We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space R n , 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.
A class of periodic initial value problems for two-dimensional NewtonBoussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved.
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