The singular limit dynamics of semilinear damped wave equations.(English)Zbl 0699.35177

The authors consider the following abstract problem: $\epsilon \ddot u+\dot u+Au=Fu,\quad u(0)=u_ 0,\quad \epsilon \dot u(0)=\epsilon v_ 0$ in a Hilbert space E, where A is a linear operator while F is nonlinear. This problem includes, e.g., the case $$Au=-u_{xx}$$, with some usual boundary conditions, and $$Fu=f(,u( ))$$ (i.e. a semilinear damped wave equation). It is shown that there exist an integer n and an $${\bar \epsilon}>0$$ such that, for every $$\epsilon\in [0,{\bar \epsilon})$$, the global attractor of the corresponding dynamical system is contained in an invariant manifold of class $$C^ 1$$ and dimension n and that for $$\epsilon$$ $$\to 0$$ both this manifold and the vector field on it converge in the $$C^ 1$$ topology towards the ones corresponding to $$\epsilon =0$$.
Reviewer: G.Moroşanu

MSC:

 35L70 Second-order nonlinear hyperbolic equations 35K57 Reaction-diffusion equations
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References:

 [1] Angenent, S, The Morse-Smale property for a semilinear parabolic equation, J. differential equations, 62, 427-442, (1986) · Zbl 0581.58026 [2] Babin, A.V; Vishik, M.I, Regular attractors of semigroups and evolution equations, J. math. pures appl., 62, 441-491, (1983), (9) · Zbl 0565.47045 [3] Babin, V; Vishik, M.I, Uniform asymptotics of the solutions of singularly perturbed evolution equations, Uspekhi mat. nauk, 42, No. 5, 231-232, (1987), [in Russian] [4] Chow, S.N; Lu, K, Invariant manifolds for flows in Banach spaces, J. differential equations, 74, 285-317, (1988) · Zbl 0691.58034 [5] Constantin, P; Foiaş, C; Nikolaenko, B; Sell, G; Temam, R; Constantin, P; Foiaş, C; Nikolaenko, B; Sell, G; Temam, R, Intégral manifolds and inertial manifolds for dissipative partial differential equations, C. R. acad. sci. Paris Sér. I math., 302, 375-378, (1987), preprint [6] Foiaş, C; Sell, G.R; Temam, R; Foiaş, C; Sell, G.R; Temam, R, Invariant manifolds for nonlinear evolutionary equations, C. R. acad. sci. Paris Sér. I math., IMA preprint series, 234, 139-141, (1986) · Zbl 0591.35062 [7] Hale, J.K, Asymptotic behavior and dynamics in infinite dimensions, Res. notes in math., 132, 1-42, (1985) · Zbl 0653.35006 [8] Hale, J.K; Raugel, G, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. differential equations, 73, 197-214, (1988) · Zbl 0666.35012 [9] Henry, D, Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001 [10] Henry, D.B, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. differential equations, 59, 165-205, (1985) · Zbl 0572.58012 [11] Irwin, M.C, Smooth dynamical systems, (1980), Academic Press San Diego, CA · Zbl 0465.58001 [12] Mallet-Paret, J; Sell, G.R; Mallet-Paret, J; Sell, G.R, Inertial manifolds for reaction-diffusion equations in higher space dimensions, Lecture notes in math., IMA preprint series, 331, 94-107, (1987) [13] Mañé, R, Reduction of semilinear parabolic equations to finite dimensional C1 flows, Lecture notes in math., 597, 361-378, (1977) [14] Mora, X, Finite-dimensional attracting manifolds in reaction-diffusion equations, Contemp. math., 17, 353-360, (1983) · Zbl 0525.35046 [15] Mora, X, Finite-dimensional attracting invariant manifolds for damped semilinear wave equations, Res. notes in math., 155, 172-183, (1987) · Zbl 0642.35061 [16] Mora, X; Solà-Morales, J, Existence and non-existence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations, (), 187-210 · Zbl 0642.35062 [17] Palis, J; Smale, S, Structural stability theorems, (), 223-231 · Zbl 0214.50702 [18] Vanderbauwhede, A; Van Gils, S.A, Center manifolds and contractions on a scale of Banach spaces, J. funct. anal., 72, 209-224, (1987) · Zbl 0621.47050
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