Abstract
IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u″+Lu+∥u∥ 2H u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast→∞. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u′) are relatively compact inD(L 1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball Bε=(u 0,u ′0 )εD(L 1/2)×H; ∥L1/2 u 0∥ 2 H +∥u 2′ <ɛ forɛ sufficiently small.
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Cazenave, T., Haraux, A. & Weissler, F.B. A class of nonlinear, completely integrable abstract wave equations. J Dyn Diff Equat 5, 129–154 (1993). https://doi.org/10.1007/BF01063738
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DOI: https://doi.org/10.1007/BF01063738