Abstract
This paper is concerned with the Cauchy problem for the Keller–Segel system
with a constant λ ≥ 0, where \({(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}\). Let
. The same method as in [9] yields the existence of a blowup solution with m (u 0; R 2) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses \({u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}\) and \({u_0 \log u_0 \in L^1 ({\bf R}^2)}\), any solution with m(u 0; R 2) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u 0; R 2) < 4π. In this paper, we prove that any solution with m (u 0; R 2) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u 0 also in the critical case m (u 0; R 2) = 8π.
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Mizoguchi, N. Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane. Calc. Var. 48, 491–505 (2013). https://doi.org/10.1007/s00526-012-0558-4
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DOI: https://doi.org/10.1007/s00526-012-0558-4