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

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. The proof of the above statement is conceptually simple, and combines several techniques from some of the authors’ and Zgliczyński’s works. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This propagation has to lead to small interval bounds, while at the same time entering the basin of attraction of the stable fixed point. For interesting parameter values the global attractor exhibits a complicated equilibrium structure, and the dynamical equation is rather stiff. This leads to a time-consuming numerical propagation of error bounds, with many integration steps. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic at various nontrivial parameter values.