Article ID Journal Published Year Pages File Type
4620641 Journal of Mathematical Analysis and Applications 2008 22 Pages PDF
Abstract

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.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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