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We consider positive solutions of cooperative parabolic Lotka-Volterrasystems 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, theseproblems possess both global solutions and solutions which blow up in finitetime. We show that any solution $(u,v)$ defined on the time interval $(0,T)$satisfies a universal estimate of the form $$u(x,t)+v(x,t)\leqC(1+t^{-1}+(T-t)^{-1}),$$ where $C$ does not depend on $x,t,u,v,T$. Inparticular, this bound guarantees global existence and boundedness forthreshold solutions lying on the borderline between blow-up and globalexistence. Moreover, this bound yields optimal blow-up rate estimates forsolutions which blow up in finite time. Our estimates are based on newLiouville-type theorems for the corresponding scaling invariant parabolicsystem and require an optimal restriction on the space dimension $n$: $n\leq5$.As an application we also prove the existence of time-periodic positivesolutions if the coefficients are time-periodic. Our approach can also be usedfor more general parabolic systems.

Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka-Volterra systems