On fractional p-Laplacian parabolic problem with general data

In this article, the problem to be studied is the following (P) {u(t) + (-Delta(s)(p))u = f(x, t) in Omega(T) Omega x (0, T), u = 0 in (R-N\Omega) x (0, T), u(x, 0) = u(x) in Omega, where Omega is a bounded domain and (-Delta(s)(p)) is the fractional p-Laplacian operator defined by (-Delta(s)(p)) u(x, t) := P.V integral(RN) vertical bar u(x, t) - u(y, t)vertical bar(p-2)(u(x, t) - u(y, t))/vertical bar x - y vertical bar(N+ps) with 1 < p < N, s is an element of(0.1) and f, u(0) being measurable functions. The main goal of this work is to prove that if (f, u(0)) is an element of L-1 L-1(Omega(T)) x L-1(Omega), 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.