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

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: $$\partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u = \partial_x \left( N(\partial_x u) \right).$$ In the authors' previous article [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 damping cases. We here give the conditions of the coefficients and the nonlinear term in 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 $t \to \infty$.