Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

TL;DR

This paper investigates the formation of high cell densities in a chemotaxis model with indirect signal production, showing finite-time blow-up under certain conditions and analyzing the convergence of solutions as a parameter approaches zero.

Contribution

It establishes the convergence of classical solutions to a hyperbolic-elliptic model and demonstrates finite-time blow-up in non-radial symmetric domains for specific parameters.

Findings

Classical solutions converge to hyperbolic-elliptic solutions as epsilon approaches zero.

Solutions can blow up in finite time in non-radial symmetric convex domains.

Solutions can exceed arbitrarily large values under certain initial conditions.

Abstract

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=\epsilon\Delta u-\nabla\cdot(u\nabla v)+ru-\mu u^\theta,\\ & 0=d_1\Delta v-\beta v+\alpha w,\\ & 0=d_2\Delta w-\delta w+\gamma u \end{array} \right. \qquad(0.1)\end{eqnarray} and the so-called strong $W^{1, q}(\Omega)$-solution of hyperbolic-elliptic-elliptic model \begin{eqnarray} \left\{ \begin{array}{llll} & u_t=-\nabla\cdot(u\nabla v)+ru-\mu u^\theta,\\ & 0=d_1\Delta v-\beta v+\alpha w,\\ & 0=d_2\Delta w-\delta w+\gamma u, \end{array} \right.\ \qquad(0.2)\end{eqnarray} in arbitrary bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq1$, where $r, \mu, d_1, d_2, \alpha, \beta, \gamma, \delta>0$ and $\theta>1$. Via applying the viscosity vanishing method, we first prove that the classical solution of (0.1) will converge to the strong $W^{1, q}(\Omega)$-solution of (0.2) as $\epsilon\rightarrow0$. After structuring the local well-pose of (0.2), we find that the strong $W^{1, q}(\Omega)$-solution will blow up in finite time with non-radial symmetry setting if $\Omega$ is a bounded convex domain, $\theta\in(1, 2]$, and the initial data is suitable large. Moreover, for any positive constant $M$ and the classical solution of (0.1), if we add another hypothesis that there exists positive constant $\epsilon_0(M)$ with $\epsilon\in(0,\ \epsilon_0(M))$, then the classical solution of (0.1) can exceed arbitrarily large finite value in the sense: one can find some points $\left(\tilde{x}, \tilde{t}\right)$ such that $u(\tilde{x}, \tilde{t})>M$.