Analysis of a Cross-Nonlinear Porous-Medium System Modeling Pressure-Driven Cell Population Dynamics
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916873711386624 |
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| author | Béjar-López, Alexis Granero-Belinchón, Rafael Pulido, Carlos Soler, Juan |
| author_facet | Béjar-López, Alexis Granero-Belinchón, Rafael Pulido, Carlos Soler, Juan |
| contents | In this work, we introduce a cross-diffusion model that couples population density and occupied area to investigate how internal pressure drives growth and motility. By blending nonlinear nonlocal interactions with porous-medium diffusion and an antidiffusive pressure term, the model captures the two-way feedback between local density fluctuations and tissue expansion or contraction. Building on Shraiman's area-growth paradigm, we enrich the framework with density-dependent spreading at the population boundary and a novel cross-diffusion term, yielding fully nonlinear transport in both equations. We prove local well-posedness for nonnegative solutions in Sobolev spaces and, under higher regularity, show both density and area remain nonnegative. Uniqueness follows when the initial density's square root lies in $H^2$, even if density vanishes on parts of the domain. We also exhibit initial data that induce finite-time blow-up, highlighting potential singularity formation. Finally, we establish that the density's spatial support remains invariant and characterize the co-evolution of occupied area and population density domains, offering new insights into pattern formation and mass transport in biological tissues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of a Cross-Nonlinear Porous-Medium System Modeling Pressure-Driven Cell Population Dynamics Béjar-López, Alexis Granero-Belinchón, Rafael Pulido, Carlos Soler, Juan Analysis of PDEs 35K61, 35Q92, 35B36, 35B44, 76S05, 74H35, 92D25, 74L15, 92C15, 92C17 In this work, we introduce a cross-diffusion model that couples population density and occupied area to investigate how internal pressure drives growth and motility. By blending nonlinear nonlocal interactions with porous-medium diffusion and an antidiffusive pressure term, the model captures the two-way feedback between local density fluctuations and tissue expansion or contraction. Building on Shraiman's area-growth paradigm, we enrich the framework with density-dependent spreading at the population boundary and a novel cross-diffusion term, yielding fully nonlinear transport in both equations. We prove local well-posedness for nonnegative solutions in Sobolev spaces and, under higher regularity, show both density and area remain nonnegative. Uniqueness follows when the initial density's square root lies in $H^2$, even if density vanishes on parts of the domain. We also exhibit initial data that induce finite-time blow-up, highlighting potential singularity formation. Finally, we establish that the density's spatial support remains invariant and characterize the co-evolution of occupied area and population density domains, offering new insights into pattern formation and mass transport in biological tissues. |
| title | Analysis of a Cross-Nonlinear Porous-Medium System Modeling Pressure-Driven Cell Population Dynamics |
| topic | Analysis of PDEs 35K61, 35Q92, 35B36, 35B44, 76S05, 74H35, 92D25, 74L15, 92C15, 92C17 |
| url | https://arxiv.org/abs/2507.23680 |