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Large global solutions of the parabolic-parabolic Keller-Segel system in higher dimensions

By Lorenzo Brandolese

Appears in collection : Jean-Morlet Chair 2022 - Conference: Nonlinear PDEs in Fluid Dynamics / Chaire Jean-Morlet 2022 - Conférence : EDP non-linéaires en dynamique des fluides

We study the global existence of the parabolic-parabolic Keller–Segel system in $\mathbb{R}^{d}$. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter $\tau$ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler, Guerra and Karch (2015) and Corrias, Escobedo and Matos (2014). Our analysis improves earlier results and extends them to any dimension d ≥ 3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in $\tau$ , when $\tau\gg 1$: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller–Segel system. For these toy models, we establish in a companion paper finite time blowup for a class of large solutions.

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Citation data

  • DOI 10.24350/CIRM.V.19916603
  • Cite this video Brandolese Lorenzo (5/12/22). Large global solutions of the parabolic-parabolic Keller-Segel system in higher dimensions. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19916603
  • URL https://dx.doi.org/10.24350/CIRM.V.19916603

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  • BILER, Piotr, BORITCHEV, Alexandre, et BRANDOLESE, Lorenzo. Large global solutions of the parabolic-parabolic Keller-Segel system in higher dimensions. arXiv preprint arXiv:2203.09130, 2022. - https://arxiv.org/abs/2203.09130

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