On the elasto-plastic filtration equation

Abstract

We study the fully nonlinear heat equation $b({\partial }_{t}u){\partial }_{t}u=\hat{I}”u$ posed in a bounded domain with Dirichlet boundary conditions. Here $b(s)=b^{-}$ if $s<0$, $b(s)=b^{+}$ if $s>0$, $b^{-}\ne b^{+}$ being two positive constants. This equation models the flow of an elastic fluid in an elasto-plastic porous medium. We are interested in the existence and uniqueness of viscosity solutions and in their asymptotic behaviour as $t\to \infty$ and when $b^{-}\to 0^{+}$ or $b^{+}\to +\infty$. We also characterize solutions of the problem as limits of a minimization dynamic game.