The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation

We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space Rn,n≥1. Here, the fractional order α is related to the diffusion-type source term behaving as the usual diffusion term on the high frequency part. It has a feature of regularity-gain and regularity-loss for α>1 and 0<α<1, respectively. We establish the global existence and time-decay rates for small-amplitude solutions to the Cauchy problem for α>0. In the case that 0<α<1, we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.