Nonlinear stability of rarefaction waves for a viscoelasticity
Consider the problem \(V_t= U_x\), \(U_t= \rho\sigma (V)_x+ a* \sigma (V)_{xt}+ h\), which is obtained from the equation of viscoelasticity \(w_{tt}= \sigma (w_x)_x+ \int_{-\infty}^t a' (t- \tau) \varphi (w_x)_x d\tau+ f(x,t)\) by setting \(U= w_t\), \(V= w_x\). Here \(\sigma\) is assumed to be smooth, increasing, and convex, the kernel \(a\) is positive, decreasing and convex, and its derivatives up to the third order have finite moments up to the third order. Also, \(\rho>0\), such that the material under consideration is a solid. The author establishes the nonlinear stability of weak rarefaction waves of this problem, in the sense that solutions tend to rarefaction waves in the corresponding elastic solid \((a=0\), \(h=0)\), if the asymptotic states that are connected are sufficiently close, \(h\) is small, and the initial data are close to the elastic rarefaction waves. In particular, global solutions of the viscoelastic problem are found that are not small at \(x= \pm\infty\) but have finite limits there. The proof uses a smooth approximation of the initial data, an integral transformation of the second equation which is used to isolate \(\sigma (V)_x\), and an energy method.
scientific article; zbMATH DE number 635876