On a parabolic problem in an infinite corner domain

Abstract

The paper deals with homogeneous problem of Solonnikov-Fasano for parabolic equations in the degenerating infinite angular domain with special boundary conditions on a moving boundary. Using heat potentials, this problem is reduced to solving a special Volterra integral equation of the second kind, to which the method of successive approximations is not applicable. The solution of the integral equation was found by the method of integral transformations. It is shown that in the class of essentially bounded functions with a given weight, the homogeneous problem of Solonnikov-Fasano, in addition to the trivial solution, has a nontrivial solution, up to a constant factor.