Continuous Dependence of Renormalized Solution for Nonlinear Degenerate Parabolic Problems in the Whole Space

Abstract

We consider the general degenerate parabolic equation: $$u_t - \Delta b(u) + div\ \tilde{F}(u) = f \qquad{\rm in} \ \ Q\ =\ ]0,T[ \, \times \, =\mathbb{R}^N , \ \ T > 0. $$ We suppose that the flux $${\tilde{F}}$$ is continuous, b is nondecreasing continuous, and both are not necessarily Lipschitz continuous functions. The well-posedness (existence and uniqueness) of the renormalized solution of the associated Cauchy problem for L 1 initial data and source term is studied in Maliki and Ouedraigo (Ann. Fac, Sci. Toulouse Math. (6) 17(3):597–611, 2008) under a structure condition $${\tilde{F}(r)=F(b(r))}$$ and an assumption on the modulus of continuity of b. In the same framework, our aim is here to establish the continuous dependence of this renormalized solution with respect to the data. The novelty is the fact that we are working in the whole space $${\Omega=\mathbb{R}^{N}}$$ with unbounded data (u 0, f) and b, $${\tilde{F}}$$ are not Lipschitz functions.