Asymptotic Behavior of Solutions to Diffusion Problems with Robin and Free Boundary Conditions

We study a nonlinear diffusion equation ut = uxx + f(u) with Robin boundary condition at x = 0 and with a free boundary condition at x = h(t), where h(t) > 0 is a moving boundary representing the expanding front in ecology models. For any f ∈ C1 with f(0) = 0, we prove that every bounded positive solution of this problem converges to a stationary one. As applications, we use this convergence result to study diffusion equations with monostable and combustion types of nonlinearities. We obtain dichotomy results and sharp thresholds for the asymptotic behavior of the solutions.