Existence and nonexistence of global solutions of some non-local degenerate parabolic systems

Abstract

This paper establishes a new criterion for global existence and nonexistence of positive solutions of the non-local degenerate parabolic system u t a m p ; = v p ( Δ u + a ∫ Ω v d x ) , v t a m p ; = u q ( Δ v + b ∫ Ω u d x ) , x ∈ Ω , t > 0 , \begin{align*} u_t&=v^p\left (\Delta u+a\int _\Omega v dx\right ), v_t&=u^q\left (\Delta v+b\int _\Omega u dx\right ),\quad x\in \Omega , t>0, \end{align*} with homogeneous Dirichlet boundary conditions, where Ω ⊂ R N \Omega \subset \mathbb {R}^N is a bounded domain with a smooth boundary ∂ Ω \partial \Omega and p , q , a , b p, q, a, b are positive constants. For all initial data, it is proved that there exists a global positive solution iff ∫ Ω φ ( x ) d x ≤ 1 / a b \int _\Omega \varphi (x) dx\leq 1/\sqrt {ab} , where φ ( x ) \varphi (x) is the unique positive solution of the linear elliptic problem − Δ φ ( x ) = 1 , x ∈ Ω ; φ ( x ) = 0 , x ∈ ∂ Ω . -\Delta \varphi (x)=1, x\in \Omega ; \varphi (x)=0, x\in \partial \Omega .