This paper is concerned with a parabolic–parabolic Keller–Segel system ut=∇·(∇u-u∇v)inΩ×(0,T),vt=Δv-αv+uinΩ×(0,T)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} u_t = \nabla \cdot ( \nabla u - u \nabla v ) &{} \quad \text{ in } \, \Omega \times (0,T), \\ v_t = \Delta v - \alpha v + u &{} \quad \text{ in } \, \Omega \times (0,T) \end{array} \right. \end{aligned}$$\end{document}with a constant α≥0\documentclass[12pt]{minimal}
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\begin{document}$$ \alpha \ge 0 $$\end{document} and nonnegative initial data in a smoothly bounded domain Ω⊂R2\documentclass[12pt]{minimal}
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\begin{document}$$ \Omega \subset \mathbb {R}^2 $$\end{document} under the Neumann boundary condition or in Ω=R2\documentclass[12pt]{minimal}
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\begin{document}$$ \Omega = \mathbb {R}^2 $$\end{document}. It was introduced as a model of aggregation of bacteria, which is mathematically translated as finite-time blowup. A solution (u, v) is said to blow up at t=T<+∞\documentclass[12pt]{minimal}
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\begin{document}$$ t = T < + \infty $$\end{document} if lim supt→T‖u(t)‖L∞=+∞\documentclass[12pt]{minimal}
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\begin{document}$$ \limsup _{t \rightarrow T} \Vert u(t) \Vert _{L^\infty } = + \infty $$\end{document}. When (u, v) blows up at t=T\documentclass[12pt]{minimal}
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\begin{document}$$ t = T $$\end{document}, the blowup is called type I if ‖u(t)‖L∞≤C(T-t)-1\documentclass[12pt]{minimal}
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\begin{document}$$ \Vert u(t) \Vert _{L^\infty } \le C (T-t)^{-1} $$\end{document} for t∈[0,T)\documentclass[12pt]{minimal}
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\begin{document}$$ t \in [0, T) $$\end{document} with some constant C>0\documentclass[12pt]{minimal}
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\begin{document}$$ C > 0 $$\end{document}, and type II otherwise. It was shown in Mizoguchi (J Funct Anal 271:3323–3347, 2016) that each blowup is type II in radial case. In this paper, we obtain the conclusion in general case.
