Global existence of solutions to a weakly coupled critical parabolic system in two-dimensional exterior domains

This paper is concerned with the existence and non-existence of global-in-time solutions of the initial-boundary value problem of the following weakly coupled parabolic system {partial derivative(t)u (x, t) - Delta u(x, t) = v(x, t)(p) partial derivative(t)v, (x, t) - Delta v(x, t) = u(x, t)(q), u(x, t) = 0, v(x, t) = 0, u(x, 0) = f (x) >= 0, v(x, 0) = g(x) >= 0, x is an element of Omega, where Omega is an exterior domain in R-2 having a smooth boundary partial derivative Omega. The given pair (p, q) with 0 < q <= p describes the effect of weakly coupled nonlinearity and (f, g) is given initial data. We determine the respective regions for existence and nonexistence of global-in-time solutions to the problem. In the case of whole space R-2 (without boundary condition) Escobedo-Herrero (1991) found the global existence for 2 +p-pq < 0 and non-existence for 2 +p-pq > 0. We emphasize that in the case of exterior domain, the critical case 2 + p -pq = 0 with (p, q) not equal(2,2) belongs to the global existence in the contrast of the case of whole space. This difference comes from the behavior of linear two-dimensional heat semigroup e(t Delta Omega) in exterior domains. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the existence and non-existence of global-in-time solutions of a weakly coupled parabolic system, providing regions for solutions in both whole space and exterior domains, highlighting critical cases and differences between them due to the behavior of linear two-dimensional heat semigroup.