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=0x=0 and free boundary condition at x=h(t)x=h(t). We prove that, for the initial data u0=σϕu0=σϕ, there exists σ∗⩾0σ∗⩾0 such that u(⋅,t)u(⋅,t) converges to a positive stationary solution which tends to 11 as x→∞x→∞ locally uniformly in [0,∞)[0,∞) when σ>σ∗σ>σ∗. In the case of σ⩽σ∗σ⩽σ∗ the solution u(⋅,t)u(⋅,t) converges to the ground state V(⋅−z)V(⋅−z) where VV is the unique even positive solution of V″+f(V)=0V″+f(V)=0 subject to V(∞)=0V(∞)=0 and zz is the root of aV′(−z)−(1−a)V(−z)=baV′(−z)−(1−a)V(−z)=b. The asymptotic behavior of the solutions is quite different from the homogeneous case b=0b=0.