Initial Boundary Value Problem For Strongly Damped Nonlinear Wave Equation

In this paper,we study the initial boundary value problem of strongly damped nonlinear wave equations u_tt-α△u_t-â–³u=f(u),x∈Ωu(x,0)=u₀(x),u_t(x,0)=u₁(x),x∈Ωu|_(?)Ω=0whereΩ(?)Rⁿ is a bounded domain.First by using new method we introducea family of potential wells.Then by using them we obtain some existence therorems of global solutions and corresponding corollaries, then obtain the existence, uniqueness of global weak solutions and strong solutions, the invariant sets of global solutions of problem, the phenomena of vacuum isolating of solutions are discovered.At the last, by using improved integral estimate we prove that the global strong solution of the problem decays to zero exponetially as t→+∞under very general assumption regarding nonlinear term.