Two-sided estimations for nonlinear parabolic systems with functionals (Q1105122)

Nonlinear parabolic systems: \(Pu=0\), \(R_ 1u=0\), \(R_ 2u=0\) are investigated, where Pu has the form: \(Pu=(P^ 1u,...,P^ mu)\) and \(P^ iu=u^ i_ t(t,x)-f^ i(t,x,u,u^ i_ x,u^ i_{xx},u(\cdot)).\) In the expression of \(P^ iu\), \((t,x)\in G\subset R^{n+1}\), \(u=(u^ 1,...,u^ m)\in U=\sqcup^ 1\times \sqcup^ 2\times...\times \sqcup^ m\), \(\sqcup^ i\) is a set of functions defined on \(G^ i\supset G\) \((i=1,...,m)\); with respect to u(\(\cdot)\) \(f^ i\) is a functional of Volterra type on \(\sqcup\) for fixed \((f,x)\in G^ i\). In an analogous manner \(R_ iu\) and \(R_ 2u\) are defined. On unbounded domains, two sided uniform estimates are proved for the solutions of differential inequalities corresponding to the mentioned systems, while the asymptotic growth of these solutions is given. The estimates provide uniqueness and stability.