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On the Temporal Decay of Solutions to a Coupled Parabolic-elliptic Equations in \({\mathbb {R}}^2\)

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This paper is devoted to the temporal decay of solutions of a coupled parabolic-elliptic equations in \({\mathbb {R}}^2\). It is proved that weak solutions of the equations decay to zero in \(L^{2,\infty }\times L^2\) without a uniform rate, and these decay estimates are optimal. Furthermore, the uniform logarithmic decay estimates of weak solutions are established when initial data are in \(L^1\cap L^2\). In addition, the temporal decay estimates of weak solutions at the sharp rate are also shown. The proofs are based on the Fourier splitting method and the generalized Ladyzhenskaya inequality for weak type spaces.

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Acknowledgements

The author was partially supported by Guangdong Basic and Applied Basic Research Foundation (No.2020A1515110299) and Natural Science Foundation of Guangzhou City (No.202102020906).

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  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, China

    Wen Tan

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Tan, W. On the Temporal Decay of Solutions to a Coupled Parabolic-elliptic Equations in \({\mathbb {R}}^2\). J Dyn Diff Equat (2022). https://doi.org/10.1007/s10884-022-10204-8

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