Inertial manifolds for parabolic differential equations: The fully nonautonomous case

Abstract

<p style='text-indent:20px;'>We study the existence of an inertial manifold for the solutions to fully non-autonomous parabolic differential equation of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \dfrac{du}{dt} + A(t)u(t) = f(t,u),\, t> s. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove the existence of such an inertial manifold in the case that the family of linear partial differential operators <inline-formula><tex-math id="M1">\begin{document}$ (A(t))_{t\in { \mathbb {R}}} $\end{document}</tex-math></inline-formula> generates an evolution family <inline-formula><tex-math id="M2">\begin{document}$ (U(t,s))_{t\ge s} $\end{document}</tex-math></inline-formula> satisfying certain dichotomy estimates, and the nonlinear forcing term <inline-formula><tex-math id="M3">\begin{document}$ f(t,x) $\end{document}</tex-math></inline-formula> satisfies the Lipschitz condition such that certain dichotomy gap condition holds.</p>