Convergence of solutions to a partially diffusive chemotaxis system with indirect signal production and phenotype switching is shown in a two-dimensional setting when the switching rate increases to infinity, thereby providing a rigorous justification of formal computations performed in the literature. The expected limit system being the classical parabolic-parabolic Keller-Segel system, the obtained convergence is restricted to a finite time interval for general initial conditions but valid for arbitrary bounded time intervals when the mass of the initial condition is appropriately small. Furthermore, if the solution to the limit system blows up in finite time, then neither of the two phenotypes in the partially diffusive system can be uniformly bounded with respect to the $L_2$-norm beyond that time.
We consider a phenotype-switching chemotaxis model for aggregation, in which a chemotactic population is capable of switching back and forth between a chemotaxing state (performing chemotactic movement) and a secreting state (producing the attractant). We show that the switching rate provides a powerful mechanism for controlling the densities of spontaneously emerging aggregates. Specifically, in two- and three-dimensional settings it is shown that when both switching rates coincide and are suitably large, then the densities of both the chemotaxing and the secreting population will exceed any …
The well-posedness of a chemotaxis system with indirect signal production in a two-dimensional domain is shown, all solutions being global unlike the classical Keller-Segel chemotaxis system. Nevertheless, there is a threshold value $M_c$ of the mass of the first component which separates two different behaviours: solutions are bounded when the mass is below $M_c$ while there are unbounded solutions starting from initial conditions having a mass exceeding $M_c$. This result extends to arbitrary two-dimensional domains a previous result of Tao \& Winkler (2017) obtained for radially symmetric s…
Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)
Global existence and boundedness for a degenerate chemotaxis system with indirect signal production via minimizing movement schemes
Osaka Metropolitan University（大阪公立大学）；Institut de Mathématiques de Toulouse (IMT) UMR5219, Université Toulouse III - Paul Sabatier（图卢兹第三大学-保罗·萨巴蒂尔大学）；Université Savoie Mont Blanc, CNRS（萨瓦-蒙布朗大学）
We consider the chemotaxis model \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v), \\ v_t = Δv - vw, \\ w_t = -δw + u \end{cases} \end{align*} in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \in \mathbb N$, where $δ\gt 0$ is a given parameter. If either $n \le 2$ or $\|v_0\|_{L^\infty(Ω)} \le \frac1{3n}$ we show the existence of a unique global classical solution $(u, v, w)$ and convergence of $(u(\cdot, t), v(\cdot, t), w(\cdot, t))$ towards a spatially constant equilibrium, as $t \to \infty$. The proof of global existence for the case $n \le 2$ relies on a bootstrap pr…
This paper deals with the classical solution of the following chemotaxis system with generalized logistic growth and indirect signal production \begin{eqnarray} \left\{ \begin{array}{llll} & u_t=εΔu-\nabla\cdot(u\nabla v)+ru-μu^θ,\\ & 0=d_1Δv-βv+αw,\\ & 0=d_2Δw-δw+γu \end{array} \right. \qquad(0.1)\end{eqnarray} and the so-called strong $W^{1, q}(Ω)$-solution of hyperbolic-elliptic-elliptic model \begin{eqnarray} \left\{ \begin{array}{llll} & u_t=-\nabla\cdot(u\nabla v)+ru-μu^θ,\\ & 0=d_1Δv-βv+αw,\\ & 0=d_2Δw-δw+γu, \end{array} \right.\ \qquad(0.2)\end{eqnarray} in arbitrary bounded domain $Ω\…
Global existence and uniqueness of solutions in a chemotaxis system with density-suppressed motility, a $θ-$logistic source, and indirect signal production under a strong Allee effect
Indian Institute of Technology Roorkee（印度理工学院鲁尔基分校）
A sharp criterion and complete classification of global-in-time solutions and finite time blow-up of solutions to a chemotaxis system in supercritical dimensions
We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding tissue. Transitions between the two cell subpopulations are influenced by subcellular (receptor binding) dynamics, thus conferring the setting a multiscale character. We prove global existenc…
