$ u_t-u_{xx}=f(t,u),   \      x \in R,  t>0,   $

and  investigate solutions in $C_0(R)$, that  is, solutions that decay
to zero as $|x|$ approaches  infinity.  The nonlinearity is assumed to
be  a $C^1$ function which  is  $\tau$-periodic  in $t$ and  satisfies
$f(t,0)=0$ and $f_u(t,0)<0$ ($t\in  R$). Our two main results describe
the structure of  $\tau$-periodic solutions and asymptotic behavior of
general solution with compact trajectories:

(1) Each nonzero $\tau$-periodic solution is of definite sign and is even
in $x$ about  its unique peak (which is independent of $t$) . Moreover,
up to shift in space, $C_0(R)$ contains at  most one $\tau$-periodic
solution of a given sign.

(2) Each solution with trajectory relatively compact
  in $C_0(R)$ converges to a single $\tau$-periodic solution.
 
In the proofs of  these statements we make extensive  use of nodal and
symmetry properties of solutions.  In particular, nodal properties are
crucial ingredients  in our study of the   linearization at a periodic
solution  and  in the description of  center   and stable manifolds of
periodic  solutions.   The  paper also contains   a section discussing
various sufficient conditions for the existence of nontrivial periodic
solutions.