Double thresholds for blowup and global existence phenomena occur in heat transfer

We consider the following problem { u t = d 1 ∆ u − a ( x ) · ∇ u + f ( u , v ) , x ∈ Ω , t > 0 , v t = d 2 ∆ v − b ( x ) · ∇ v + g ( u , v ) , x ∈ Ω , t > 0 , u ( x , 0 )= u 0 ( x ) , v ( x , 0 )= v 0 ( x ) , x ∈ Ω subject to Dirichlet(or Neumann) boundary condition. Here Ω ⊂ R N ( N ≥ 1 ) is a bounded smooth domain. Besides some results on blowup and global existence of the solution, we find more interesting results as follows. 1. There exist double thresholds for blowup and global existence of the solution. Under certain conditions, if f ( u , v ) = f 1 ( v ) g 1 ( u ) and g ( u , v ) = f 2 ( v ) g 2 ( u ) , then the first watershed is ∫ c 1 + ∞ du g 1 ( u ) = + ∞ and ∫ c 2 + ∞ dv f 2 ( v ) = + ∞ , and the second watershed is ∫ c ̵̃ 1 + ∞ dU f ̵̃ ( F ̵̃ − 1 ( K G ̵̃ ( U ))) = + ∞ and ∫ c ̵̃ 2 + ∞ dV g ̵̃ ( G ̵̃ − 1 ( 1 ϵ F ̵̃ ( V ))) = + ∞ . Here f ̵̃ , g ̵̃ , F ̵̃ and G ̵̃ will be defined in Section 2.2. 2. If there exist nonnegative smooth functions h ( u ), l ( v ) and H ( s ) such that f ( u , v ) h ′ ( u ) l ( v ) + g ( u , v ) h ( u ) l ′ ( v ) = H [ h ( u ) l ( v ) ] ≥ 0 , then the watershed for blowup in finite time and global existence of the solution is