Morrey spaces norms and criteria for blowup in chemotaxis models

Piotr Biler; Grzegorz Karch; Jacek  Zienkiewicz; Piotr Biler; Grzegorz Karch; Jacek  Zienkiewicz

doi:10.3934/nhm.2016.11.239

Networks and Heterogeneous Media

2016, Volume 11, Issue 2: 239-250. doi: 10.3934/nhm.2016.11.239

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Morrey spaces norms and criteria for blowup in chemotaxis models

1.
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50--384 Wrocław
2.
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław

Received: 01 April 2015 Revised: 01 July 2015
Primary: 35Q92; Secondary: 35B44, 35K55, 35K40.

Two-dimensional Keller--Segel models for the chemotaxis with fractional (anomalous) diffusion are considered. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Similarly, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is shown for radially symmetric solutions of chemotaxis in several dimensions. Those conditions are, in a sense, complementary to the ones guaranteeing the global-in-time existence of solutions.
- Chemotaxis,
- parabolic-elliptic Keller-Segel model,
- blowup of solutions,
- fractional diffusion
Citation: Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models[J]. Networks and Heterogeneous Media, 2016, 11(2): 239-250. doi: 10.3934/nhm.2016.11.239

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Abstract

Two-dimensional Keller--Segel models for the chemotaxis with fractional (anomalous) diffusion are considered. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Similarly, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is shown for radially symmetric solutions of chemotaxis in several dimensions. Those conditions are, in a sense, complementary to the ones guaranteeing the global-in-time existence of solutions.

References

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