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

Abstract We consider the Cauchy problem for the linear beam equation with two variable coefficients: u t t + b ( t ) u t + u x x x x − a ( t ) u x x = 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] .