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 ? ???????? ???????? ut = 1 & UDelta;u - & chi;1 backward difference & BULL; (u backward difference v) - & chi;2 backward difference & BULL; (u backward difference z) + & mu;1u(1 - u - a1w), 0 = & UDelta;v -v + w & sigma;1 , wt = 2 & UDelta;w - & chi;3 backward difference & BULL; (w backward difference z) - & chi;4 backward difference & BULL; (w backward difference v) + & mu;2w(1 - w - a2u), 0 = & UDelta;z - z + u & sigma;2 (1)and strong W1,q(0)-solution of hyperbolic-elliptic-hyperbolic-elliptic model ? ???????? ???????? ut = -& chi;1 backward difference & BULL; (u backward difference v) - & chi;2 backward difference & BULL; (u backward difference z) + & mu;1u(1 - u - a1w), 0 = & UDelta;v -v +w & sigma;1 , wt = -& chi;3 backward difference & BULL; (w backward difference z) - & chi;4 backward difference & BULL; (w backward difference v) + & mu;2w(1 - w - a2u), 0 = & UDelta;z -z + u & sigma;2, (2)in a smoothly bounded domain 0 & SUB; Rn. By the viscosity vanishing method and logarithmic Sobolev inequality, we first built the local well-pose of the strong W1,q(0)-solution in the sense that the classical solution of (1) will converge to the strong W1,q(0)-solution of (2) as 1, 2 \ 0 and then get a blow-up result with nonradial symmetry setting:& BULL; if & sigma;1 = & sigma;2 = 1, then for suitably larger & chi;2, & chi;4 and initial data, the strong W1,q(0)-solution (u, w) blows up in finite time with Lq(0) x Lq(0)-norm.& BULL; if & sigma;1 = & sigma;2 > 1, then the suitably larger initial data is enough to ensure the strong W1,q(0)-solution (u, w) blows up in finite time with Lq(0)xLq(0)-norm.The upper bound estimates of blow-up time and rate are obtained. We also hypothesize that if 1, 2 are suitably small, the classical solution of (1) can exceed arbitrarily large finite value.