Continuous dependence of renormalized solution for convection-diffusion problems involving a nonlocal operator

Abstract

In Ouédraogo A. et al (cf. [30]), it is provided existence and uniqueness results of L1-renormalized entropy solution for the Cauchy problem associated to the following vast class of nonlinear anisotropic degenerate parabolic-hyperbolic equations involving a nonlocal diffusion term: \begin{equation*}\label{problem_(CP)} \partial_{t}u+\nabla.F(u)-\displaystyle\sum_{i,j=1}^{N}\partial^{2}_{x_{i}x_{j}}A_{ij}(u) -{\cal L}_{\mu}[u]= f(u) \hskip0.3 cm\hbox{in }\ \ Q=(0,T)\times \RR^N \ \text{with}\ T>0\ \text{and}\ N\geq 1. \end{equation*} Our goal is to complement this previous work with a continuous dependence result of the L1-solution with respect to the data set (F,a,μ,f, u0). The strategy is to follow the approach developed by Karlsen and Ulusoy in [28]. However, we must manage the difficulties due to the fact that we are working in the whole space RN with an only integrable initial datum u0 and the term source f depends on the unknown function u.