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

We study a nonlinear diffusion equation of the form with free boundary conditions and for some . Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. When , 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 . In this case shrinking (i.e. ) may happen, which is quite different from the case . Moreover, we show that, under certain conditions on , shrinking is equivalent to vanishing (i.e. ), both of them happen as tends to some finite time. On the other hand, every bounded and positive time-global solution converges to a nonzero stationary solution as . As applications, we consider monostable, bistable and combustion types of nonlinearities, and obtain a complete description on the asymptotic behavior of the solutions.