ABSTRACT
The main object of this report is convergence to an equilibrium of weak global energy-bounded solutions of the wave equation u t t + c u t − Δ u + f ( x , u ) = 0 in ℝ + × Ω, u = 0 on ℝ + × ∂ Ω https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/equ_chapter_21_255_1.tif"/> where Ω is a bounded smooth domain of ℝ N, c > 0 and f : Ω × ℝ → ℝ ( x , s ) ↦ f ( x , s ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/equ_chapter_21_255_2.tif"/> is a function analytic in s satisfying some relevant growth conditions. Such a result has been obtained recently in a joint work with M.Jendoubi [8] as a culminating point of our attempts to understand in depth the fundamental theorem of L. Simon [17] on convergence of bounded trajectories of semilinear parabolic systems with analytic nonlinearities. The convergence problem for gradient or gradient-like systems has already a rather long history starting around 1960 with some investigations of S. Lojasiewicz on the steepest descent problem in differential geometry. Even in finite dimensions the question is more complicated that it may look at first sight. S. Lojasiewicz [12, 13] was led to the idea that analyticity of the potential is the key to convergence and, under this condition, established a topological property of stationary points which allows to derive convergence via a simple differential inequality. Much later, in 1982, J. Palis and W. de Melo [15] somehow confirmed the necessity of the Lojasiewicz condition which may fail even for C ∞ gradient systems. Very soon after, in 1983, L. Simon achieved his remarkable work on parabolic problems. In the mean time, The convergence property had been established for various gradient and quasi-gradient flows by B.Aulbach [1], H. Matano [14] and T.J. Zelenyak [18]. The dimension condition used in the proofs of Aulbach and Matano has been later exploited in a quite general context by Hale-Raugel [3] and allows conver gence proofs under standard regularity assumptions, in particular analyticity is not needed. However, apart from the case of thin domains, in higher space dimensions 256this hypothesis is not very natural. Also the counterexample of [15] suggests that we have to be careful even when dealing with the heat equation. And maybe there is something really deep hidden behind this strange-looking Lojasiewicz Lemma. The hyperbolic case studied below as well as a recent result of E. Feireisl and F. Simondon [2] for degenerate parabolic equations seem to confirm once more the interest of this simple intuitive idea: analyticity of the potential has a tendancy to kill small oscillations, even if the solution itself is not analytic!
