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

In this paper we study the degenerate parabolic $p$-Laplacian,${\mathrm{\partial }}_{t}u-v^{-1}\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }\sqrt{Q}\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}Q\mathrm{\nabla }u)=0$, 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.