# Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model

## TL;DR

This paper uses computer-assisted methods to rigorously prove the existence of heteroclinic connections in a complex PDE model of diblock copolymers, revealing intricate dynamical behaviors not seen in simpler related models.

## Contribution

The paper provides the first rigorous proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model, highlighting its multistability and complex attractor structure.

## Key findings

- Existence of heteroclinic connections between homogeneous and energy-minimizing states.
- Not all solutions near the homogeneous state converge to the global minimizer.
- Presence of stable states with higher energy trapping solutions.

## Abstract

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state, which represents a perfect copolymer mixture, and all local and global energy minimizers. In this way, we show that not every solution originating near the homogeneous state will converge to the global energy minimizer, but rather is trapped by a stable state with higher energy. This phenomenon can not be observed in the one-dimensional Cahn-Hillard equation, where generic solutions are attracted by a global minimizer.