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

Abstract We consider the chemotaxis system { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + g ( u ) , x ∈ Ω , t > 0 , 0 = Δ v − v + u , x ∈ Ω , t > 0 , in a smooth bounded domain Ω ⊂ R n , where χ > 0 and g generalizes the logistic function g ( u ) = A u − b u α 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 u 0 ∈ L 1 ( 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 u 0 ∈ L ∞ ( Ω ) has small norm in L γ ( 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 L 1 initial data. The paper closes with numerical experiments that illustrate some of the theoretically established results.