Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 43 min ago
PreprintarXiv licencearXiv

A fully nonlinear parabolic segregation problem

  • Emily Casey
  • Cornelia Mihaila
  • María Soria-Carro
  • Clara Torres-Latorre
Open at sourcePDFDOICite

Abstract

We study a two-phase parabolic free boundary problem arising as the segregation limit of a competition-diffusion system with strong interactions and fully nonlinear diffusion. The positive and negative phases are governed by different equations and are coupled through global viscosity inequalities. We construct viscosity solutions that are Lipschitz in space and 1/2-Hölder in time, and establish differentiability in space across the free boundary, including at singular points. At regular points, we prove that the solution has a parabolic C1,α asymptotic expansion, and deduce that the free boundary is locally a C1,α hypersurface. The proof relies on a parabolic Alt-Caffarelli-Friedman monotonicity formula to classify blow-up limits, followed by an improvement of flatness argument.

Related

  1. Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
    Jiangwen Wang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Geometric approaches for improved regularity in fully nonlinear parabolic models
    Junior da Silva Bessa, João Vitor da Silva, Mayra Soares
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. Deep Stochastic Residual Methods for Fully Nonlinear Parabolic Equations
    Zhenhua Zhao, Jihao Long
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingOptimisation & controlarXiv
  4. A doubly nonlinear parabolic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source
    Bogdan Maxim
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv
  5. Global solutions to the fully nonlinear obstacle problem
    Yi Cao, Dongsheng Li
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{casey2026fully,
  title = {{A fully nonlinear parabolic segregation problem}},
  author = {Emily Casey and Cornelia Mihaila and María Soria-Carro and Clara Torres-Latorre},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38480},
  eprint = {2610.38480},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38480},
}
RIS
TY  - UNPB
TI  - A fully nonlinear parabolic segregation problem
AU  - Emily Casey
AU  - Cornelia Mihaila
AU  - María Soria-Carro
AU  - Clara Torres-Latorre
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38480
UR  - https://arxiv.org/abs/2610.38480
AB  - We study a two-phase parabolic free boundary problem arising as the segregation limit of a competition-diffusion system with strong interactions and fully nonlinear diffusion. The positive and negative phases are governed by different equations and are coupled through global viscosity inequalities. We construct viscosity solutions that are Lipschitz in space and 1/2-Hölder in time, and establish differentiability in space across the free boundary, including at singular points. At regular points, we prove that the solution has a parabolic C^1,α asymptotic expansion, and deduce that the free boundary is locally a C^1,α hypersurface. The proof relies on a parabolic Alt-Caffarelli-Friedman monotonicity formula to classify blow-up limits, followed by an improvement of flatness argument.
ER  -