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)∈Ω×R+u∣(?)Ω=O,△u∣(?)Ω=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)∣三a∣s∣,if ∣s∣≥1,(2)ρ(∣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(σ)dσ.(F2)(?) μ＞0,s.t.0≤F(s)≤ μsf(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.