Convergence Rates To Diffusion Waves Of Solutions To Nonlinear Evolution Equations With Damping And Conservative Form

In this paper, we study the global existence and the asymptotic behavior of the solutions to the Cauchy problem for the following nonlinear evolution equations with conservation form, damping and differing end stateswith initial data(ψ, θ)(x,0) = (ψ0(x),θ0(x)) → (ψ±,θ±) as x → ± ∞ (I)where a and z/ are positive constants such that a < 1, v < a(l - a). By applying modifying Hisiao-Liu [3] correction function θ(x,t) defined by (2.13) and using the energy method, we show sup(|(ψ,θ)(x,t)| + |(ψx,θx)(x,t)|) → 0 as t → ∞ and the solutions decay with exponential rates. The similar problem is studied by Tang and Zhao [14] for the case of the initial data with the same end states, i.e.(ψ±,θ±) = (0,0).The above problems will be discussed as the following four parts. Part 1. Related results for the similar models wih proper initial data . Main results and methods despription.Part 2. Prelimilary results for the modified system.Part 3. The local and global existence of solutions by a Prior estimates.Part 4. Decay rates based on Part 2.