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Chemotaxis; global existence; Lyapunov functional; Adams’ inequality
We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than $(8\pi)^2$, whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known $8\pi$ problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel system and our system and discuss the difference.
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