On Potential Wells And Applications To Strongly Damped Nonlinear Wave Equations

This paper deals with the initial boundary value problem of a class of strongly damped wave equationsWhere,Ω(?)R~n is a bounded domain and f∈C with f(u)u≥0.The existence of the weak solutions of the above problem is discussed by combining the Galerkin method and the family of new potential wells that defined in this paper, and the new existence terms are obtained. Besides, by utilyzing some important inequalities such as Holder inequality, Gronwall inequalities and the family of potetial wells, the existence of the strong solution of the problem is analyzed, and the new existence terms and uniqueness are gained then. Then, the invariance of the family of potential wells under the flow of (l)-(3) is reaserched, vacuum isolating property of the solutions is gained, that means all solutions of the equations may appear in the inside of a small ball or outside of a large ball, rather than the band-shape intervenient regine. Thus, a non-solution region called vacuum isolating region is formed. Finally, the asymptotic behaviors for solutions of the problem were studied by using integral estimate method. The rezults indicate that the solutions of the problems decay to zero according to the exponent of t.