ON THE TIME-DELAYED FRACTIONAL MOBILE-IMMOBILE EQUATIONS: REGULARITY AND LARGE-TIME BEHAVIOR

Abstract

We consider the Cauchy problem driven by nonlinear fractional mobile-immobile equations (Fr-MIMEs) involving time-varying delays in Hilbert spaces. Based on fixed-point arguments, smoothness of the resolvent operators, and local estimates, we show global solvability and regularity results for both linear and semilinear problems in which the nonlinearity terms are supposed to satisfy sublinear or superlinear growth conditions. Several qualitative aspects concerning the large-time behavior of solutions, such as dissipativity and stability, are investigated by establishing a new Halanay-type inequality. Moreover, thanks to the technique of measure of noncompactness, the existence of decay solutions with polynomial rates is also proved.