A Logarithmic Decay of the Energy for the Hyperbolic Equation with Supercritical Damping

作     者：LI Xiaolei GUO Bin LI Xiaolei;GUO Bin

作者机构：School of Mathematics and StatisticsChangchun University of Science and TechnologyChangchun 130022China School of MathematicsJilin UniversityChangchun 130012China 

出 版 物：《Journal of Partial Differential Equations》 (偏微分方程（英文版）)

年 卷 期：2024年第37卷第2期

页      面：150-165页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by the Scientific and Technological Project of jilin Province's Education Department in Thirteenth Five-Year(JKH20180111KI) supported by NSFC(11301211) 

主　　题：Energy decay estimate asymptotic behavior p(x)-Laplacian operator supercritical damping 

摘      要：We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic behavior of solutions,an inevitable step is to deal with the integralΩ|ut|^(m−2)utudx.A usual technique is to apply Young’s inequality and Sobolev embedding inequality to use the energy function and its derivative to control this integral for the subcritical or critical ***,for the supercritical case,the failure of the Sobolev embedding inequality makes the classical method be *** do this,our strategy is to prove the rate of the integral RΩ|u|^(m)dx grows polynomially as a positive power of time variable t and apply the modified multiplier method to obtain the energy functional decays *** results improve and extend our previous work[12].Finally,some numerical examples are also given to authenticate our results.