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One considers mixed boundary value problems for a quasilinear hyperbolic equation with a weak, as well as strong, dissipation. The nonlinear function in the equation is assumed Lipschitz continuous. For each of these problems one obtains the conditions on the Lipschitz constant that ensure the existence of inertial manifolds.

We completely describe the characteristic frequencies of zeros of a sum of two harmonic oscillations with distinct frequencies and partly describe the characteristic frequencies of zeros of sums of arbitrarily many oscillations. As a corollary, we show that the set of characteristic frequencies of zeros of a linear equation can be infinite and moreover have continuum measure even in the case of an equation with constant coefficients.

Exponential dichotomy properties are studied for non-autonomous quasilinear partial differential equations that can be written as an ordinary dif- ferential equation du/dt + Au = F (u, t) in a Hilbert space H. It is assumed that the non-linear function F (u, t) is essentially subordinated to the linear operator A; namely, the gap property from the theory of inertial manifolds must hold. Integral manifolds M+ and M− attracting at an exponential rate an arbitrary solution of this equation as t → +∞ and t →− ∞, respectively, are constructed. The general results established are applied to the study of the dichotomy properties of solutions of a one-dimensional reaction-diffusion system and of a dissipative hyperbolic equation of sine-Gordon type. Bibliography: 18 titles.

The paper deals with a quasilinear nonautonomous parabolic equation. A finite-dimensional Lipschitz integral manifold is constructed in a neighborhood of an equilibrium point of the limiting autonomous equation. It is proved that the integral manifold attracts exponentially all the solutions of the initial nonautonomous equation, passing through this neighborhood. Bibliography: 13 titles.