Inertial Manifolds and Spectral Gap Properties for Wave Equations with Weak and Strong Dissipation
Sufficient conditions for the existence of an inertial manifold for the equation \(u_{tt}-2\gamma _{s} \varDelta u_t +2\gamma _{w} u_t - \varDelta u = f(u)\) , \(\gamma _{s} > 0\) , \(\gamma _{w} \ge 0\) are found. The nonlinear function \(f\) is suppo
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Inertial Manifolds and Spectral Gap Properties for Wave Equations with Weak and Strong Dissipation Natalia Chalkina
Abstract Sufficient conditions for the existence of an inertial manifold for the equation u tt − 2γs Δu t + 2γw u t − Δu = f (u), γs > 0, γw ≥ 0 are found. The nonlinear function f is supposed to satisfy Lipschitz property. The proof is based on construction of a new inner product in the phase space in which the conditions of a general theorem on the existence of inertial manifolds for an abstract differential equation in a Hilbert space are satisfied.
14.1 Introduction In the theory of nonlinear evolution partial differential equations, great attention is paid to long-time behavior of dynamic systems. Some way of such description relates with notion of an inertial manifold (see [5, 6, 9]). Let us consider an initial-value problem for an abstract differential equation in a Hilbert space, d y+Ay = F(y), y∈H, dt y t=0 = y0 ∈ H .
(14.1) (14.2)
Here A is a linear operator and F is a nonlinear operator. Suppose problem (14.1), (14.2) has a unique solution y for any y0 ∈ H . Hence, this problem generates a continuous semigroup {S(t) | t ≥ 0}, acting in the space H by the formula S(t)y0 = y(t) ∈ H . Definition 14.1 A Lipschitz finite dimensional manifold M ⊂ H is an inertial manifold for the semigroup S(t) if it is invariant (i.e., S(t)M = M , ∀t ≥ 0) and it satisfies the following asymptotic completeness property: N. Chalkina (B) Department of Mechanics and Mathematics, Nikulinskaya, 15-2, Moscow 119602, Russia e-mail: [email protected] M. Z. Zgurovsky and V. A. Sadovnichiy (eds.), Continuous and Distributed Systems, Solid Mechanics and Its Applications 211, DOI: 10.1007/978-3-319-03146-0_14, © Springer International Publishing Switzerland 2014
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∀y0 ∈ H ∃ y˜0 ∈ M such that S(t)y0 − S(t) y˜0 H ≤ q(y0 H )e−ct , t ≥ 0, where the positive constant c and the monotonic function q are independent of y0 . Inertial manifolds enable one to reduce the study of the behavior of an infinite-dimensional dynamical system to the investigation of this problem for some finite-dimensional dynamical system generated by original system on an inertial manifold. For the abstract equation of the form (14.1), there are known sufficient conditions under which there is an inertial manifold in the Hilbert space H (see [3]). Let us present these conditions. Let A be a linear closed (possibly unbounded) operator with dense domain D(A) in H and let the spectrum σ (A) of A be disjoint from the strip {m < ζ < M}, where M ≥ 0, M > m. Denote by P the orthogonal projection to the invariant subspace of A corresponding to the part of the spectrum σ ∩ { ζ ≤ m} and write Q = Id − P. Assume that the space P(H ) is finite-dimensional. Theorem 14.1 Let the space H be equiped with an inner product in such a way that the space P(H ) and Q(H ) are orthogonal and the following relations hold: (Ay, y) ≤ m|y|2 (Ay, y) ≥ M|y|2
∀y ∈ P(H ), ∀y ∈ Q(H ) ∩ D(A).
(14.3)
Moreover, let F(y) be a nonlinear function such that F(0
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