The Initial Boundary Value Problem Of The Double Dispersion Wave Equation With Nonlinear Damping Term

In this paper,we investigate the initial-boundary value problem of double dispersion wave equation with nonlinear damping term utt-?Autt-?u+?2u-Ag(ut)-?f(u)= 0,(x,t)??×R+u?(?)?=O,?u?(?)?=0 u(x,0)= u0(x),ut(x,0)= u1(x)Suppose g satisfies the following conditions(G1)g?C1(R),g(0)=0.(G1)exists a strictly increasing odd function p? C1(R).St.(1)?s???g(s)??a?s?,if ?s??1,(2)?(?s?)?|g(s)|?ap-1(?s?),if?s??1.Suppose f satisfies the following conditions(F1)f? C11(R),f(0)= 0,let F(s)=?s0 f(?)d?.(F2)(?) ??0,s.t.0?F(s)? ?sf(s),Vs E?R.(F3)(?) C>0,1?p??(n= 1,2),1<q?n/n-2(n?2),s.t.(?)s E? R,?f1(s)??C(1+?S?q-1).By the means of Faedo-Galerkin method,we obtain the existence and uniqueness of global solution,and the regularity of solution.Moreover,we derive the decay esti-mation of the global solutions by using the convexity method established by Fatiha Alabau-Boussouira,and the result is applied to a specific nonlinear double dispersion wave equation.