Abstract
An initial boundary value problem is considered for an equation of Sobolev type that is four-dimensional in terms of spatial variables. The region of interest here is a cylindrical shell in \({{\mathbb{R}}^{4}}\) . In other words, a single coordinate direction (variable \({{x}_{4}}\) ) parallel to the cylinder axis is singled out, and the base of the cylinder lies in the subspace that specifies the other coordinate directions. Note that spherical layers from \({{\mathbb{R}}^{3}}\) with identical cavity radii underlie the bases of the cylindrical shell. In that case, we may consider solutions characterized by spherical radial symmetry with respect to the three variables. For that domain, accurate solutions may then be obtained as functional series in terms of eigenfunctions of the auxiliary boundary problem. Finally, the theorem of unique solubility is established by means of integral identities.
