A note on uniqueness of entropy solutions to degenerate parabolic equations in $\mathbb{R}^N$.

The original publication is available at www.springerlink.com DOI: 10.1007/s00030-009-0042-9International audienceWe study the Cauchy problem in $\mathbb{R}^N$ for the parabolic equation $u_t+\text{div} F(u)=\Delta \varphi(u)$, which can degenerate into a hyperbolic equation for some intervals of values of $u$. In the context of conservation laws (the case $\varphi\equiv 0$), 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 $L^\infty$ initial datum, under the isotropic condition on the flux $F$ known for conservation laws. The only assumption on the diffusion term is that $\varphi$ is a non-decreasing continuous function