Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping (Q1361430)

The Cauchy problem for \(V_{tt}- a(V_x)_x+\alpha V_t= 0\), \((x,t)\in\mathbb{R}\times (0,\infty)\), \(\alpha>0\), \(a'(0)>0\), is considered. It is shown that the asymptotic profile is given by the solution \(\varphi\) to \((*)\) \(\alpha\varphi_t- a'(0)\varphi_{xx}= 0\), \(\varphi(x,0)= V(x,0)+ 1/\alpha V_t(x,0)\), and that \[ |(V-\varphi), (V- \varphi)_x, (V- \varphi)_t(t))|_{L^\infty}={\mathcal O}(t^{-1}, t^{-3/2}, t^{-2}). \] The proof uses the energy method and the fundamental solution for \((*)\).