Frequency-localization Duhamel principle and its application to the optimal decay of dissipative systems in low dimensions

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• 作者：Jiang Xu^a ; ^{jiangxu_79@nuaa.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Shuichi Kawashima^b ; ^{kawashim@math.kyushu-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35L60 ; 35L45 ; 35F25 ; 35B40
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 September 2016
• 年：2016
• 卷：261
• 期：5
• 页码：2670-2701
• 全文大小：468 K

文摘

Recently, a time-decay framework $L^2({\mathbb{R}}^n)\cap {\dot{B}}_{2,\infty }^{-s}({\mathbb{R}}^n)(s>0)$ has been given by [49] for linearized dissipative hyperbolic systems, which allows to pay less attention to the traditional spectral analysis. However, owing to interpolation techniques, those decay results for nonlinear hyperbolic systems hold true only in higher dimensions (n≥3)$(n\ge 3)$, and the analysis in low dimensions (say, n=1,2$n=1,2$) was left open. We try to give a satisfactory answer in the current work. First of all, we develop new time-decay properties on the frequency-localization Duhamel principle, and then it is shown that the classical solution and its derivatives of fractional order decay at the optimal algebraic rate in dimensions n=1,2$n=1,2$, by using a new technique which is the so-called “piecewise Duhamel principle” in localized time-weighted energy approaches compared to [49]. Finally, as direct applications, explicit decay statements are worked out for some relevant examples subjected to the same dissipative structure, for instance, damped compressible Euler equations, the thermoelasticity with second sound, and Timoshenko systems with equal wave speeds.