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We study nonlinear diffusion problems of the form uₜ=uₗₗ+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 (ₓu (t, x) =1) and vanishing (ₓu (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.
Du et al. (Wed,) studied this question.