Nonlinear stability of travelling wave solutions for viscoelastic materials with fading memory (Q1913667)

The authors give a proof of existence and stability of shock-wave solutions to the system of equations describing wave propagation in a viscoelastic medium with memory, \[ v_t=u_x, \qquad u_t=[\sigma(v)]_x+ \int^t_{-\infty} a'(t-\tau)[\eta(v)]_x d\tau. \] They consider a class of smooth initial conditions satisfying the Rankine-Hugoniot boundary conditions at infinity. Using a priori estimates for the solutions and the continuation principle, they prove that any initial condition from this class gives rise to a unique smooth global solution and, provided that the initial profile is taken sufficiently close to a traveling wave \(u=u(x-ct)\), \(v=v(x-ct)\), the solution asymptotically coincides with the traveling (shock) wave at \(t\to \infty\) in the sense of the maximum norm.