Potential Wells And Its Applications To Wave Equations With Nonlinear Damping And Source Terms

In this paper, we study the initial boundary value problem of wave equations with nonlinear damping and source termsu_tt-â–³u+a|u_t|^m-1u_t=b|u|^p-1u, x∈Ω,t＞0 (1)u(x,O)=u₀(x), u_t(x,0)=u₁(x),x∈Ω(2)u(x,t)=0,x∈(?)Ω,t≥0 (3)where a≥0, b≥0Ω(?)Rⁿ is a bounded domain, 1＜m≤p＜∞for N=1,2;1＜m≤p≤N+2/N-2 for N≥3. First by using new method we introduce a family of potential wells which include the single well as a special case. Then by using it we obtain some new existence therorems of global solutions and relating corollaries, the invariant sets of global solutions of probem (1)-(3) and the phenomena of vacuum isolating of solutions are discovered. At the last we derive a precise decay estimate of the global solutions using a difference inequality of M. Nakao.