Regularity and large‐time behavior of solutions for fractional semilinear mobile–immobile equations

Abstract

This paper addresses the regularity and large‐time behavior of solutions for the fractional semilinear mobile–immobile equations where the nonlinearity term admits various kind of growth conditions. Concerning the associated linear Cauchy problem, a variation of parameters formula of mild solution via the relaxation functions and the eigenfunction expansions is established and the ‐regularity in time of this solution is proved. In addition, based on the theory of completely positive functions, local estimates, and fixed‐point arguments, some results on existence, regularity, and stability of solutions to above‐mentioned semilinear problem are shown. Furthermore, we prove a result on convergence to equilibrium of solutions with polynomial rate in the case when the nonlinearity function is globally Lispschitzian.