Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity

We consider the chemotaxis system with nonlinear diffusion and singular sensitivity { u(t) = Delta u(m) - del (u/v del(v)), x is an element of Omega, t > 0, v(t )= Delta v - uv, x is an element of Omega, t > 0, partial derivative u/partial derivative v = partial derivative u/partial derivative v = 0, x is an element of Omega, t > 0, u(x, 0) = u(0)(x), v(x,0) = v(0)(x), x is an element of Omega in a smooth bounded domain Omega subset of R-n, n >= 2. In this work it is shown that for all reasonably regular initial data u(0) >= 0 and v(0) > 0, the corresponding Neumann initial-boundary value problem possesses a global generalized solution provided that m > 1 + n-2/2n. (C) 2018 Elsevier Ltd. All rights reserved.