Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains Part I. Existence, Uniqueness and Upper Bounds

TL;DR

This paper studies the existence, uniqueness, and bounds of solutions to nonlinear fractional diffusion equations on bounded domains, covering various operators and nonlinearities, and establishing fundamental properties for further analysis.

Contribution

It introduces a broad class of solutions for fractional nonlinear diffusion equations, proving their existence, uniqueness, and deriving key bounds, extending classical results to fractional and degenerate cases.

Findings

Established existence and uniqueness of solutions.

Derived upper bounds and smoothing effects.

Provided weighted-$L^1$ estimates for solutions.

Abstract

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian $(-\Delta)^s$, $0<s<1$, in a bounded domain with zero Dirichlet boundary conditions, but it also includes many other examples since our theory only needs some basic properties that are typical of "linear heat semigroups." The nonlinearity $F$ is assumed to be increasing and is allowed to be degenerate, the prototype is the power case $F(u)=|u|^{m-1}u$, with $m>1$. In this paper we propose a suitable class of solutions of the equation, and cover the basic theory: we prove existence, uniqueness of such solutions, and we establish upper bounds of two forms (absolute bounds and smoothing effects), as well as weighted-$L^1$ estimates. The class of solutions is very well suited for that work. The standard Laplacian case $s=1$ is included and the linear case $m=1$ can be recovered in the limit. In a companion paper [12], we will complete the study with more advanced estimates, like the upper and lower boundary behaviour and Harnack inequalities, for which the results of this paper are needed.