Existence of traveling waves in elastodynamics with variable viscosity and capillarity

Motivated by our earlier works, Thanh (2010) [3] and [4], we study the global existence of traveling waves associate with a Lax shock of a model of elastodynamics where the viscosity and capillarity are functions of the strain. The system is hyperbolic and may not be genuinely nonlinear. The left-hand and right-hand states of a Lax shock correspond to a stable node and a saddle point. By defining a Lyapunov-type function and using its level sets, we estimate the attraction domain of the stable node. Then we show that the saddle point lies on the boundary of the attraction domain of the stable node. Moreover, exactly one stable trajectory enters this attraction domain. This gives a stable-to-saddle connection for 1-shocks (a saddle-to-stable connection for 2-shocks), and therefore defines exactly one traveling wave connecting the two states of the Lax shock.