The Cauchy Problem for a Strongly Degenerate Quasilinear Equation

Abstract

We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation $u_t=\mathrm{div}\mathbf{a}(u,Du)$, where $\mathbf{a}(z,\xi )={\nabla }_{\xi }f(z,\xi )$, and $f$ is a convex function of $\xi$ with linear growth as $\Vert \xi \Vert \to \infty$, satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.