On some Improperly Posed Problem for a Degenerate Nonlinear Parabolic Equation

Abstract

We consider the non-characteristic Cauchy problem for the degenerate nonlinear parabolic equation$|u|^{\alpha}u_t – \Delta u – \gamma |u|^{–\beta} u = 0$ under some assumptions on$\alpha, \beta$ and $\gamma$. The problem is improperly posed in the sense of Hadamard. We derive for such solutions an estimate in terms of the Cauchy data and a prescribed bound of the solution.