Nonlinear degenerate parabolic equations with time dependent singular coefficients

We are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T ) with initial condition u (., 0) = u 0 (.) ⩾ 0 and vanishing on the boundary ∂Ω × (0, T ), given by\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ \frac{\partial u}{\partial t}=u^\alpha \nabla \cdot \big (u^\beta |\nabla u|^{p-2}\nabla u\big )+V(x,t)u^{p-1+\alpha +\beta } $$ \end{document} where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega \in \mathbf {R}^N$\end{document} (resp. a Carnot Carathéodory metric ball in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {R}^{2N+1})$\end{document} with smooth boundary and the time dependent singular potential function V ( x , t ) ∈ L 1 loc (Ω × (0, T )), \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\alpha , \beta \in \mathbf {R}$\end{document} , 1 < p < N , p − 1 + α + β > 0. We find the best lower bounds for p + β and provide proofs for the nonexistence of positive solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim