Degenerate parabolic p-Laplacian equations: existence, uniqueness and asymptotic behavior of solutions

In this paper we study the degenerate parabolic p-Laplacian, ₜ u - v^-1 div (|Q u|^p-2 Q u) =0, where the degeneracy is controlled by a matrix Q and a weight v. With mild integrability assumptions on Q and v, we prove the existence and uniqueness of solutions on any interval 0, T. If we further assume the existence of a degenerate Sobolev inequality with gain, the degeneracy again controlled by v and Q, then we can prove both finite time extinction and ultracontractive bounds. Moreover, we show that there is equivalence between the existence of ultracontractive bounds and the weighted Sobolev inequality.