Decay and scattering of small solutions of a generalized Boussinesq equation
Let \(u\) be a solution of the Boussinesq equation: \[ u_{tt}-u_{xx}+ \bigl( u_{xx} +f(u)\bigr)_{xx}=0 \] for \(x\in\mathbb{R}\), and denote \(u_0=u|_{t=0}\), \(v_0=v_t |_{t=0} \). Then, if \((u_0,v_0)\) is small enough in the energy norm, and if \(f(\tau)= {\mathcal O} (|\tau |^p)\) as \(\tau\to 0\) with \(p>5\), it is proven that \(u\) decays uniformly like \(|t |^{-1/3}\) when \(t\to\pm\infty\). Moreover, in this case it is also possible to construct a scattering operator for the Boussinesq equation, which acts on a neighborhood of 0 in the energy space.
scientific article; zbMATH DE number 1045907