Abstract
In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space \({L^{\frac{n}{2}}(\mathbb{R}^{n})}\) , then the L q-norm (\({\frac{n}{2} \leq q \leq \infty}\)) of the βth order spatial derivative of mild solutions are majorized by \({K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}}\) for some constants K 1 and K 2. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space \({\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)}\) (\({\frac{n}{2} < p < n}\)).
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Zhao, J., Liu, Q. & Cui, S. Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system. Nonlinear Differ. Equ. Appl. 19, 1–18 (2012). https://doi.org/10.1007/s00030-011-0115-4
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DOI: https://doi.org/10.1007/s00030-011-0115-4