On fractional p-laplacian parabolic problem with general data

TL;DR

This paper investigates the existence and properties of weak and entropy solutions for a fractional p-Laplacian parabolic problem with general data, extending the understanding of such nonlinear nonlocal PDEs.

Contribution

It proves the existence of weak and entropy solutions for the fractional p-Laplacian parabolic problem with minimal data regularity and analyzes their qualitative properties.

Findings

Existence of weak solutions for L^1 data.

Existence of nonnegative entropy solutions with nonnegative data.

Quantitative and qualitative properties depending on p.

Abstract

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } \O_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u & \ge & 0 & \text{ in }\ren \times (0,T),\\ u(x,0) & = & u_0(x) & \mbox{ in }\O, \end{array}% \right. $$ where $\Omega$ is a bounded domain, and $(-\D^s_{p})$ is the fractional p-Laplacian operator defined by $$ (-\D^s_{p})\, u(x,t):=P.V\int_{\ren} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,dy$$ with $1<p<N$, $s\in (0,1)$ and $f, u_0$ are measurable functions. The main goal of this work is to prove that if $(f,u_0)\in L^1(\O_T)\times L^1(\O)$, problem $(P)$ has a weak solution with suitable regularity. In addition, if $f_0, u_0$ are nonnegative, we show that the problem above has a nonnegative entropy solution. In the case of nonnegative data, we give also some quantitative and qualitative properties of the solution according the values of $p$.