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

TL;DR

This paper establishes universal bounds, blow-up rate estimates, and existence of periodic solutions for cooperative parabolic Lotka-Volterra systems, using Liouville-type theorems and applicable in dimensions up to five.

Contribution

It introduces new universal estimates and Liouville theorems for these systems, enabling analysis of global solutions, blow-up behavior, and periodic solutions.

Findings

Universal estimate: u+v ≤ C(1 + t^{-1} + (T - t)^{-1})

Optimal blow-up rate estimates for finite-time blow-up solutions

Existence of time-periodic positive solutions under periodic coefficients

Abstract

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.