Nonlinear Diffusion Problems with Free Boundaries: Convergence, Transition Speed, and Zero Number Arguments

Abstract

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 $g'(t)=-\mu u_x(t, g(t))$, $h'(t)=-\mu u_x(t,h(t))$, 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 $\lim_{t\to\infty} h(t)/\ln t=\lim_{t\to\infty} -g(t)/\ln t=c_1$, and for combustion type of $f$, there exists a uniquely determi...