Asymptotic behavior for a finitely degenerate semilinear pseudo-parabolic equation

TL;DR

This paper analyzes the long-term behavior and blow-up characteristics of solutions to a finitely degenerate semilinear pseudo-parabolic equation, providing new estimates and convergence results.

Contribution

It develops novel estimates for solutions, determines bounds on blow-up time and rate, and proves strong convergence to stationary solutions.

Findings

Exponential decay of energy functional established.

Upper bounds for blow-up time and rate determined.

Global solutions converge to stationary solutions as time approaches infinity.

Abstract

This paper investigates the initial boundary value problem of a finitely degenerate semilinear pseudo-parabolic equation associated with H\"{o}rmander's operator. Based on the global existence of solutions in previous literature, the exponential decay estimate of the energy functional is obtained. Moreover, by developing some novel estimates about solutions and using the energy method, the upper bounds of both blow-up time and blow-up rate and the exponential growth estimate of blow-up solutions are determined. In addition, the lower bound of blow-up rate is estimated when a finite time blow-up occurs. Finally, it is established that as time approaches infinity, the global solutions strongly converge to the solution of the corresponding stationary problem. These results complement and improve the ones obtained in the previous literature.