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 \( u_t = \div \, \a(u,Du) \), where \( \a(z,\xi) = \nabla_\xi f(z,\xi) \), 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.