We consider the parabolic system u_t - Δu = u^r v^p, v_t - Δv = u^q v^s, x ∈ Ω, t ∈ (0,∞), complemented by the homogeneous Dirichlet boundary conditions and the initial conditions (u,v)(.,0) = (u₀,v₀) in Ω, where Ω is a smooth bounded domain in R^N and u₀,v₀ ∈ L^∞(Ω) are nonnegative functions. We find conditions on p, q, r, s guaranteeing a priori estimates of nonnegative classical global solutions. More precisely every such solution is bounded by a constant depending on suitable norm of the initial data. Our proofs are based on bootstrap in weighted Lebesgue spaces, universal estimates of auxiliary functions (see [3]) and estimates of the Dirichlet heat kernel (see [1]).
We also present results from [2] on the elliptic system - Δu = a(x) |x|^-κ v^q, - Δv = b(x) |x|^-λ u^p, x ∈ Ω, complemented by the homogeneous Dirichlet boundary condition, where Ω is a smooth bounded domain with the origin lying on its boundary, a,b ∈ L^∞(Ω) are nonnegative and not identically zero. Under some assumptions on p, q, κ, λ we prove a priori estimates and existence of positive very weak solutions of the system.