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 , 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).