Global existence, uniform decay and exponential growth for a class of semi-linear wave equation with strong damping

In this paper, we consider the nonlinearly damped semi-linear wave equation associated with initial and Dirichlet boundary conditions. We prove the existence of a local weak solution and introduce a family of potential wells and discuss the invariants and vacuum isolating behavior of solutions. Furthermore, we prove the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively, and the asymptotic behavior of solutions for the cases of potential well family with 0 < E(0) < d. At last we show that the energy will grow up as an exponential function as time goes to infinity, provided the initial data is large enough or E(0) < 0.