Existence of Positive Stationary Solutions and Threshold Results for a Reaction–Diffusion System

The systemu1t−Δu1=u1u2−bu1,u2t−Δu2=au1inΩ×(0, T), whereΩ nis a smooth bounded domain, with homogeneous Dirichlet boundary conditionsu1=u2=0 on ∂Ω×(0, T) and initial conditionsu1(x, 0),u2(x, 0), is studied. First, it is proved that there is at least one positive stationary solution if 2 n<6. Second, it is proved that every positive stationary solution is a thereshold whenΩis a ball.