Large time behavior of solutions to the nonlinear pseudo-parabolic equation

In this paper, we investigate the initial value problem for the nonlinear pseudo-parabolic equation. Global existence and optimal decay estimate of solution are established, provided that the initial value is suitably small. Moreover, when n ⩾ 2 and the nonlinear term f ( u ) disappears, we prove that the global solutions can be approximated by the linear solution as time tends to infinity. When n = 1 and the nonlinear term f ( u ) disappears, we show that as time tends to infinity, the global solution approaches the nonlinear diffusion wave described by the self-similar solution of the viscous Burgers equation.