On fractional p-Laplacian parabolic problem with general data

By Boumediene Abdellaoui, Ahmed Attar, Rachid Bentifour, I. Peral. 2016

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In this article, the problem to be studied is the following $$\begin{aligned} (P) \left\{ \begin{array}{llll} u_t+(-\Delta ^s_{p}) u = f(x,t) &{}\quad \text { in } \Omega _{T}\equiv \Omega \times (0,T), \\ u = 0 &{}\quad \text { in }({\mathbb {R}}^N{\setminus }\Omega ) \times (0,T), \\ u(x,0) = u(x) &{}\quad \text{ in } \Omega , \end{array} \right. \end{aligned}$$(P)ut+(-Δps)u=f(x,t)inΩT≡Ω×(0,T),u=0in(RN\Ω)×(0,T),u(x,0)=u(x)inΩ,where $$\Omega $$Ω is a bounded domain and $$(-\Delta ^s_{p})$$(-Δps) is the fractional p-Laplacian operator defined by $$\begin{aligned} (-\Delta ^s_{p})\, u(x,t):=P.V\int _{{\mathbb {R}}^N} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,\mathrm{d}y \end{aligned}$$(-Δps)u(x,t):=P.V∫RN|u(x,t)-u(y,t)|p-2(u(x,t)-u(y,t))|x-y|N+psdywith $$1<p<N$$1<p<N, $$s\in (0,1)$$s∈(0,1) and $$f, u_0$$f,u0 being measurable functions. The main goal of this work is to prove that if $$(f,u_0)\in L^1(\Omega _T)\times L^1(\Omega )$$(f,u0)∈L1(ΩT)×L1(Ω), problem (P) has a weak solution with suitable regularity. In addition, if $$f_0, u_0$$f0,u0 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4c8e4b0e43b3e4fc11f4a53695c1997bd6325df7