IBVP for anisotropic fractional type degenerate parabolic equation

Abstract

This paper is a generalization of the author's previous work [14]. We extend the argument [14] for any uniformly elliptic operator in divergence form $\mathcal{L}u=-div(A(x)\nabla u)$, more precisely, we study a fractional type degenerate elliptic equation posed in bounded domains $$ \partial_t u=div(u\,A(x)\nabla \mathcal{L}^{-s}u) $$ where $\mathcal{L}^{-s}$ is the inverse $s$-fractional elliptic operator for any $s \in (0,1)$. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition.