Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds

Exponential dichotomy properties are studied for non-autonomous quasilinear partial differential equations that can be written as an ordinary differential 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 $t\to+\infty$ and $t\to-\infty$, 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.