On some differential inclusions and their applications

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

Łukaszewicz, Grzegorz ; Ton, Bui An
On some differential inclusions and their applications
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The existence of a solution of the evolution inclusion % $$u' + \partial φ(t,u) + g(t,u)-F(u) \ni 0 \ \ \text{on}\ \ (0,T), u(0)=ξ$$ % is established. For each t in [0,T], $φ$ (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 % $$u' + \partial φ(t,u)-F(u) \ni 0 \ \ \text{on}\ \ (0,T), \ \ u(0)=u(T)$$ % 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.

Mathematics Subject Classification (1991): Primary 47H04, 35K55; Secondary 35K20, 35R70
Mathematical Reviews Number: MR1317465

Received: 1993-08-05