Abstract

This paper deals with the parabolic–elliptic Keller–Segel system on , involving a source term of logistic type defined in terms of the mass capacity M 0 and the total mass of the individuals. We exhibit that the qualitative behaviour of solutions is decided by the mass capacity M 0 and the initial mass m 0. For general solutions, the existence of a global weak solution is proved under the assumption that both M 0 and m 0 are less than 8π, whereas there exist solutions blowing up in finite time under the hypotheses of either with any integrable initial data or accompanied with large initial data. Moreover, gives rise to a compromise that solutions exist globally and blow up as time goes to infinity. For radially symmetric solutions, we introduce a strategy of relegating the lack of mass conservation via a transformation to the density and then obtain that there are stationary solutions given by with λ > 0. Subsequently we prove that if the initial data is strictly below for some λ > 0, then the solution vanishes in as . If the initial data is strictly above for some λ > 0, then the solution either blows up in finite time or has a mass concentration at the origin as time goes to infinity. Finally, our results are complemented by numerical simulations that demonstrate the asymptotic behaviour of solutions.

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