Discrete approximations and stability properties of inertial manifolds for dissipative differential equations

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We have gotten the following results.1) We proved the existence and uniqueness of renormalized solution to the degenerate quasilinear elliptic equation with the convection termβ(u) - div{a (x, ∇u) + h (u) } ∋ f (x ∈ Ω), u (x) = 0 (x ∈ ∂Ω) .Here, β is a maximal monotone graph in R, h : R → RィイD1dィエD1 is a continuous function, a : Ω×RィイD1dィエD1 → RィイD1dィエD1 is a Caratheodory function satisfying an appropriate monotone and coercive condition, and f ∈ LィイD11ィエD1 (Ω) . Since this problem is not sufficiently known in case Ω is unbounded, we gave a result of it.2) We proved the existence, smoothness and CィイD11ィエD1-approximation of the inertial manifolds for the evolution equation (E) : du/dt = Au + F (t, u) in Banach spaces, and applied the results to studying the asymptotic behavior of Kuramoto-Sivashinsky equation. Namely, if AィイD2nィエD2 → A, FィイD2nィエD2 → F and DFィイD2nィエD2 → DF, then the inertial manifold for the approximate equation (EィイD2nィエD2) : du/dt = AィイD2nィエD2u + FィイD2nィエD2 (t,u) , converges to that of (E).3) By using the above results about initial manifolds, we improved the theorems of the local center unstable manifolds for the evolution equation (E) . Moreover, we try to apply these theorems to the study of the asymptotics of the nonlinear heat equation UィイD2tィエD2 = Δu + F (u, ∇u) on unbounded domains.