Convergence of an approximate solution of the Showalter–Sidorov–Dirichlet problem for the modified Boussinesq equation

In this paper, we obtain necessary and sufficient conditions for the existence of a unique solution of the Showalter–Sidorov–Dirichlet problem for a second-order, semilinear Sobolev-type equation. For the initial-boundary-value problem considered, using the Galerkin method, we construct an approximate solution as an expansion in the system of eigenfunctions of the homogeneous Dirichlet problem for the Laplace operator. The proof of the $*$-weak convergence of the Galerkin approximations to the exact solution is based on a priori estimates, embedding theorems, and the Gronwall lemma.