Classification of asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients: Effective damping case

Shuji Yoshikawa,Yuta Wakasugi

doi:10.1016/j.jde.2020.10.008

Classification of asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients: Effective damping case

Shuji Yoshikawa, Yuta Wakasugi

https://doi.org/10.1016/j.jde.2020.10.008

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Journal: Journal of Differential Equations	Publication Date: Oct 21, 2020
Citations: 1

Affiliation: Oita University, Hiroshima University

#Variable Coefficients #Asymptotic Profile + Show 8 more

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Abstract

We consider the Cauchy problem for the linear beam equation with two variable coefficients:utt+b(t)ut+uxxxx−a(t)uxx=0,(t,x)∈R+×R. The purpose of this study is to classify the behavior of solution with respect to (α,β) when the coefficients a and b behave as a(t)∼(1+t)α and b(t)∼(1+t)β. In this article we shall give the asymptotic profile formula in effective damping cases, by using the method of scaling variables developed by Gallay and Raugel [9].

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