Asymptotic behavior of solutions of a reaction–diffusion equation with inhomogeneous Robin boundary condition and free boundary condition. (April 2016) - Liu, Xiaowei
Abstract: 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 = σ ϕ, there exists σ ∗ ⩾ 0 such that u ( ⋅, t ) converges to a positive stationary solution which tends to 1 as x → ∞ locally uniformly in [ 0, ∞ ) when σ > σ ∗ . In the case of σ ⩽ σ ∗ the solution u ( ⋅, t ) converges to the ground state V ( ⋅ − z ) where V is the unique even positive solution of V ″ + f ( V ) = 0 subject to V ( ∞ ) = 0 and z is the root of a V ′ ( − z ) − ( 1 − a ) V ( − z ) = b . The asymptotic behavior of the solutions is quite different from the homogeneous case b = 0 .