L2-blowup estimates of the wave equation and its application to local energy decay

IF 0.5 4区数学 Q4 MATHEMATICS, APPLIED

Journal of Hyperbolic Differential Equations Pub Date : 2021-11-03 DOI:10.1142/s021989162350008x

R. Ikehata

引用次数: 10

Abstract

We consider the Cauchy problems in [Formula: see text] for the wave equation with a weighted [Formula: see text]-initial data. We derive sharp infinite time blowup estimates of the [Formula: see text]-norm of solutions in the case of [Formula: see text] and [Formula: see text]. Then, we apply it to the local energy decay estimates for [Formula: see text], which is not studied so completely when the [Formula: see text]th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the [Formula: see text]-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].

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波动方程的L2爆破估计及其在局部能量衰减中的应用

我们在[公式：见正文]中考虑波动方程的Cauchy问题，该波动方程具有加权[公式：参见正文]-初始数据。在[公式：见正文]和[公式：看正文]的情况下，我们导出了解的[公式：见正文]-范数的尖锐的无限时间放大估计。然后，我们将其应用于[公式：见正文]的局部能量衰减估计，当初始速度的[公式：参见正文]时刻没有消失时，这一估计就没有得到充分的研究。导出它们的想法受到了[R.Ikehata，具有强阻尼的波动方程的渐近轮廓，J.Differ.Equ.257（2014）2159-2177；R.Ikehata和M.Onodera，关于强阻尼波动方程解的[Former:见正文]-范数的大时间行为的注释，Differ。积分等于。30（2017）505–520]。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Hyperbolic Differential Equations 数学-物理：数学物理

CiteScore

1.10

自引率

0.00%

发文量

审稿时长

24 months

期刊介绍： This journal publishes original research papers on nonlinear hyperbolic problems and related topics, of mathematical and/or physical interest. Specifically, it invites papers on the theory and numerical analysis of hyperbolic conservation laws and of hyperbolic partial differential equations arising in mathematical physics. The Journal welcomes contributions in: Theory of nonlinear hyperbolic systems of conservation laws, addressing the issues of well-posedness and qualitative behavior of solutions, in one or several space dimensions. Hyperbolic differential equations of mathematical physics, such as the Einstein equations of general relativity, Dirac equations, Maxwell equations, relativistic fluid models, etc. Lorentzian geometry, particularly global geometric and causal theoretic aspects of spacetimes satisfying the Einstein equations. Nonlinear hyperbolic systems arising in continuum physics such as: hyperbolic models of fluid dynamics, mixed models of transonic flows, etc. General problems that are dominated (but not exclusively driven) by finite speed phenomena, such as dissipative and dispersive perturbations of hyperbolic systems, and models from statistical mechanics and other probabilistic models relevant to the derivation of fluid dynamical equations. Convergence analysis of numerical methods for hyperbolic equations: finite difference schemes, finite volumes schemes, etc.