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

TL;DR

This paper proves the global existence and decay rates of solutions for a fractional pseudo-parabolic equation on the whole space, addressing different behaviors depending on the fractional order and introducing a time-weighted energy method.

Contribution

It establishes the global existence and decay rates for small solutions of the fractional pseudo-parabolic equation, handling both regularity-gain and regularity-loss cases with novel techniques.

Findings

Global existence of solutions for all positive fractional orders.

Decay rates depend on the fractional order and solution regularity.

Introduction of a time-weighted energy method for weakly dissipative cases.

Abstract

We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space $R^n,n\geq 1$. Here, the fractional order $\alpha$ 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 $0<\alpha < 1$ and $\alpha> 1$, respectively. We establish the global existence and time-decay rates for small-amplitude classical solutions to the Cauchy problem for $\alpha>0$. In the case that $0<\alpha < 1$ , we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.