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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.

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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[1] R. Abraham, J. E. Marsden and T. Ratiu,Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences75, Springer-Verlag, Berlin, 1988. · Zbl 0875.58002
[2] S. Agmon, and L. Nirenberg,Properties of solutions of ordinary differential equations in Banach Space, Comm. Pure Appl. Math.16, (1963), 121–239. · Zbl 0117.10001 · doi:10.1002/cpa.3160160204
[3] R. Benguria, H. Brézis and E. H. Lieb,The Thomas-Fermi-Von Weizsacker theory of atoms and molecules, Comm. Math. Phys.79 (1980), 167–180. · Zbl 0478.49035 · doi:10.1007/BF01942059
[4] M. Berger, P. Gauduchon et E. Mazet,Le spectre d’une variété riemannienne, Lecture Notes in Math.194, Springer-Verlag, Berlin, 1971. · Zbl 0223.53034
[5] M. F. Bidaut-Véron and L. Véron,Nonlinear elliptic equation on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math.106 (1991), 489–539. · Zbl 0755.35036 · doi:10.1007/BF01243922
[6] M. Bouhar and L. Veron,Integral representation of solutions of semilinear elliptic systems in cylinders and applications, Nonlinear Analysis23 (1994), 275–296. · Zbl 0817.35022 · doi:10.1016/0362-546X(94)90048-5
[7] H. Brezis and L. Nirenberg,Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36 (1983), 437–477. · Zbl 0541.35029 · doi:10.1002/cpa.3160360405
[8] A. Calsina, X. Mora and J. Solà-Morales,The dynamical approach to elliptic problems in cylindrical domains and a study of their parabolic singular limit, J. Differential Equations102 (1993), 244–304. · Zbl 0803.35048 · doi:10.1006/jdeq.1993.1030
[9] L. A. Caffarelli, B. Gidas and J. Spruck,Behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math.42 (1989), 271–297. · Zbl 0702.35085 · doi:10.1002/cpa.3160420304
[10] X. Y. Chen, H. Matano and L. Véron,Anisotropic singularities of solutions of nonlinear elliptic equations in R 2, J. Functional Analysis83 (1989), 50–97. · Zbl 0687.35020 · doi:10.1016/0022-1236(89)90031-1
[11] J. Fabbri et L. Véron,Equations elliptiques non linéaires singulières au bord dans des domaines non réguliers, C.R. Acad. Sci. Paris, série I320 (1995), 941–946. · Zbl 0823.35024
[12] S. Gallot, D. Hulin and J. Lafontaine,Riemannian Geometry, Springer-Verlag, Universitext, 1987.
[13] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[14] A. Gmira and L. Véron,Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J.64 (1991), 271–324. · Zbl 0766.35015 · doi:10.1215/S0012-7094-91-06414-8
[15] B. Guerch and L. Véron,Local properties of stationary solutions of some nonlinear singular Schrödinger equations, Revista Matematica Iberoamericana7 (1991), 65–114. · Zbl 0739.35016
[16] T. Kato,Schrödinger operators with singular potentials, Israel J. Math.13 (1973), 135–148. · Zbl 0246.35025 · doi:10.1007/BF02760233
[17] H. Matano,Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Tokyo30 (1983), 645–673. · Zbl 0545.35042
[18] R. Ossermann,On the inequality \(\Delta\)u(u), Pacific J. Math.7 (1957), 1641–1647. · Zbl 0083.09402
[19] R. Temam,Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences68, Springer-Verlag, New York, 1988. · Zbl 0662.35001
[20] L. Véron,Equations d’évolution semi-linéaires du second ordre dans L 1 Rev. Roum. Math. Pura Appl.27 (1982), 95–123.
[21] L. Véron,Singular solutions of some nonlinear elliptic equations, Nonlinear Anal.5 (1981), 225–242. · Zbl 0457.35031 · doi:10.1016/0362-546X(81)90028-6
[22] L. Véron,Comportement asymptotique des solutions d’équations elliptiques semi-linéaires dans R N, Ann. Mat. Pura Appl.127 (1981), 25–50. · Zbl 0467.35013 · doi:10.1007/BF01811717
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