1999 Fiscal Year Final Research Report Summary
Discrete approximations and stability properties of inertial manifolds for dissipative differential equations
Project/Area Number |
10640217
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Single-year Grants |
Section | 一般 |
Research Field |
Global analysis
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Research Institution | Waseda University |
Principal Investigator |
KOBAYASHI Kazuo Waseda University, School of Education, Professor, 教育学部, 教授 (30139589)
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Project Period (FY) |
1999
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Keywords | inertial manifold / dissipative PDE / evolution equation / finite difference |
Research Abstract |
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 term β(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.
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Research Products
(8 results)