Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production

In this paper, a quasilinear chemotaxis system of parabolic-elliptic type {ut = del . (phi(u)del u) - del . (u del v) x epsilon Omega, t > 0, (0.1) 0 = Delta v - mu(t) + f(u), mu(t) := 1 vertical bar Omega vertical bar integral Omega f(u(.,t)), x epsilon Omega, t > 0 is considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain Omega C R-n, n 1. The diffusivity phi(xi) epsilon C-2([0, infinity)) is a positive function and we particularly suppose that i,o(0 = Co(1 for all xi >= 0 with C-0 > 0 and m epsilon R. f is a suitably regular positive function given by f (xi) = K(1 + xi)(kappa) for all xi >= 0 with parameters K > 0 and kappa > 0. It is shown that >= there exist regular initial data 7/0 such that if kappa + m < 2n/, then all solutions of (0.1) are global and bounded; if kappa + m > 2/n, however, there exists nonnegative radially symmetric initial data u(0) such that the corresponding solution of (0.1) blows up in finite time. The critical exponent 2/n seems to be essentially optimal which was given by [30]. (C) 2019 Elsevier Inc. All rights reserved.