Pointwise convergence problem of the one-dimensional Boussinesq-type equation

In this paper, we consider the initial data problem of the one-dimensional Boussinesq-type equation. Inspired by the work of Compaan et al. (Int. Math. Res. Not. 2021, 599-650), by using the Fourier restriction norm method, we investigate the pointwise convergence of the one-dimensional Boussinesq equation and cubic Boussinesq equation. We also only use the Strichartz estimates instead of Fourier restriction norm method to establish convergence conclusions of the one-dimensional cubic Boussinesq equation. The key ingredients are some Strichartz estimates and high-low frequency decomposition related to the nonlinear part of the integral form solution in the Duhamel form and the maximal function estimate related to inhomogeneous estimate.