Paper detail

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

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 $Ω\subset\mathbb{R}^n$, $n\geq1$, where $r, μ, d_1, d_2, α, β, γ, δ>0$ and $θ>1$. Via applying the viscosity vanishing method, we first prove that the classical solution of (0.1) will converge to the strong $W^{1, q}(Ω)$-solution of (0.2) as $ε\rightarrow0$. After structuring the local well-pose of (0.2), we find that the strong $W^{1, q}(Ω)$-solution will blow up in finite time with non-radial symmetry setting if $Ω$ is a bounded convex domain, $θ\in(1, 2]$, 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 $ε_0(M)$ with $ε\in(0,\ ε_0(M))$, 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 $u(\tilde{x}, \tilde{t})>M$.