Existence and Non-Existence of Finite-Dimensional Globally Attracting Invariant Manifolds in Semilinear Damped Wave Equations

Inertial manifolds of damped semilinear wave equations

“…The desired attracting invariant manifolds wili then be sought for as graphs of mappings giving the fast components as a function of the slow ones. In [8], this was done for e ^ 0 by working on the first-order system (1.9) as a perturbation of the one corresponding to F = 0, and decomposing the variable according to the spectrum of A c . Here, we adopt a somewhat different approach, the différences lying both in the way of decomposing the variable and of choosing the « unperturbed » linear system.…”

“…In the following, the orthogonal projections of E onto E t (i = l> 2) will be denoted as P n and the corresponding parts of A will be denoted as A t . Parallel to this décomposition of E, the spaces E a /a = -5 décompose also orthogonally as E a = Ef © E% 9 where Ef -P t E a . The spaces E l7 El /2 , El consist all in the same n-dimensional vector space provided with different but equivalent inner products.…”

“…For E = 0, the state variable is thus decomposed into the two components u x and w 2 , which are to be considered as taking values respectively in E x 2 and E 2 2 . For e ^ 0, the state variable will be considered as decomposed into the four components u x , u 2 ) and in fact to C\W l9 El) 9 where W 1 is a certain bounded domain in E x . Here, this result will be accompanied by an extension to small non-zero values of s. Specifically, it will be shown that, under the same gap condition there exists an ê>0 such that, for e e (0, ê), the global attractor of the hyperbolic system (1.5) is also contained in a local invariant manifold of class C 1 and dimension n, M £ , which is described by a set of relations giving u 2 , ü 1 , and ü 2 as functions of u x :…”

“…by Xavier MORA Q), Joan SOLÀ-MORALES ( 2 ) In the present communication we give an account of the results obtained by the authors in [8], [9], and [10], with some slight improvements in what refers to [9]. We are concerned with the qualitative dynamics of a onedimensional semilinear damped wave équation and its dependence with respect to the coefficient of the second-order time derivative, hereafter denoted by e 2 , this parameter being considered to vary right up to the limiting value e 2 = 0, in which case the équation turns into a semilinear diffusion one.…”

“…In the finitedimensional case, our result is included essentially in that of Hartman (1960) [6, Theorem (I)], which instead of our condition (3.2) requires only that Lbea contraction. For the proofs of the following statements, the reader is referred to [9] …”