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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.

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.