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PreprintCC BY 4.0arXiv

Self-similar smoothing of a discontinuity by degenerate cross-diffusion

  • Michael C. Dallaston
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Abstract

We consider the dynamics of an idealised cross-diffusion model of biological invasion in which the diffusion of an invading population is inhibited by a resident population, which is in turn degraded by the former. We formulate and numerically solve the problem that describes the self-similar smoothing of an initially piecewise-constant invading population. In the limit the height ahead of the initial discontinuity vanishes, the speed of propagation also vanishes, but only logarithmically slowly. This result is confirmed by a matched asymptotic analysis. The singular nature of this limit indicates that the system does not permit the existence of compactly supported solutions that exhibit a moving front or interface, which has important implications for the simulation of more complex models that also feature this degenerate diffusive mechanism.

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Cite

BibTeXRIS
BibTeX
@misc{dallaston2026selfsimilar,
  title = {{Self-similar smoothing of a discontinuity by degenerate cross-diffusion}},
  author = {Michael C. Dallaston},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.15228},
  eprint = {2606.15228},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15228},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Self-similar smoothing of a discontinuity by degenerate cross-diffusion
AU  - Michael C. Dallaston
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2606.15228
UR  - https://arxiv.org/abs/2606.15228
AB  - We consider the dynamics of an idealised cross-diffusion model of biological invasion in which the diffusion of an invading population is inhibited by a resident population, which is in turn degraded by the former. We formulate and numerically solve the problem that describes the self-similar smoothing of an initially piecewise-constant invading population. In the limit the height ahead of the initial discontinuity vanishes, the speed of propagation also vanishes, but only logarithmically slowly. This result is confirmed by a matched asymptotic analysis. The singular nature of this limit indicates that the system does not permit the existence of compactly supported solutions that exhibit a moving front or interface, which has important implications for the simulation of more complex models that also feature this degenerate diffusive mechanism.
ER  -