Decay Rates And Convergence Of Solutions To System Of One-Dimensional Viscoelastic Model With Damping

In this paper, we will consider the asymptotic behavior of solutions to the system of one-dimensional viscoelastic model with damping and prove that the corresponding solutions time-asymptotically behave like nonlinear diffusion wave as in [4,11]. In addition, we also show that the system of one-dimensional viscoelastic model with damping is a viscosity approximation of a hyperbolic conservation laws with damping.In this paper, we establish the following results:(i) For any fixed ε>> 0, we will prove the existence of the global smooth solutions (v ε(x,t),uε(x,t)) to the Cauchy problem (1.1),(1.2) when the initial data satisfy some smoothness and smallness assumptions (see Theorem 2.2 and Theorem 2.3). Moreover, the solutions asymptotically converge to the nonlinear diffusion wave defined by (1.4) uniformly with respect to e.(it) It is expected that the system (1.1) is a viscosity approximation of the following hyperbolic system with dampingwith initial dataTo be precise, we show that there exists a subsequence {(uεk(x,t),uεk(x,t))} of {(vε(x,t), uε(x,t))} obtained in (t) and a pair of smooth functions (v(x,t),u(x,t)) such thatMoreover, the limit functions (w(x,t),u(x,t)) is a global smooth solution to Cauchy problem (1.5),(1.6).