ON STABILITY AND REGULARITY FOR SEMILINEAR ANOMALOUS DIFFUSION EQUATIONS PERTURBED BY WEAK-VALUED NONLINEARITIES

Various diffusion processes in media with memory are depicted by anomalous diffusion equations. We are concerned with a class of anomalous diffusion equations with the nonlinearity taking values in Hilbert scales of negative order. The fundamental questions on global solvability, stability and regularity of solutions to the Cauchy problem governed by the mentioned equations are taken into account. We obtain some solvability results under different assumptions on the regularity of the nonlinear function. When the nonlinearity becomes less singular, the asymptotic stability of solutions is proved. Provided that the kernel function is 2-regular and sectorial, we show the Ho center dot lder continuity of the obtained solutions. Our approach is based on the resolvent theory, fixed point argument and embeddings of fractional Sobolev spaces. The applicability of our results is presented for some anomalous diffusion problems, where the nonlinear function is of polynomial or convection type.