Keller-Segel chemotaxis models: a review. (English) Zbl 1464.35001
Summary: We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global existence, boundedness and blow-up of solutions to parabolic-elliptic type models. Thereafter we describe the global existence, boundedness and blow-up of solutions to parabolic-parabolic models. The numerical analysis of these models is still at a rather early stage only. We recollect quite a few of the known results on numerical methods also and direct the attention to a number of open problems in this domain.
MSC:
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
| 35A35 | Theoretical approximation in context of PDEs |
| 35D30 | Weak solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Keller-Segel models; renormalized solutions; local existence; global existence; boundedness; stabilization; finite difference method; finite element method; finite volume method; discontinuous Galerkin methodSoftware:
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