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End-point maximal regularity and its application to two-dimensional Keller–Segel system

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Abstract

We prove the maximal L ρ regularity of the Cauchy problem of the heat equation in the Besov space \({\dot{B}_{1,\rho}^0(\mathbb{R}^n)}\), 1 < ρ ≤ ∞, which is not UMD space. And as its application, we establish the time local well-posedness of the solution of two dimensional nonlinear parabolic system with the Poisson equation in \({\dot{B}_{1,2}^0(\mathbb{R}^2)}\) , where the equation is considered in the space invariant by a scaling and particularly the natural free energy is well defined from the initial time. The small data global existence is also obtained in the same class.

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Takayoshi Ogawa

  2. Faculty of Science, Shizuoka University, Shizuoka, 422-8529, Japan

    Senjo Shimizu

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  1. Takayoshi Ogawa
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  2. Senjo Shimizu
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Correspondence to Senjo Shimizu.

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Ogawa, T., Shimizu, S. End-point maximal regularity and its application to two-dimensional Keller–Segel system. Math. Z. 264, 601–628 (2010). https://doi.org/10.1007/s00209-009-0481-3

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