Dichotomy gap conditions for the existence of smooth inertial manifolds for non-instantaneous impulsive parabolic equations

Abstract

This paper deals with the fully non-autonomous semilinear parabolic equation $\frac{dx}{dt}+A\left(t\right)x=f(t,x)$ under certain non-instantaneous impulsive effects. Using the Lyapunov–Perron method, we provide dichotomy gap conditions, where the dichotomy gap ${\Lambda }_n-{\lambda }_n$ is compared to ${\sup }_{t\in R}{\int }_{t-1}^t{\left(\phi \right(\tau \left)\right)}^{2}d\tau$, for the existence of an inertial manifold for the non-instantaneous impulsive parabolic equations. Moreover, we investigate the regularity of the inertial manifolds under the condition on the regularity of the nonlinear term. We will discuss a consequence of the obtained results for the parabolic equations without impulses. Finally, we apply our abstract results to a nonautonomous reaction-diffusion equation.