INVARIANT SETS AND THE HUKUHARA-KNESER PROPERTY FOR PARABOLIC SYSTEMS

In this paper we are concerned with the nonlinear boundary value problem forparabolic system(Lu=f(x,t,u,▽u),x∈Ω,0

INVARIANT SETS AND THE HUKUHARA-KNESER PROPERTY FOR PARABOLIC SYSTEMS

In this paper we are concerned with the nonlinear boundary value problem for parabolic system $$\\left\{ {\begin{array}{*{20}{c}} {Lu = f(x,t,u,Du),x \in \Omega ,0 < t \le T,}\{Bu = g(x,t,u),x \in \partial \Omega ,0 \le t \le T,}\{u(x,0) = h(x),x \in \bar \Omega } \end{array}\begin{array}{*{20}{c}} {}&{(1)} \end{array}} \right.\]$$ where $\Lu = ({L_1}{u_1}, \cdots ,{L_N}{u_N})\]$ with $\{L_k}\]$ the second order uniformly parabolic operators which may be different from one another, and $\Bu = ({B_1}{u_1}, \cdots ,{B_N}{u_N})\]$ wther the Dirichlet boundary operators or the regular oblique derivative ones. We have proved that a certain form of convex set is invariant for (1), that there exist solutions to (1) if $\f = f(x,t,u,p)\]$ has an almost quadratic growth in p, and that the set of solutions possesses the Hukuhara- Kneser property.