Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity

By Kentarou Fujie, T. Senba. 2015

85 citations recorded by the source corpus.

This paper is concerned with the parabolic-elliptic Keller-Segel system with signal-dependent sensitivity $\chi(v)$, \begin{align*} \begin{cases} u_t=\Delta u - \nabla \cdot ( u \nabla \chi(v)) &\mathrm{in}\ \Omega\times(0,\infty), \\ 0=\Delta v -v+u &\mathrm{in}\ \Omega\times(0,\infty), \end{cases} \end{align*} under homogeneous Neumann boundary condition in a smoothly bounded domain $\Omega \subset \mathbb{R}^2$ with nonnegative initial data $u_0 \in C^{0}(\overline{\Omega})$, $\not\equiv 0$. In the special case $\chi(v)=\chi_0 \log v\, (\chi_0>0)$, global existence and boundedness of the solution to the system were proved under some smallness condition on $\chi_0$ by Biler (1999) and Fujie, Winkler and Yokota (2015). In the present work, global existence and boundedness in the system will be established for general sensitivity $\chi$ satisfying $\chi'>0$ and $\chi'(s) \to 0 $ as $s\to \infty$. In particular, this establishes global existence and boundedness in the case $\chi(v)=\chi_0\log v$ with large $\chi_0>0$. Moreover, although the methods in the previous results are effective for only few specific cases, the present method can be applied to more general cases requiring only the essential conditions. Actually, our condition is necessary, since there are many radial blow-up solutions in the case $\inf_{s>0} \chi^\prime (s) >0$.

Subfields: Cancer Research, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/50a300f3112e64dfe7e2281e11ed0fdf22818077

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Citations in the corpus

A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data (2017, background): “015 and χ0(n) := √ 2 n for n ≥ 3, global bounded classical solutions exist for all reasonably regular positive initial data ([11], [2], [4], [24], [13], [21]); in the corresponding parabolic-elliptic analogue, the same conclusion holds with χ0(2) = ∞ ([5]) and with χ0(n) := 2 n−2 when n ≥ 3 and the ”
Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, result): “1) seems complex enough so as to allow for partial results identifying conditions sufficient for global boundedness up to now, with an exception formed by the particular choice χ(v) = χ0 v of a singular sensitivity consistent with the Weber-Fechner law for which besides various results on global sol”
A critical blow-up exponent in a chemotaxis system with nonlinear signal production (2018, background): “1) upon replacing the first equation therein with ut = ∆u − χ∇ · ( u v ∇v), the size of the parameter χ is known to play a crucial role with regard to the occurrence of blow-up: On the one hand, exclusively global smooth solutions exist if n ≥ 2 and χ < χ⋆(n) with some χ⋆(n) ∈ (0,∞] depending on whe”