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arXiv · 1508.07871

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

Matteo Bonforte·Juan Luis Vázquez
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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, $x\inΩ\subset \mathbb{R}^N$, 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 $(-Δ)^s$, $0 1$. 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.

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Matteo Bonforte, Juan Luis Vázquez. 2015-09-30. Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains Part I. Existence, Uniqueness and Upper Bounds. https://arxiv.org/abs/1508.07871

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