (z) ≥ 0 and vanishing on the boundary ∂Ω × (0, T), where Ω is a Carnot-Carathéodory metric ball in ℝ^d+k and the time-dependent singular potential function is V (z, t) ∈ L ¹_loc (Ω × (0, T)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.