{"title":"Optimal decay estimate and asymptotic profile for solutions to the generalized Zakharov–Kuznetsov–Burgers equation in 2D","authors":"Ikki Fukuda , Hiroyuki Hirayama","doi":"10.1016/j.nonrwa.2024.104130","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the Cauchy problem for the generalized Zakharov–Kuznetsov–Burgers equation in 2D. This is one of the nonlinear dispersive–dissipative equations, which has a spatial anisotropic dissipative term <span><math><mrow><mo>−</mo><mi>μ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow></math></span>. In this paper, we prove that the solution to this problem decays at the rate of <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></math></span> in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-sense, provided that the initial data <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> satisfies <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm of the solution. As a result, we prove that the given decay rate <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></math></span> of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schr<span><math><mover><mrow><mi>o</mi></mrow><mrow><mo>̈</mo></mrow></mover></math></span>dinger equation, we derive the explicit asymptotic profile for the solution.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Real World Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824000701","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the Cauchy problem for the generalized Zakharov–Kuznetsov–Burgers equation in 2D. This is one of the nonlinear dispersive–dissipative equations, which has a spatial anisotropic dissipative term . In this paper, we prove that the solution to this problem decays at the rate of in the -sense, provided that the initial data satisfies and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the -norm of the solution. As a result, we prove that the given decay rate of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schrdinger equation, we derive the explicit asymptotic profile for the solution.
期刊介绍:
Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems.
The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.