# Global attractors for the Benjamin-Bona-Mahony equation with memory

## TL;DR

This paper proves the existence of a global attractor for a nonlinear Benjamin-Bona-Mahony equation with memory effects, under small external forces, using a novel approach involving invariant sets.

## Contribution

It introduces a new method for establishing global attractors for integrodifferential equations with memory, expanding understanding of long-term dynamics in such systems.

## Key findings

- Existence of a global attractor in the weak energy space.
- The approach applies under small external force conditions.
- The method involves constructing attractors on invariant sets.

## Abstract

We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation $$ u_t - u_{txx} + u_x - \int_0^\infty g(s) u_{xx}(t-s) {\rm d} s + u u_x = f $$ where the dissipation is entirely contributed by the memory term. Under a suitable smallness assumption on the external force $f$, we show that the related solution semigroup possesses the global attractor in the natural weak energy space. The result is obtained by means of a nonstandard approach based on the construction of a suitable family of attractors on certain invariant sets of the phase space.