Rigorous integration of semilinear parabolic PDEs: forward evolution bounds and projected boundary value problems

Abstract

In this work, we present a rigorous integrator for initial and boundary value problems for semilinear parabolic PDEs. The method builds on earlier semigroup-based approaches while improving the bounds used in the fixed-point argument and extending the class of problems that can be treated. In particular, it controls the solution operator without requiring backward integration of the variational equation and allows projected boundary value problems to be studied through a simplified construction of an approximate inverse for the linearized operator. To illustrate the scope of the method, we present three applications to the two-dimensional Ohta--Kawasaki equation. For two initial value problems, we prove global existence and obtain explicit enclosures valid for all sufficiently large times around numerical approximations of stripe and spot equilibria. These results use a fixed-point formulation on an unbounded time interval, avoiding the construction of trapping regions. We also rigorously solve a projected boundary value problem with endpoint conditions chosen from linear approximations of the unstable and stable spaces of two saddle equilibria. This provides a step toward the rigorous study of heteroclinic saddle-to-saddle connections.