On the dynamic of the nonlinear reaction diffusion equation u_t=∆F(u)+f(x,u,∇u) under Robin boundary data

Let Omega be a bounded smooth domain in R^3. Under Robin boundary data, the nonlinear reaction diffusion equation u_t=∆F(u)+f(x,u,∇u), (x, t) in Omega x R^+, is studied. Existence, uniqueness, stability and longtime behaviour of solutions are analyzed. An application to the dynamic of a population inhabiting a strongly heterogeneous environment is considered.