Sharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problems

Abstract

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. For monostable, bistable, and combustion types of nonlinearities, Du and Lou [``Spreading and vanishing in nonlinear diffusion problems with free boundaries,” J. Eur. Math. Soc. (JEMS), to appear] obtained a rather complete description of the long-time dynamical behavior of the problem and revealed sharp transition phenomena between spreading ($\lim_{t\to\infty}u(t,x)=1$) and vanishing ($\lim_{t\to\infty}u(t,x)=0$). They also determined the asymptotic spreading speed of the fronts by making use of semiwaves when spreading happens. In this paper, we give a much sharper estimate for the spreading speed of the fronts than that in the above-mentioned work of Du and Lou, and we describe how the solution approaches the semiwave when spreading happens.