Chemotaxis with logistic source: Very weak global solutions and their boundedness properties

We consider the chemotaxis system {u(t) = Delta u - X-del - (u del v) + g(u), x is an element of Omega, t > 0, 0 = Delta V - v + u, x is an element of Omega, t > 0, in a smooth bounded domain Omega subset of R-n, where X > 0 and g generalizes the logistic function g(u) = Au - bu(alpha) with alpha > 1, A >= 0 and b > 0. A concept of very weak solutions is introduced, and global existence of such solutions for any nonnegative initial data u(0) is an element of L-1(Omega) is proved under the assumption that alpha > 2 - 1/n. Moreover, boundedness properties of the constructed solutions are studied. Inter alia, it is shown that if b is sufficiently large and u(0) is an element of L-infinity(Omega) has small norm in L-gamma(Omega) for some gamma > n/2 then the solution is globally bounded. Finally, in the case that additionally alpha > n/2 holds, a bounded set in L-infinity(Omega) can be found which eventually attracts very weak solutions emanating from arbitrary L-1 initial data. The paper closes with numerical experiments that illustrate some of the theoretically established results. (C) 2008 Elsevier Inc. All rights reserved.