Convergence to diffusion waves of solutions to nonlinear viscoelastic model with fading memory (Q1198814)

The author studies the large time behaviour of solutions to the one- dimensional Cauchy problem: \(u_ t-v_ x=0\), \(v_ t-\sigma_ x=0\), \(x\in\mathbb{R}^ 1\), \(t>0\), which describes the motion of an unbounded, homogeneous viscoelastic bar with fading memory, that is, \(\sigma=f(u(t,x))+\int^ t_{-\infty} a'(t-\tau)g(u(x,\tau))d\tau\), where \(a(s)\) is a given kernel on \(0\leq s<\infty\) with derivative \(a'\), and \(f(u)\), \(g(u)\) are given smooth material functions. The history and initial data are given by \(u(x,t)=\eta(x,t)\), \(t\leq 0\), \(v(x,0)=v_ 0(x)\). The assumption is that \(g'(u)>0\), \(p'(u)>0\), \(p''(u)<0\), \(a\), \(a'\), \(a''\in L^ 1(0,\infty)\), \(\int^ \infty_ 0 t| a(t)| dt<\infty\), \(L(a')(z)\neq 0\) for any \(z\in\mathbb{C}\) with \(\text{Re }z\geq 0\), \(a\) is strongly positive definite, \(L(a)\) is a rational fraction, where \(L(a)\) is the Laplace transform of \(a\) and the strongly positive definiteness of \(a\) means that \(\int^ t_ 0 y(s)\int^ s_ 0 b(s- \tau)y(\tau)d\tau ds\geq 0\) for any \(t\geq 0\) and for every \(y\in C[0,\infty)\), where \(b(t)=a(t)-ce^{-t}\) with some constant \(c>0\). For example, \(a(t)=\sum^ N_{j=1}\alpha_ j\exp(-\mu_ j t)\), \(t\geq 0\) or \(e^{-\mu t}\cos\beta t\), with \(\alpha_ j\), \(\mu_ j>0\) \((j=1,\dots,N)\) and \(\mu>0\). Roughly speaking, the result is that the solution \((u,v)\) decays a constant state \((\bar u,\bar v)\) is \(L^ 2\) at rate \((1+t)^{-1/4}\) and in \(L^ \infty\) at rate \((1+t)^{-1/2}\). Moreover, \((u,v)\) is asymptotically approximated in \(L^ 2\) at rate \((1+t)^{-1/2}\) by a combination of \((\bar u,\bar v)\) and diffusion waves, which are solutions of Burgers equations. The rates obtained in the paper are optimal.