On a generalized volume-filling chemotaxis system with nonlinear signal production. (English) Zbl 1497.35070
The article considers a generalization of the parabolic-elliptic chemotaxis system
\[
\begin{cases} u_t = \nabla \cdot((1+u)^{-m}\nabla u) - \nabla \cdot(u(1+u)^{q-1}\nabla v)\\
0 = \Delta v + (1+u)^{\kappa} - \frac1{|\Omega|}\int_\Omega (1+u)^{\kappa} \end{cases}
\]
in smooth bounded domains \(\Omega\subset \mathbb{R}^n\) and under homogeneous Neumann boundary conditions.
It is shown that solutions emanating from nonnegative \(C^1\) initial data are global and bounded if \[ m+\kappa+q < \frac{n+2}{n}, \] whereas if \(\Omega=B_R\) and \[ \kappa+q>\frac{n+2}{n}\quad (m\ge 0) \quad \text{ or } \quad m+\kappa+q > \frac{n+2}{n} \quad (m<0), \] then radially symmetric initial data leading to finite-time blow-up can be found.
The blow-up proof is based on a study of the behaviour of \[ \Phi(t)=\int_0^{s_0} s^{-\alpha} (s_0-s) w(s,t) ds \] for \[ w(s,t)= n \int_0^{s^{\frac{1}{n}}} \rho^{n-1} u(\rho,t) d\rho \] in the tradition of W. Jäger and S. Luckhaus [Trans. Am. Math. Soc. 329, No. 2, 819–824 (1992; Zbl 0746.35002)] and M. Winkler [Z. Angew. Math. Phys. 69, No. 2, Paper No. 40, 25 p. (2018; Zbl 1395.35048)].
It is shown that solutions emanating from nonnegative \(C^1\) initial data are global and bounded if \[ m+\kappa+q < \frac{n+2}{n}, \] whereas if \(\Omega=B_R\) and \[ \kappa+q>\frac{n+2}{n}\quad (m\ge 0) \quad \text{ or } \quad m+\kappa+q > \frac{n+2}{n} \quad (m<0), \] then radially symmetric initial data leading to finite-time blow-up can be found.
The blow-up proof is based on a study of the behaviour of \[ \Phi(t)=\int_0^{s_0} s^{-\alpha} (s_0-s) w(s,t) ds \] for \[ w(s,t)= n \int_0^{s^{\frac{1}{n}}} \rho^{n-1} u(\rho,t) d\rho \] in the tradition of W. Jäger and S. Luckhaus [Trans. Am. Math. Soc. 329, No. 2, 819–824 (1992; Zbl 0746.35002)] and M. Winkler [Z. Angew. Math. Phys. 69, No. 2, Paper No. 40, 25 p. (2018; Zbl 1395.35048)].
Reviewer: Johannes Lankeit (Hannover)
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 92C17 | Cell movement (chemotaxis, etc.) |
Keywords:
chemotaxis; finite-time blow-up; global existence; volume-filling effect; nonlinear signal production; parabolic-elliptic chemotaxis systemReferences:
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