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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.

##### MSC:
 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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