Periodic initial-value problem for BBM-equation
The BBM or regularized long-wave equation \[ u_t+u_x+uu_x-u_{xxt}=0,\;x\in\mathbb{R},\;t>0,\quad u(x,0)=\varphi(x),\;x\in\mathbb{R},\tag{1} \] was originally proposed as an alternative to the Korteweg-de Vries equation in describing small-amplitude, long surface wave propatation. Its well-posedness in \(H^1(\mathbb{R})\) and \(L_2(\mathbb{R})\) have been studied by many authors. We consider the BBM-equation while the initial data \(\varphi\) is a periodic function on \(\mathbb{R}\). The result is that if \(\varphi\) is Lebesgue measurable and square-integrable within one period interval, then equation (1) is globally well posed in time \(t\).
scientific article; zbMATH DE number 2167473
scientific article