Invariant sets and the Hukuhara-Kneser property for parabolic systems (Q801479)

This paper is concerned with the nonlinear boundary value problem for the parabolic system \((P)\quad Lu=f(x,t,u,\nabla u),\quad x\in \Omega,\quad 0<t\leq T,\quad Bu=g(x,t,u),\quad x\in \partial \Omega,\quad 0\leq t\leq T,\quad u(x,0)=h(x),\quad x\in {\bar \Omega}\) where \(Lu=(L_ 1u_ 1,L_ 2u_ 2,...,L_ Nu_ N)\) with \(L_ k's\) second order uniformly parabolic operators and \(Bu=(B_ 1u_ 1,B_ 2u_ 2,...,B_ Nu_ N)\) with \(B_ k\) either the Dirichlet boundary operator or the regular oblique derivative one. Invariant sets, existence, uniqueness and the so- called Hukuhara-Kneser property are discussed. In problem (P), not only may \(L_ k\) be different from one another, but the boundary conditions be nonlinear as well. In proving existence of solutions, f(x,t,u,p) is assumed to have an almost quadratic growth in p, rather than to be bounded.