On the elasto-plastic filtration equation

Abstract

We study the fully nonlinear heat equation b(∂tu)∂tu = Δu posed in a bounded domain with Dirichlet boundary conditions. Here b(s) = b− if s < 0, b(s) = b+ if s > 0, b− ̸= 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 → ∞ and when b− → 0+ or b+ → +∞. We also characterize solutions of the problem as limits of a minimization dynamic game.