A nonlinear fractional diffusion equation: Well-posedness, comparison results, and blow-up

Abstract

This paper aims the study of the existence of solutions for the semilinear fractional diffusion equationut(t,x)=∂t∫0tgα(s)Δu(t−s,x)ds+|u(t,x)|ρ−1u(t,x),in(0,T)×Ω;u(t,x)=0,on(0,T)×∂Ω;u(x,0)=u0(x),inΩ, where gα(t)=tα−1Γ(α), for α∈(0,1), Δ is the Laplace operator, and Ω is a sufficiently smooth domain in RN. We prove a local well-posedness result to this problem with initial data in Lq(Ω), the unique continuation of the solution and the persistence of continuous dependence on the initial data for the continued solution. Furthermore, we prove a comparison principle for mild solutions and we derive from this the existence of positive solutions. We also give sufficient conditions to obtain the blowing up behavior of the solution. Finally, by working in Besov spaces, how to proceed in order to recover most of the results, when initial conditions are assumed to be in interpolation spaces.