Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient

Abstract We consider the Cauchy problem for the linear beam equation: u t t + u t + u x x x x − a ( t ) u x x = 0 , ( t , x ) ∈ R + × R , where a ( t ) ∼ ( 1 + t ) α . The purpose of this study is to clarify the behavior of solution depending on the rate α . Here we shall give the asymptotic behavior of the solution in the case α > − 1 ∕ 2 , by using the method of scaling variables developed by Gallay and Raugel (1998).

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