Supersolutions for a class of nonlinear parabolic systems

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including{∂tu=Δu+vp,x∈Ω,t>0,∂tv=Δv+uq,x∈Ω,t>0,u=v=0,x∈∂Ω,t>0,(u(x,0),v(x,0))=(u0(x),v0(x)),x∈Ω, where p≥0p≥0, q≥0q≥0, Ω is a (possibly unbounded) smooth domain in RNRN and both u0u0 and v0v0 are nonnegative and locally integrable functions in Ω. The supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow-up rate of the solutions.