Logarithmic nature of the long-time asymptotics of solutions of a Sobolev-type nonlinear equations with cubic nonlinearities

Abstract

The Cauchy problem of the form $$ \begin{cases} i \partial_{t}(u-\partial_{x}^{2}u)+\partial_{x}^{2}u -a \partial_{x}^{4}u=u^{3}, & t>0, x\in\mathbb{R}, u(0,x) =u_{0}(x),& x\in\mathbb{R}, \end{cases} $$ is considered for a Sobolev-type nonlinear equation with cubic nonlinearity, where $a>1/5$, $a\neq1$. It is shown that the asymptotic behaviour of the solution is characterized by an additional logarithmic decay in comparison with the corresponding linear case. To find the asymptotics of solutions of the Cauchy problem for a nonlinear Sobolev-type equation, factorization technique is developed. To obtain estimates for derivatives of the defect operators, $\mathbf{L}^{2}$-estimates of pseudodifferential operators are used. Bibliography: 20 titles.