NONLINEAR DIFFUSION PROBLEMS WITH FREE BOUNDARIES: CONVERGENCE, TRANSITION SPEED, AND ZERO NUMBER ARGUMENTS.

This paper continues the investigation of [Du and Lou, J. Eur. Math. Soc. (JEMS), arXiv:1301.5373, 2013], where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form u t = u xx + f(u) for x over a varying interval (g(t), h(t)) was examined. Here x = g(t) and x = h(t) are free boundaries evolving according to g'(t) = −μu x (t, g(t)), h'(t) = −μu x (t, h(t)), and u(t, g(t)) = u(t, h(t)) = 0. We answer several intriguing questions left open in that investigation. First we prove the conjectured convergence result for the general case that f is C 1 and f(0) = 0. Second, for bistable and combustion types of f, we determine the asymptotic propagation speed of h(t) and g(t) in the transition case. More presicely, we show that when the transition case happens, for bistable type of f there exists a uniquely determined c 1 > 0 such that lim t→∞ h(t)/ ln t = lim t→∞ −g(t)/ ln t = c 1 , and for combustion type of f, there exists a uniquely determined c 2 > 0 such that lim t→∞ h(t)/ √ t = lim t→∞ −g(t)/ √ t = c 2 . Our approach is based on the zero number arguments of Matano and Angenent and on the construction of delicate upper and lower solutions. [ABSTRACT FROM AUTHOR]

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