On a parabolic-elliptic chemotaxis system with periodic asymptotic behavior

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain urn:x-wiley:mma:media:mma5423:mma5423-math-0001. We consider a growth term of logistic type in the equation of “u” in the form μu(1 − u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense

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where f ∗ is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ∗, if the constant chemotactic sensitivity χ satisfies
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we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.