Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part II: Semilinear equations

In this paper, we deal with analysis of the initial-boundary value problems for the semilinear time-fractional diffusion equations, while the case of the linear equations was considered in the first part of the present work. These equations contain uniformly elliptic spatial differential operators of the second order and the Caputo type fractional derivative acting in the fractional Sobolev spaces as well as a semilinear term that depends on the spatial variable, the unknown function and its gradient. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. For these problems, we first prove uniqueness and existence of their solutions. Under some suitable conditions, we then show the non-negativity of the solutions and derive several comparison principles. We also apply the monotonicity method by upper and lower solutions to deduce some a priori estimates for solutions to the initial-boundary value problems for the semilinear time-fractional diffusion equations. Finally, we consider some initial-boundary value problems for systems of the linear and semilinear time-fractional diffusion equations and prove non-negativity of their solutions under the suitable conditions.