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

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=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 $W^{1,q}(\Omega )$-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 $\Omega \subset {\mathbb{R}}^n$, $n\ge 1$, 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 $\epsilon \to 0$. 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 $u(\tilde{x},\tilde{t})>M$.

Discrete and Continuous Dynamical Systems-Series B

References in corpus (1)

• Analysis of a chemotaxis model with indirect signal absorption