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In this article we investigate the asymptotic profile of solutions for theCauchy problem of the nonlinear damped beam equation with two variablecoefficients: \[ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u +\partial_x^4 u  = \partial_x \left( N(\partial_x u) \right). \] In the authors' previousarticle [17], the asymptotic profile of solutions for linearized problem ($N\equiv 0$) was classified depending on the assumptions for the coefficients$a(t)$ and $b(t)$ and proved the asymptotic behavior in effective dampingcases. We here give the conditions of the coefficients and the nonlinear termin order that the solution behaves as the solution for the heat equation: $b(t)\partial_t u - a(t) \partial_x^2 u=0$ asymptotically as $t \to \infty$.

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity