ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35L Shi, Yuming, ENS-Cachan, France and Fudan University, China A Foias-Saut type of expansion for a dissipative wave equation Consider in a bounded domain the nonlinear wave equation u_{tt} − △u + au_{t} + bu = f(u) complemented with the boundary condition u = 0, where a > 0, f(u) = O(u^{2}) as u → 0. If the spectrum σ(L) of the generator L of the linear semigroup associated to this problem satisfies a nonresonant condition, then every small solution u(t) admits a Foias-Saut type of expansion ∑_{μ∈Π(L)} e^{−μt}w_{μ}(x), where Π(L) is the additive semigroup generated by σ(L), such that ∀Λ > 0, u(t) − ∑_{μ∈Π(L),ℜμ≤Λ} e^{−μt}w_{μ}(x) = o(e^{−Λt}), as t → +∞, and ∀μ ∈ σ(L), {w_{μ}(x), μw_{μ}(x)} belongs to the eigenspace of μ.