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Asymptotic estimates for a conservative dynamical system related to a nonlinear elliptic equation. (Asymptotiques d’un système dynamique conservatif associé à une équation elliptique non linéaire.) (French) Zbl 0844.58072
Summary: Let \(M\) be a compact Riemannian manifold, \(h\) an odd function such that \(h(r)/r\) is nondecreasing with limit 0 at 0. Let \(f(r) = h(r) - \lambda r\) and assume there exist nonnegative constants \(A\) and \(B\) and a real \(p > 1\) such that \(f(r) > Ar^p - B\).
We prove that any nonnegative solution \(u\) of \(u_{tt} + \Delta_g u = f(u)\) on \(M \times \mathbb{R}^+\) satisfying Dirichlet or Neumann boundary conditions on \(\partial M\) converges to a (stationary) solution of \(\Delta_g \Phi = f(\Phi)\) on \(M\) with exponential decay of \(|u - \Phi |_{C^2 (M)}\). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small \(\lambda\); further, we show that for every \(\lambda\) the map \((u(0, \cdot)\), \(u_t (0, \cdot)) \mapsto (u(t, \cdot)\), \(u_t (t, \cdot))\) defines a dynamical system on \(W^{1/2} (M) \cap C (M) \times L^2 (M)\) which possesses a compact maximal attractor.

MSC:
58J05 Elliptic equations on manifolds, general theory
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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