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

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We study the Cauchy problem in \({\mathbb{R}^N}\) for the parabolic equation $$u_t+{\rm div}\,F(u)=\Delta\varphi(u),$$ which can degenerate into a hyperbolic equation for some intervals of values of u. In the context of conservation laws (the case 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.