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.