Limit Sets for Multidimensional Nonlinear Transport Equations

We propose a hyperbolic generalisation of the well-known reaction diffusion equation and study the structure of its ω-limit sets. Under a dissipativity condition on the nonlinearity, we show quasi-convergence of the flow, saying that the ω-limit set of any orbit is non-empty, compact, connected, and contained in the set of equilibrium solutions.