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

We consider the Cauchy problem for the linear beam equation: utt+ut+uxxxx−a(t)uxx=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).