Asymptotic behavior for a finitely degenerate semilinear pseudo-parabolic equation

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.