On fractional p-Laplacian parabolic problem with general data

In this article, the problem to be studied is the following (P)ut+(-Δps)u=f(x,t)inΩT≡Ω×(0,T),u=0in(RN\Ω)×(0,T),u(x,0)=u(x)inΩ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (P) \left\{ \begin{array}{llll} u_t+(-\Delta ^s_{p}) u = f(x,t) &{}\quad \text { in } \Omega _{T}\equiv \Omega \times (0,T), \\ u = 0 &{}\quad \text { in }({\mathbb {R}}^N{\setminus }\Omega ) \times (0,T), \\ u(x,0) = u(x) &{}\quad \text{ in } \Omega , \end{array} \right. \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a bounded domain and (-Δps)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta ^s_{p})$$\end{document} is the fractional p-Laplacian operator defined by (-Δps)u(x,t):=P.V∫RN|u(x,t)-u(y,t)|p-2(u(x,t)-u(y,t))|x-y|N+psdy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta ^s_{p})\, u(x,t):=P.V\int _{{\mathbb {R}}^N} \,\dfrac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}} \,\mathrm{d}y \end{aligned}$$\end{document}with 1<p<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<N$$\end{document}, s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0,1)$$\end{document} and f,u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f, u_0$$\end{document} being measurable functions. The main goal of this work is to prove that if (f,u0)∈L1(ΩT)×L1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f,u_0)\in L^1(\Omega _T)\times L^1(\Omega )$$\end{document}, problem (P) has a weak solution with suitable regularity. In addition, if f0,u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_0, u_0$$\end{document} 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.