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

We consider the chemotaxis system{ut=Δu−χ∇⋅(u∇v)+g(u),x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0, in a smooth bounded domain Ω⊂RnΩ⊂Rn, where χ>0χ>0 and g   generalizes the logistic function g(u)=Au−buαg(u)=Au−buα with α>1α>1, A⩾0A⩾0 and b>0b>0. A concept of very weak solutions is introduced, and global existence of such solutions for any nonnegative initial data u0∈L1(Ω)u0∈L1(Ω) is proved under the assumption that α>2−1n. Moreover, boundedness properties of the constructed solutions are studied. Inter alia, it is shown that if b   is sufficiently large and u0∈L∞(Ω)u0∈L∞(Ω) has small norm in Lγ(Ω)Lγ(Ω) for some γ>n2 then the solution is globally bounded. Finally, in the case that additionally α>n2 holds, a bounded set in L∞(Ω)L∞(Ω) can be found which eventually attracts very weak solutions emanating from arbitrary L1L1 initial data. The paper closes with numerical experiments that illustrate some of the theoretically established results.