Finite-time blow-up in a quasilinear system of chemotaxis

We consider an elliptic–parabolic system of the Keller–Segel type which involves nonlinear diffusion. We find a critical exponent of the nonlinearity in the diffusion, measuring the strength of diffusion at points of high (population) densities, which distinguishes between finite-time blow-up and global-in-time existence of uniformly bounded solutions. This critical exponent depends on the space dimension n ⩾ 1, where apart from the physically relevant cases n = 2 and n = 3 also the result obtained in the one-dimensional setting might be of mathematical interest: here, namely, finite-time explosion of solutions occurs although the Lyapunov functional associated with the system is bounded from below. Additionally this one-dimensional case is an example to show that L ∞ estimates of solutions to non-uniformly parabolic drift–diffusion equations cannot be expected even when boundedness of the gradient of the drift term is presupposed.