Parabolic Equation with Nonlocal Diffusion without a Smooth Inertial Manifold

TL;DR

This paper presents a one-dimensional parabolic integro-differential equation with nonlocal diffusion that lacks finite-dimensional asymptotic dynamics, challenging previous assumptions about the long-term behavior of such equations.

Contribution

It constructs a natural example of a parabolic equation without a smooth inertial manifold, expanding understanding of the dynamics of nonlocal diffusion equations.

Findings

The constructed equation does not have an asymptotically finite-dimensional attractor.

This example is more natural within the class of parabolic evolutionary equations.

It demonstrates limitations of existing theories on inertial manifolds for nonlocal equations.

Abstract

We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier.