Asymptotics for the modified Boussinesq equation in one space dimension (Q1688011)

The authors consider the Cauchy problem for the modified 1D Boussinesq equation \[ \begin{cases} w_{tt}=a^2\partial^2_xw-\partial^4_xw+\partial^2_x(w^3), \quad (t,x)\in \mathbb{R}^2, \\ w(0,x)=w_0(x), \quad w_t(0,x)=w_1(x), \quad x\in \mathbb{R}, \end{cases} \] where \(a>0\). This equation describes the propagation of long waves in shallow water. By using the so-called factorization technique, the authors investigate the long time behavior of the solutions of the above Cauchy problem. Specifically, they show that for ``small'' initial data there exists a unique solution on \([0,\infty)\) of the Cauchy problem satisfying the estimate \(|w(t)|_{L^{\infty}}\leq C\varepsilon t^{-1/2}\).