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

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.