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A rigorous integrator and global existence for higher-dimensional semilinear parabolic PDEs via semigroup theory

Gabriel William Duchesne, Jean-Philippe Lessard, Akitoshi Takayasu

TL;DR

This paper presents a constructive framework for global existence of solutions to higher-dimensional semilinear parabolic PDEs by reformulating the IVP as an infinite system of ODEs in Fourier space and solving it with a Newton-like fixed-point method built from a rigorously bounded evolution operator. The approach couples a Fourier-Chebyshev spectral discretization with a Banach-algebra setting to control nonlinearities and to obtain explicit, verifiable bounds that ensure local inclusions and multi-step long-time existence. It is then applied to the 3D/2D Swift-Hohenberg equation, where trapping regions and semigroup estimates yield global existence with convergence to nontrivial equilibria, and to the 2D Ohta-Kawasaki equation, where derivatives in nonlinear terms are handled within the same CAP framework. The results showcase a parallelizable, scalable method for rigorous forward integration in higher spatial dimensions, providing new insights into pattern formation and global dynamics via computer-assisted proofs.

Abstract

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift-Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta-Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

A rigorous integrator and global existence for higher-dimensional semilinear parabolic PDEs via semigroup theory

TL;DR

This paper presents a constructive framework for global existence of solutions to higher-dimensional semilinear parabolic PDEs by reformulating the IVP as an infinite system of ODEs in Fourier space and solving it with a Newton-like fixed-point method built from a rigorously bounded evolution operator. The approach couples a Fourier-Chebyshev spectral discretization with a Banach-algebra setting to control nonlinearities and to obtain explicit, verifiable bounds that ensure local inclusions and multi-step long-time existence. It is then applied to the 3D/2D Swift-Hohenberg equation, where trapping regions and semigroup estimates yield global existence with convergence to nontrivial equilibria, and to the 2D Ohta-Kawasaki equation, where derivatives in nonlinear terms are handled within the same CAP framework. The results showcase a parallelizable, scalable method for rigorous forward integration in higher spatial dimensions, providing new insights into pattern formation and global dynamics via computer-assisted proofs.

Abstract

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift-Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta-Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.
Paper Structure (34 sections, 19 theorems, 315 equations, 17 figures, 1 table)

This paper contains 34 sections, 19 theorems, 315 equations, 17 figures, 1 table.

Table of Contents

  1. Keywords:
  2. AMS subject classifications:
  3. Introduction
  4. Newton-like operator to solve initial value problems
  5. Function space and Banach algebra
  6. The zero-finding problem for the initial value problem
  7. Exact inverse of the linearization and definition of the Newton-like operator
  8. Rigorous integration of initial value problems
  9. Explicit uniform bounds of the evolution operator
  10. Computing ${W_{m,q}^{\mathcal{S}_{J}}}$ : rigorous numerics for the fundamental solution matrix
  11. Generation of the evolution operator and $W^{(\infty)}_{{q}}(t,s)$ bounds
  12. The other bounds
  13. Validation theorem for local inclusion of the solution
  14. The bound $\varepsilon$.
  15. The bound $\delta$.
  16. ...and 19 more sections

Key Result

Theorem 3.2

Let $(t, s) \in {\mathcal{S}_{J}}$ and $\bar{a}$ be fixed. Let ${\bm{m}}\ge 0$ be the size of the Fourier projection such that $\mu_{{\bm{k}}}<0$ holds for ${\bm{k}}\not\in{\bm{F_m}}$. For such ${\bm{m}}\ge 0$, define $\mu_\ast$ as and assume that there exists a constant ${W_{m,q}^{\mathcal{S}_{J}}}>0$ such that Assume also that $\bar{U}^{(\infty)}(t,s)$ satisfies Moreover, let us assume the ex

Figures (17)

  • Figure 1: Initial data ($t_0=0$).
  • Figure 2: After $50$ steps ($t_{25}=12.5$).
  • Figure 3: After $100$ steps ($t_{100}=25$).
  • Figure 4: After $150$ steps ($t_{150}=37.5$).
  • Figure 5: After $287$ steps ($t_{287}=71.75$).
  • ...and 12 more figures

Theorems & Definitions (47)

  • Remark 2.1
  • Remark 2.2: Choice of basis functions
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • proof
  • Corollary 3.4
  • ...and 37 more