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Parametric Resonance in Wave Maps

In this note we concern with the wave maps from the Lorentzian manifold with the periodic in time metric into the Riemannian manifold, which belongs to the one-parameter family of Riemannian manifolds. That family contains as a special case the Poincare upper half-plane model. Our interest to such maps is motivated with some particular type of the Robertson-Walker spacetime arising in the cosmology. We show that small periodic in time perturbation of the Minkowski metric generates parametric resonance phenomenon. We prove that, the global in time solvability in the neighborhood of constant solutions is not a stable property of the wave maps.