Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire |
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Authors: | Jean-René Licois |
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Institution: | (1) Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200 Tours, France |
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Abstract: | LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar
P-B. We prove that any non-negative solutionu ofu
tt+Δgu=f(u) onM x ℝ+ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ
g
Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C
2(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that
for every λ the map (u(0,·),u
t(0,·))→(u(t,·), u
t(t,·)) defines a dynamical system onW
1/2(M)⊂C(M)×L
2(M) which possesses a compact maximal attractor.
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Keywords: | |
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