## On some differential inclusions and their applications

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 2, Page 369--391.

Åukaszewicz, Grzegorz ; Ton, Bui An
On some differential inclusions and their applications
The existence of a solution of the evolution inclusion % $$u' + \partial Ï(t,u) + g(t,u)-F(u) \ni 0 \ \ \text{on}\ \ (0,T), u(0)=Î¾$$ % is established. For each t in [0,T], $Ï$ (t, $\cdot$) is a proper l.s.c. convex function from H to [0, $\infty$] \; and $F$ is an upper hemicontinuous set-valued mapping of $L^2(0,T;H)$ into its closed convex subsets. The time periodic problem % $$u' + \partial Ï(t,u)-F(u) \ni 0 \ \ \text{on}\ \ (0,T), \ \ u(0)=u(T)$$ % is studied. Applications to the heat equation with mixed boundary conditions and to the coupled Navier Stokes and heat equations with convection, dissipation and control terms in non-cylindrical domains are given.