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

Abstract 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 { ∂ t u ( x , t ) − Δ u ( x , t ) = v ( x , t ) p , ( x , t ) ∈ Ω × ( 0 , ∞ ) , ∂ t v ( x , t ) − Δ v ( x , t ) = u ( x , t ) q , ( x , t ) ∈ Ω × ( 0 , ∞ ) , u ( x , t ) = 0 , v ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , ∞ ) , u ( x , 0 ) = f ( x ) ≥ 0 , v ( x , 0 ) = g ( x ) ≥ 0 , x ∈ Ω , where Ω is an exterior domain in R 2 having a smooth boundary ∂Ω. 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 − p q 0 and non-existence for 2 + p − p q ≥ 0 . We emphasize that in the case of exterior domain, the critical case 2 + p − p q = 0 with ( p , q ) ≠ ( 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 Δ Ω in exterior domains.