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Recall that in the paper \"Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, {252}(2012), 1213--1228\", Kozono, Sugiyama & Wachi studied global well-posedness of $n$($\\ge3$) dimensional Keller-Segel system and posted a question about the even local in time existence for the Keller-Segel system with $L^1(R^2)\\times{L}^\\infty(R^2)$ initial data. 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Recall that in the paper \"Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, {252}(2012), 1213--1228\", Kozono, Sugiyama & Wachi studied global well-posedness of $n$($\\ge3$) dimensional Keller-Segel system and posted a question about the even local in time existence for the Keller-Segel system with $L^1(R^2)\\times{L}^\\infty(R^2)$ initial data. 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