Qualitative analysis of semilinear Cattaneo equations

The linear Cattaneo equation appears in heat transport theory to describe heat wave propagation with finite speed. It can also be seen as a generalization of a correlated random walk. If the system admits nonconservative forces (or reactions), then a nonlinear Cattaneo system is obtained. Here we consider asymptotic behavior of solutions of the nonlinear Cattaneo system. Following Brayton and Miranker we define a Lyapunov function to show global existence of solutions and to show that each omega-limit set is contained in the set of all stationary solutions.