On fractional p-laplacian parabolic problem with general data
Abstract
In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } Ø_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminusØ) \times (0,T), \\ u & \ge & 0 & \text{ in }\ren \times (0,T),\\ u(x,0) & = & u_0(x) & \mbox{ in }Ø, \end{array}% \right. $$ 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 $(f,u_0)\in L^1(Ø_T)\times L^1(Ø)$, 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 $p$.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2016
- DOI:
- 10.48550/arXiv.1612.01301
- arXiv:
- arXiv:1612.01301
- Bibcode:
- 2016arXiv161201301A
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35K59;
- 35K65;
- 35K67;
- 35K92;
- 35B09