A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system

Abstract

The repulsive chemotaxis-consumption system {u(t) = del. D(u)del u) + del . (u del v), 0 = Delta v - uv, is considered along with the boundary conditions (D(u)del u + u del v) . nu vertical bar(partial derivative Omega) = 0 and v vertical bar(partial derivative Omega) = M in a ball Omega subset of R-2. Under the assumption that D suitably generalizes the function 0 <= xi bar right arrow (xi + 1)(-alpha) for some alpha > 0, it is firstly shown that for each nontrivial radially symmetric u(0) is an element of W-1,W-infinity (Omega), one can find M-star(u(0)) > 0 with the property that whenever M > M-star(u(0)), a corresponding initial-boundary value problem admits a classical solution blowing up in finite time. This is complemented by a second statement which asserts that when inf D and M are positive, for any such initial data a global bounded classical solution exists.