Asymptotic behaviour of nonlinear dispersive models with variable coefficients (Q1913397)

The paper is concerned with the asymptotic behavior of solutions of the nonlinear dispersive equation \[ Mu_t+a(t)u_t+b(t)u^pu_x=0\quad\text{for}\quad (x,t)\in\mathbb{R}\times\mathbb{R}^+\quad\text{and}\quad u(x,0)=\varphi(x),\tag{\(*\)} \] where \(M\) is a Fourier multiplier operator defined as \(\widehat{Mf}(y)=m(y)\widehat f(y)\), and \(p\) is a positive integer. The authors were motivated by the decay estimate of solutions of the generalized Benjamin-Bona-Mahony equation: \(u_t+u_x+u^pu_x-u_{xxt}=0\) in [\textit{J. Albert}, J. Math. Anal. Appl. 141, No. 2, 527-537 (1989; Zbl 0697.35116)], and derived an analogous one of solutions \(u(x,t)\) of \((*)\) with sufficiently small initial data \(\varphi(x)\) under suitable assumptions on \(m(y)\), \(a(t)\), \(b(t)\) and \(p\). Roughly, the obtained estimate is of the type \[ |u(\cdot,t)|_{L^\infty}=O\Biggl(\Biggl(1+\int^t_0 a(s)ds\Biggr)^{-\alpha}\Biggr)\quad\text{as}\quad t\to+\infty, \] where \(\int^\infty_0f(t)dt\) is assumed to be positively infinite and \(\alpha\) is a positive constant related to constants in the assumptions.