Asymptotic behavior of solutions of a reaction diffusion equation with inhomogeneous Robin boundary condition and free boundary condition

This paper studies the long time behavior of solutions of a reaction-diffusion model with inhomogeneous Robin boundary condition at x = 0 and free boundary condition at x = h(t). We prove that, for the initial data u(0) = sigma phi, there exists sigma* >= 0 such that u(center dot, t) converges to a positive stationary solution which tends to 1 as x -> infinity locally uniformly in [0, infinity) when sigma > sigma*. In the case of sigma <= sigma* the solution u(center dot, t) converges to the ground state V(center dot - z) where V is the unique even positive solution of V" + f (V) = 0 subject to V(infinity) = 0 and z is the root of aV' (-z) - (1- a)V(-z) = b. The asymptotic behavior of the solutions is quite different from the homogeneous case b = 0. (C) 2015 Elsevier Ltd. All rights reserved.