Analysis of a pseudo-parabolic equation by potential wells

Abstract

In this paper, we consider a pseudo-parabolic equation, which was studied extensively in recent years. We generalize and extend the existing results in the following three aspects. First, we consider the vacuum isolating phenomenon with the initial energy $$J(u_0)$$ satisfying $$J(u_0)\le 0$$ and $$0<J(u_0)<d$$ , respectively, where d is a positive constant denoting the potential well depth. By means of potential well method, we find that there are two explicit vacuum regions which are annulus and ball, respectively. Second, we study the asymptotic behaviors of the solutions and the energy functional. Generally speaking, we establish the exponential decay of the solutions and energy functional when the solutions exist globally, and the concrete decay rate is given. As for the blow-up solutions, we prove that the solutions grow exponentially and obtain the behavior of energy functional as the time t tends to the maximal existence time. We get further two necessary and sufficient conditions for the solutions existing globally and blowing up in finite time, respectively, under the assumption that $$J(u_0)<d$$ . Finally, we give a new blow-up condition with eigenfunction method; it should be point out that this initial condition is independent of the initial energy. Under this condition, an upper bound estimation of the blow-up time is obtained and we prove that the solutions grow exponentially; the grow speed is given specifically.