Global solutions to the equation of viscoelasticity with fading memory
The author analyses the viscoelastic one-dimensional model \[ v_ t - u_ x = 0,\;u_ t - S_ x = 0,\;S = \Phi (v) + \int^ t_{-\infty} \dot a(t - \tau) \psi (v) (\tau)d \tau, \] where \(v\) and \(u\) are the deformation gradient and the velocity, respectively; \(S\) is the stress; \(\Phi (v)\) and \(\psi (v)\) are given smooth functions of \(v\) such that \(\Phi (0) = \psi (0) = 0\). When the kernel \(a(t)\) has integrable singularities of first order, then the problem has a unique global solution if the initial conditions are smooth and small. This solution is convergent to the equilibrium state, when the kernel is strongly positive definite. This result improves the results of \textit{W. J. Hrusa} and \textit{J. A. Nohel} [J. Differ. Equations 59, 388-412 (1985; Zbl 0564.35075)], who proved only the local existence of the solution.
scientific article; zbMATH DE number 166905