Asymptotic behavior for wave equations with damping term

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We considered the initial value problem of the abstract wave equation with dissipation whose coefficient tends to 0 as t tends to infinity. It is known that the solution of the wave equation with a constant the dissipative term is asymptotically free if the constant of the dissipation decays with the polynomial order less than-1. On the other hand, in the case that the coefficient of the dissipation is a positive constant, it is known that the difference between the solution of the abstract wave equation and the solution of the corresponding abstract heat equation decays faster than each of the solution does (diffusion phenomenon). In this research, we showed the decay estimate of the difference between the solution of the abstract wave equation with decaying dissipative term and the solution of the corresponding abstract parabolic equation. As applications, we obtained the estimate of the difference between the solution of the dissipative wave equation with Dirichlet and Robin boundary conditions.Next, we considered the abstract quasilinear dissipative hyperbolic equation of Kirchhoff type. The unique existence of the global solution was known for sufficiently small initial data. We can easily show that this solution tends to the solution of the corresponding heat equation with a constant coefficient as the time tends to infinity. Then we considered the case that the dissipative Kirchhoff equation with parameter tends to the corresponding quasilinear parabolic equation. For every initial data, it is known that if the dissipative Kirchhoff equation is sufficiently close to the corresponding parabolic equation, the unique global solvability of the both equations and the estimate of difference between the solutions of the two equations. However the known estimates are local in time. We showed the global decay rate estimate combined with the asymptotic behavior.