On an integral equation of the problem of heat conduction with domain boundary moving by law of t = x 2In the article it is shown that the homogeneous Volterra integral equation of the second kind, to which the homogeneous boundary value problem of heat conduction in the degenerating domain is reduced, has a nonzero solution. The boundary of the domain moves with a variable velocity. It is shown that the norm of the integral operator acting in classes of continuous functions is equal to 1. Mellin transformation is applied to the obtained integral equation. It is proved that for certain values of the spectral parameter the eigenvalues of the integral equation will be simple.

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This paper considers an initial boundary value problem for a one-dimensional Boussinesq-type equation in a domain, that is, a trapezoid. Using the methods of the theory of monotone operators, we establish theorems on their unique weak solvability in Sobolev classes.