On the Cauchy problem for the wave equation in a two-dimensional domain with data on the boundary

Abstract

The subject of the paper is the Cauchy problem for the wave equation in a space-time cylinder $\Omega \times \mathbb{R}$, $\Omega \subset {\mathbb{R}}^2$, with the data on the surface $\partial \Omega \times I$, where $I$ is a finite time interval. The algorithm for solving the Cauchy problem with the data on $S\times I$, $S\subset \partial \Omega$, was obtained previously. Here we adapt this algorithm to the special case $S=\partial \Omega$ and show that in this situation, the solution is determined with higher stability in comparison with the case $S⫋\partial \Omega$.