A priori estimates of global solutions of superlinear parabolic systems - Július Pačuta (Electronic Journal of Qualitative Theory of Differential Equations, 2016)
We consider the parabolic system $ u_{t}-Δ u = u^{r}v^{p}$, $v_{t}-Δ v = u^{q}v^{s}$ in $Ω×(0,∞)$, complemented by the homogeneous Dirichlet boundary conditions and the initial conditions $(u,v)(·,0) = (u_{0},v_{0})$ in $Ω$, where $Ω $ is a smooth bounded domain in $ \mathbb{R}^{N} $ and $ u_{0},v_{0}\in 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 and estimates of the Dirichlet heat kernel.