Invariant Manifolds for Retarded Semilinear Wave Equations

Dynamics in Infinite Dimensions

“…The method of proof of Theorem 9.0.10, developed in Magalhiies [131]' was applied to retarded semilinear wave equations by Taboada and You [196], in situations that include the Sine-Gordon equation with a retarded perturbation Then, with the methods developed in [131]' Taboada and You [196] proved the Theorem 9.0.12 below. …”

“…on [-h [196) p.347 for the definition) and the Lipschitz constant L(g) of the perturbation 9 are sufficiently small.…”

“…3) with boundary and initial conditions (9.4) and(9.5). ([196]) For suitable small h > 0, the retarded evolution equation(9.9) dw -dt = Aw(t) + f(w(t)) + g(Wt) t ~ 0associated to the retarded Sine-Gordon equation(9.3)-(9.4) has a globally invariant manifoldMg = {¢(.,~,f + Kg) : ~ E U x H},where Kg is a mapping from U x H into itself, determined by g as in Theorem 9.0.12.We also mentioned above that Theorem 9.0.12 can be applied to the dissipative Klein-Gordon equation with a retarded perturbation and boundary conditions (9.6). A typical example of such a mapping M is where ° :S hI < ... < hm = h and the J-Li are the hysteresis functions appearing in the theory of shape memory alloys.…”

A Contribution to Nonautonomous Inertial Manifolds

“…by autonomous evolution equations (1.1), two-parameter semiflows as generated by nonautonomous evolution equationṡ (1.2) or random dynamical systems as generated by random evolution equations. In [10,11], the notion of inertial manifolds was extended to nonautonomous dynamical systems: Obviously, this notion is a natural extension of the above-mentioned notion for autonomous evolution equations and it is also an extension of the notion of inertial manifolds for nonautonomous differential equations [8,13,16], retarded parabolic differential equations [2,15], or differential equations with random or stochastic perturbations [1,[3][4][5]7].…”

Cone invariance and squeezing properties for inertial manifolds for nonautonomous evolution equations

“…The notion of inertial manifolds mentioned above is translated and extended to more general classes of differential equations like nonautonomous differential equations [GV97,WF97,LL99], retarded parabolic differential equations [TY94,BdMCR98], or differential equations with random or stochastic perturbations [Chu95,BF95,CL99,CS01,DLS01]. Here spectral gap conditions are also utilized, where the special form of this condition may depend on the assumptions on the nonlinearity and, in particular, on the delay.…”