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

We consider the chemotaxis system ut = u −χ∇ · (u∇v)+ g(u), x ∈ Ω, t > 0, 0 = v − v + u, x ∈ Ω, t > 0, in a smooth bounded domain Ω ⊂ Rn, where χ > 0 and g generalizes the logistic function g(u) = Au − buα with α > 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 u0 ∈ L1(Ω) is proved under the assumption that α > 2 − 1 n . Moreover, boundedness properties of the constructed solutions are studied. Inter alia, it is shown that if b is sufficiently large and u0 ∈ L∞(Ω) has small norm in Lγ (Ω) for some γ > n 2 then the solution is globally bounded. Finally, in the case that additionally α > n 2 holds, a bounded set in L∞(Ω) can be found which eventually attracts very weak solutions emanating from arbitrary L1 initial data. The paper closes with numerical experiments that illustrate some of the theoretically established results.