Exact Robin bifurcation for a nonlocal heat equation

This repository contains a self-contained research preprint on

[ u_t=u_{xx}+\delta\frac{e^u}{\left(\int_{-1}^{1}e^u,dx\right)^p} ]

with homogeneous Robin boundary conditions. The paper derives every equilibrium, proves the exact multiplicity diagram, classifies the linearized spectrum, and establishes global convergence when (p\geq1).

The central parameterization is

[ \delta_{p,\beta}(a) =2^{p+1}a^{2-p}\sinh^p(a)\cosh^{p-2}(a) \exp!\left(\frac{2(p-1)a\tanh a}{\beta}\right). ]

For (0<p<1), this curve has exactly one fold. For (p\geq1), it is strictly increasing. At the critical exponent,

[ \delta_{1,\beta}(a)=4a\tanh a, ]

so the parameter curve is independent of the cooling coefficient even though the equilibrium profile still depends on it.

Repository contents

• HANDOFF.md: canonical project status, proof risks, exact next gates, and research directions. Start here when resuming the project.
• exact_robin_nonlocal_bifurcation.tex: complete manuscript source.
• references.bib: bibliography used by the manuscript.
• proof_audit.md: theorem-by-theorem static audit and edge cases.
• novelty_and_submission_notes.md: closest-result comparison, novelty risks, and journal-positioning notes.
• robin_bifurcation_verification.py: independent numerical transcription checks for later execution.
• robin_bifurcation_verification.main.kts: equivalent Kotlin checks for later execution.
• RESOURCE_CONSTRAINED_FUTURE_GATE.sh: future build and verification gate.
• submission/arxiv_metadata.md: draft arXiv metadata.

Verification status

The mathematical proof has received two manual static passes. The source and verification programs have not been compiled or executed during the resource-constrained preparation window. The manuscript proofs are authoritative; the programs only check transcription and representative parameter regimes.

After resource constraints are lifted, run:

./RESOURCE_CONSTRAINED_FUTURE_GATE.sh --verify

Publication and novelty remain subject to independent expert review and a complete MathSciNet, zbMATH, Scopus, and thesis search.

The ordered route from this repository to submission is recorded in HANDOFF.md.

Closest published work

The closest paper proves existence of a critical parameter and gives upper and lower bounds for the same Robin slab model, while reserving exact multiplicity for later work:

• M. Al-Refai and N. I. Kavallaris, On computation of bounds of the bifurcation parameter for a non-local elliptic equation with increasing nonlinearity, 2013.

Earlier Dirichlet results establish global existence, blow-up, and asymptotic behavior but do not provide the Robin classification proved here.