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

我们得到了以下的结果。1）我们将还原化解决方案的存在和唯一性，将其转化为与吸收期的四线性弹性方程β（u） - div{a （x，μ u） + h （u） } μ f （x μ Ω）， u （x） = 0 （x μ μ Ω）。这里， β是R， h ： R → R-D1 d μ D1是一个连续函数， a ：Ω× R-D1d-D1 → R-D1d-D1 is a Caratheodory function satisfying an appropriate monotone and coercive condition， and f-L-D11-D1 （Ω）。由于在Ω的情况下，这个问题并不完全为人所知，我们得到了它的结果。2）我们为进化平等（E）提供了存在、光滑性和C-D11-D1近似的非线性描述（E） ： du/dt = Au + F （t， u）在Banach空间，并应用了研究Kuramoto-Sivashinsky equation的不对称行为的结果。Namely， if A y D2n y D2 → A， F y D2n y D2 → F and DF y D2n y D2 → DF，然后the inertial manifold for the approximate equation（E-D-2） ：du/dt = AイD2nイエD2u + FイイD2nイエD2 （t，u） ， converges to that of （E）.3） By using the above results about initial manifolds，我们为进化均衡（E）提出了一个不稳定的地方中心理论。Moreover，我们将尝试应用这些理论来研究非线性热等式的不对称性，D2 = Δu + F （u，u）在无边界域上。

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.