A Rigorous Integrator and Global Existence for Higher-Dimensional Semilinear Parabolic PDEs via Semigroup Theory
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Bibliographic record
Abstract
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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it