ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: Kiguradze, Tariel, Tbilisi I. Javakhishvili State University, Georgia On periodic solutions of nonlinear second order hyperbolic equations

u_{tt} − u_{xx} = g_{0}(u) + g_{1}(u)u_{t} + g_{2}(u)u_{x} + g(t, x, u)  (1)

is considered, where g_{i} : R → R (i = 0, 1, 2) and g : R^{3} → R are continuously differentiable functions and g(t + ω_{1}, x, z) ≡ g(t, x, z) ≡ g(t, x + ω_{2}, z). The sufficient conditions are found under which this equation has a solution u such that

u(t + ω_{1}, x) = u(t, x), u(t, x + ω_{2}) = u(t, x).  (2)

In particular, it is shown that if following conditions take place

g_{i}(z) ≥ 0 (i = 1, 2), inf{z ∈ R : g_{1}(z) − g_{2}(z)} > 0, lim inf_{|z|→+∞} g_{0}((z) < 0,

lim_{|z|→+∞} \frac{g^{2}(x, y, z)}{g_{0}(z)} = 0 uniformly with respect to (t, x) ∈ R^{2},

then problem (1), (2) has at least one solution.