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Invariant manifolds for retarded semilinear wave equations. (English) Zbl 0815.34067
The authors study the second order equation \(u_{tt} (t) + Au(t) = R(u)\), \(u(0) = u_ 0\), \(u_ t (0) = u_ 1\), in a Hilbert space \(H\), where \(A\) is selfadjoint, positive definite and has a compact resolvent. The nonlinear operator \(R(u)\) maps \(D(A^{1/2})\) into \(H\) in a locally bounded and locally Lipschitz continuous fashion, and has a Fréchet derivative \(R'(u)\) locally Lipschitz continuous in \(u\). Finally, the global solvability condition \(\int^ t_ 0 (R(u(s))\), \(u_ t(s)) ds \leq C(u_ 0, u_ 1) < \infty\) holds for every \(u(\cdot) \in C([0,b); U) \cap C^ 1 ([0,b); H)\) for every \(b < \infty\). Using a nonlinear variation-of-constants formula, the authors show existence of an invariant manifold for the equation \(u_{tt} (t) + Au(t) = R(u) + M(u^ t)\), where \(M\) is a nonlinear operator acting on sections \(u^ t(\theta) = u(t + \theta)\), \(\theta \in [-h,0])\). This invariant manifold is inertial. There are applications to the sine-Gordon and the Klein-Gordon equation under this type of retarded perturbation.

34K30 Functional-differential equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
34C30 Manifolds of solutions of ODE (MSC2000)
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