Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

TL;DR

This paper studies an initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations in bounded domains, proving the existence of solutions under mixed boundary conditions for a range of fractional orders.

Contribution

It introduces a novel analysis of anisotropic fractional degenerate parabolic equations with mixed boundary conditions and establishes existence results for solutions across all fractional orders in (0, 1).

Findings

Existence of solutions for measurable, bounded, non-negative initial data.

Solvability holds for any fractional order s in (0, 1).

Analysis of nonlocal anisotropic diffusion effects.

Abstract

We consider an initial mixed-boundary value problem for anisotropic fractional type degenerate parabolic equations posed in bounded domains. Namely, we consider that the boundary of the domain splits into two parts. In one of them, we impose a Dirichlet boundary condition and in the another one a Neumann condition. Under this mixed-boundary condition, we show the existence of solutions for measurable and bounded non-negative initial data. The nonlocal anisotropic diffusion effect relies on an inverse of a s-fractional type elliptic operator, and the solvability is proved for any s in (0, 1).