On a Volterra equation of the second kind with ‘incompressible’ kernel

Abstract

Solving the boundary value problems of the heat equation in noncylindrical domains degenerating at the initial moment leads to the necessity of research of the singular Volterra integral equations of the second kind, when the norm of the integral operator is equal to 1. The paper deals with the singular Volterra integral equation of the second kind, to which by virtue of ‘the incompressibility’ of the kernel the classical method of successive approximations is not applicable. It is shown that the corresponding homogeneous equation when $|\lambda|>1$ has a continuous spectrum, and the multiplicity of the characteristic numbers increases depending on the growth of the modulus of the spectral parameter $|\lambda|$ . By the Carleman-Vekua regularization method (Vekua in Generalized Analytic Functions, 1988) the initial equation is reduced to the Abel equation. The eigenfunctions of the equation are found explicitly. Similar integral equations also arise in the study of spectral-loaded heat equations (Amangaliyeva et al. in Differ. Equ. 47(2):231-243, 2011).