Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with arbitrary functionals (Q1176901)

The paper looks for a new object for which an analogue of the maximum principle for solutions of parabolic equations is valid. The case of noncylindrical domains and nonlocal parabolic inequalities of the type \[ u^ i_ t(x,t)\leq f^ i(x,t,u(x,t), u^ i_ x(x,t), u^ i_{xx}(x,t);[u])\hbox { for a.e. } (x,t) \] \(i=1,\ldots,m\); \(u=(u^ 1,\ldots,u^ m)\), with some additional nonlocal assumptions is discussed. Here \(f^ i(\cdots;[u])\) are functionals with respect to \(u\).