  This paper continues the investigation of Du and Lou (J. European Math Soc,
to appear), 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 the paper of Du and Lou.First we prove the conjectured
convergence result in the paper of Du and Lou 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 determined $c_2>0$ such that
$\lim_{t\to\infty} h(t)/\sqrt t=\lim_{t\to\infty} -g(t)/\sqrt t=c_2$. Our
approach is based on the zero number arguments of Matano and Angenent, and on
the construction of delicate upper and lower solutions.
