On the Solvability of Heat Boundary Value Problems in Sobolev Spaces
Jenaliyev M.T. Kosmakova M.T. Tuleutaeva Z.M.
August 2022Pleiades Publishing
Lobachevskii Journal of Mathematics
2022#43Issue 82133 - 2144 pp.
Abstract: In a domain, degenerating into a point at the initial moment of time, we consider a boundary value problem (BVP) of heat conduction that is a two-dimensional BVP with respect to space variables. The problem is studied for solvability in Sobolev Hilbert spaces. The correctness of the set problem is shown. The solvability of the heat BVP in a truncated cone is preliminary proved. Next, a system of embedded truncated cones is constructed, the union of which in the limit gives the domain (the cone) of the solution of the original boundary value problem. For each cone, a boundary value problem of heat conduction is posed, similar to the original problem, and its unique solvability is shown. Then, from the solutions of these problems, a sequence is compiled, and the terms of the sequence are continued by zero to complement the domain of these solutions up to the original cone. Using the methods of functional analysis, it is proved that the limit of this sequence is the only solution to the boundary problem under study.
boundary value problem , degenerating domain , heat equation , Hilbert space , Sobolev space
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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Karaganda Buketov University, Karaganda, Kazakhstan
Karaganda Technical University, Karaganda, Kazakhstan
Institute of Mathematics and Mathematical Modeling
Karaganda Buketov University
Karaganda Technical University
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