A note on uniqueness of entropy solutions to degenerate parabolic equations in ℝN

Abstract

We study the Cauchy problem in ℝN for the parabolic equation ut + div F (u) = ΔΦ(u), which can degenerate into a hyperbolic equation for some intervals of values of u. In the context of conservation laws (the case φ ≡ 0), it is known that an entropy solution can be non-unique when F′ has singularities. We show the uniqueness of an entropy solution to the general parabolic problem for all L∞ initial datum, under the isotropic condition on the flux F known for conservation laws. The only assumption on the diffusion term is that φ is a non-decreasing continuous function. © 2009 Birkhäuser Verlag Basel/Switzerland.