The Nonlinear Telegraph Equation With Nonlinear Damping

We consider the existence and uniform decays to the cauchy problem of the following nonlinear telegraph equation in the case that the initial data are small. utt-αΔutt-Au-Δut=Δf(u)t,x€Rn,t>0,(0.3) u(x,0)=uo(x), ut(x,0)=u1(x),x∈Rn,(0.4) where u(x, t) denotes the unknown function,f(u)∈Ck(R) and|f(l)(u)|≤|u|m-l for0≤l≤k≤m,andm≤2,u0and u1are the given initial value functions, αand β are two positive constants, the subscript t indicates the partial derivative with respect to t, Δ is the n-dimensional Laplacian. First of all, We will introduce the physical meaning and research results of the equation. Second, We will give the notations and inequalities. Third, We establish the decay estimate and existence of the solution to the linearized equation. Then we give the existence and uniform decays to the cauchy problem of the nonlinear telegraph equation in the case that the initial data are small.