  This paper deals with a semilinear parabolic system with free boundary in one
space dimension. We suppose that unknown functions $u$ and $v$ undergo
nonlinear reactions $u^q$ and $v^p$, and exist initially in a interval $\{0\leq
x\leq s(0)\}$, but expand to the right with spreading front $\{x=s(t)\}$, with
$s(t)$ evolving according to the free boundary condition $s'(t)=-\mu (u_x+\rho
v_x)$, where $p,\, q,\, \mu, \,\rho$ are given positive constants. The main
purpose of this paper is to understand the existence, uniqueness, regularity
and long time behavior of positive solution or maximal positive solution.
Firstly, we prove that this problem has a unique positive solution $(u,v,s)$
defined in the maximal existence interval $[0,T_{\max})$ when $p,\,q\geq 1$,
while it has a unique maximal positive solution $(u,v,s)$ defined in the
maximal existence interval $[0,T_{\max})$ when $p<1$ or $q<1$. Moreover,
$(u,v,s)$ and $T_{\max}$ have property that either (i) $T_{\max}=+\infty$, or
(ii) $T_{\max}<+\infty$ and
  $$ \limsup_{T\nearrow
T_{\max}}\|u,\,v\|_{L^{\infty}([0,T]\times[0,s(t)])}=+\infty.$$ Then we study
the regularity of $(u,v)$ and $s$. At last, we discuss the global existence
($T_{\max}=+\infty$), finite time blow-up ($T_{\max}<+\infty$), and long time
behavior of bounded global solution.
