Under suitable assumptions on parameters, the chemotaxis system admits a unique positive entire solution that is globally asymptotically stable.
Winkler, Blow-up in a higher-dimensional chemotaxi s system despite logistic growth restriction, Journal of Mathematical Analysis and Applications , 384(2011), 261-272
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Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments
Under suitable assumptions on parameters, the chemotaxis system admits a unique positive entire solution that is globally asymptotically stable.