ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35 Shirikyan, Armen, Moscow State University, Russia Asymptotic behaviour of solutions to second-order hyperbolic equations Let Ω ⊂ R_{x}^{n} be a bounded domain. Consider the equation

u_{tt} + g(u_{t}) − Δu = f(t, x), x ∈ Ω,

supplemented with the Dirichlet boundary condition. Here g(p) is a monotone increasing continuous function on R and f(t, x) is a time-bounded function with range in L^{2}(Ω). Sufficient conditions are found for the existence, uniqueness, and asymptotic stability of a time-bounded solution. Under the additional condition that f(t, x) is an almost periodic function in time, the almost periodicity of the constructed solution is established. Similar results are obtained for some other classes of hyperbolic equations. References: A. R. Shirikyan, Asymptotic behaviour of solutions to secondorder hyperbolic equations with non-linear damping term, Memorie Mat. Appl. , to appear. L. R. Volevich and A. R. Shirikyan, Bounded and almost periodic, in time, solutions of nonlinear hyperbolic equations of high order. Trans. Mosc. Math. Soc. 58 (1997), 89–135.