Emergence of large densities in a chemotaxis system with signaling loops, nonlinear signal productions and competitions sources under nonradial symmetry case

This paper deals with classical solution of system $ \begin{array}{*{20}{l}}{\begin{eqnarray} { } \left\{ \begin{array}{ll} u_t = \epsilon_1\Delta u-\chi_{1}\nabla\cdot(u\nabla v)-\chi_{2}\nabla\cdot(u\nabla z)+\mu_1 u(1-u-a_1w), &\\ 0 \Delta v-v+w^{\sigma_1}, w_t \epsilon_2\Delta w-\chi_3\nabla\cdot(w\nabla z)-\chi_4\nabla\cdot(w\nabla v)+\mu_2 w(1-w-a_2u), z-z+u^{\sigma_2} & \end{array} \right. \end{eqnarray}}&{\;\;\;\;(1)}\end{array} and strong W^{1, q}( \Omega) $-solution hyperbolic-elliptic-hyperbolic-elliptic model$ -\chi_{1}\nabla\cdot(u\nabla -\chi_3\nabla\cdot(w\nabla z-z+u^{\sigma_2}, \end{eqnarray}}&{\;\;\;(2)}\end{array} $in a smoothly bounded domain \Omega\subset \mathbb{R}^n $. By the viscosity vanishing method logarithmic Sobolev inequality, we first built local well-pose in sense that (1) will converge to (2) as \epsilon_1, \epsilon_2\searrow0 then get blow-up result nonradial symmetry setting:● if \sigma_1 \sigma_2 1 $, for suitably larger \chi_2, \chi_4 initial data, (u, w) blows up finite time L^q( \Omega)\times $-norm.● \sigma_2>1 data is enough ensure $-norm.The upper bound estimates rate are obtained. We also hypothesize \epsilon_2 small, can exceed arbitrarily large value.