Qualitative properties of solution to a class of finitely degenerate semilinear pseudo-parabolic equations

In this paper, we investigate the initial-boundary value problem for finitely degenerate pseudo-parabolic equation associated with the Hörmander type operator. We first specify the conditions under which the weak solutions of the equation exist globally and finite time blow up when the initial energy \(J(u_0)>d\) $J(u_0)>d$ . Especially, the life span of the blow-up solution and the blow-up result with arbitrarily high initial energy level are derived. Additionally, we show the solution of degenerate pseudo-parabolic problem will converge to the steady state solution. Finally, we obtain the existence of ground state solution to the stationary problem. We prove that there exists initial value such that the instability occurs as a blow-up in finite time and the corresponding solution exists globally, respectively.