To the solution of the Solonnikov-Fasano problem with boundary moving on arbitrary law x = γ(t).

Muvasharkhan Jenaliyev, M.I. Ramazanov, A.O. Tanin

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS, 2021

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DOI 10.31489/2021m1/37-49

Data confidence 70 medium

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Abstract

In this paper we study the solvability of the boundary value problem for the heat equation in a domain that degenerates into a point at the initial moment of time. In this case, the boundary changing with time moves according to an arbitrary law x = γ(t). Using the generalized heat potentials, the problem under study is reduced to a pseudo-Volterra integral equation such that the norm of the integral operator is equal to one and it is shown that the corresponding homogeneous integral equation has a nonzero solution.