Asymptotic Behavior of Solutions of a Reaction Diffusion Equation with Free Boundary Conditions

We study a nonlinear diffusion equation of the form ut = uxx + f (u) (x ε [g(t), h(t)]) with free boundary conditions g'(t) = -ux(t, g(t)) + α and h'(t) = -ux(t, h(t)) - α for some α > 0. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. When α = 0, the problem was recently investigated by Du and Lin (SIAM J Math Anal 42:377-405, 2010) and Du and Lou (J Euro Math Soc arXiv:1301.5373). In this paper we consider the case α > 0. In this case shrinking (i.e. h(t)-g(t) → 0) may happen, which is quite different from the case α = 0. Moreover, we show that, under certain conditions on f, shrinking is equivalent to vanishing (i.e. u → 0), both of them happen as t tends to some finite time. On the other hand, every bounded and positive time-global solution converges to a nonzero stationary solution as t → ∞. As applications, we consider monostable, bistable and combustion types of nonlinearities, and obtain a complete description on the asymptotic behavior of the solutions.