On the non-uniqueness of solution to the homogeneous boundary value problem for the heat conduction equation in an angular domain

Abstract

The article deals with the homogeneous boundary value problem for the heat equation in a degenerating angular domain. Using a simple layer potentials the posed problem is reduced to pseudo-Volterra integral equation of the second kind. The obtained integral equation is solved by the method of regularization. For this purpose the characteristic part of the integral equation is allocated. Non - applicability of the method of successive approximations is substantiated for its solving. We have proven the lemma on reducing the obtained integral equation to an equation with a diference kernel and have written its solution. We give an estimate for the resolvent of the equation with a diference kernel, and the conditions for boundedness of its solutions. The explicit representation of the solution to the equation with a diference kernel leads the initial integral equation to a Volterra equation of the second kind with a weak singularity, which has a unique solution. The solution is written in the operator form. It is shown that the posed homogeneous boundary value problem has a nontrivial solution up to a constant factor in the class of essentially bounded functions with defined weight. Classes of uniqueness for solution to the posed boundary value problem are define.