Uniqueness and non-uniqueness of solutions of the boundary value problems of the heat equation

The article addresses the singular Volterra integral equation of the second kind which has the "incompressible" kernel. It is shown that the corresponding homogeneous equation on broken vertical bar lambda broken vertical bar exp{arg lambda arg A E [-pi, pi] has a continuous spectrum, and the multiplicity of the characteristic numbers grows with increasing A I The equation is reduced to Abel equation by the regularization method. The eigenfunctions of the equation are found in an explicit form. We prove the solvability theorem of the inhomogeneous equation in a case when the right-hand side of the equation belongs to a certain class.