Existence of positive solutions for some nonlinear parabolic systems in exteriors domains

We prove some existence results of positive continuous solutions to the semilinear parabolic system Δ u − ∂ u ∂ t = λ p ( x , t ) g ( v ) , Δ v − ∂ v ∂ t = μ q ( x , t ) f ( u ) in an unbounded domain D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f, g are nonnegative continuous monotone on ( 0 , ∞ ) and the potentials p, q are nonnegative and satisfy some hypotheses related to the parabolic Kato class J ∞ ( D ) .