  We study a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)\ (x\in
[g(t),h(t)])$ with free boundary conditions $g'(t)=-u_x(t,g(t))+\alpha$ and
$h'(t)=-u_x(t,g(t))-\alpha$ for some $\alpha>0$. Such problems may be used to
describe the spreading of a biological or chemical species, with the free
boundaries representing the expanding fronts. When $\alpha=0$, the problem was
recently investigated by \cite{DuLin, DuLou}. In this paper we consider the
case $\alpha>0$. In this case shrinking (i.e. $h(t)-g(t)\to 0$) may happen,
which is quite different from the case $\alpha=0$. Moreover, we show that,
under certain conditions on $f$, shrinking is equivalent to vanishing (i.e.
$u\to 0$), both of them happen as $t$ tends to some finite time. On the other
hand, every bounded and positive time-global solution converges to a nonzero
stationary solution as $t\to \infty$. As applications, we consider monostable
and bistable types of nonlinearities, and obtain a complete description on the
asymptotic behavior of the solutions.
