Approximation of the inertial manifold for a nonlocal dynamical system

We consider inertial manifolds and their approximation for a class of partial differential equations with a nonlocal Laplacian operator $-(-\mathrm{\Delta }{)}^{\frac{\alpha }{2}}$, with $0<\alpha <2$. The nonlocal or fractional Laplacian operator represents an anomalous diffusion effect. We first establish the existence of an inertial manifold and highlight the influence of the parameter $\alpha$. Then we approximate the inertial manifold when a small normal diffusion $\epsilon \mathrm{\Delta }$ (with $\epsilon \in (0,1)$) enters the system, and obtain the estimate for the Hausdorff semi-distance between the inertial manifolds with and without normal diffusion.