Existence and asymptotic behavior of positive continuous solutions for some nonlinear parabolic systems

We show that positive continuous solutions exist for the nonlinear parabolic system Δu−∂u∂t=λp(x,t)g(v), Δv−∂v∂t=μq(x,t)f(u) on Rn×(0,∞),n≥3Rn×(0,∞),n≥3 with boundary conditions u(x,0)=φ(x),v(x,0)=ψ(x)u(x,0)=φ(x),v(x,0)=ψ(x) for nonnegative constants λλ and μμ, provided that the functions ff, gg are nonnegative continuous and nondecreasing on (0,∞)(0,∞). Also the potentials pp, qq are nonnegative and are required to satisfy some hypotheses related to the parabolic Kato class P∞(Rn)P∞(Rn).