  We consider positive solutions of cooperative parabolic Lotka-Volterra
systems with equal diffusion coefficients, in bounded and unbounded domains.
The systems are complemented by the Dirichlet or Neumann boundary conditions.
Under suitable assumptions on the coefficients of the reaction terms, these
problems possess both global solutions and solutions which blow up in finite
time. We show that any solution $(u,v)$ defined on the time interval $(0,T)$
satisfies a universal estimate of the form $$u(x,t)+v(x,t)\leq
C(1+t^{-1}+(T-t)^{-1}),$$ where $C$ does not depend on $x,t,u,v,T$. In
particular, this bound guarantees global existence and boundedness for
threshold solutions lying on the borderline between blow-up and global
existence. Moreover, this bound yields optimal blow-up rate estimates for
solutions which blow up in finite time. Our estimates are based on new
Liouville-type theorems for the corresponding scaling invariant parabolic
system and require an optimal restriction on the space dimension $n$: $n\leq5$.
As an application we also prove the existence of time-periodic positive
solutions if the coefficients are time-periodic. Our approach can also be used
for more general parabolic systems.
