The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

Alexander Kozhanov, Dr. Sci. (Phys.–Math.), Prof., S. L. Sobolev Institute of Mathematics SB RAS, 4 Koptyuga Ave., Novosibirsk, 630090, Russian Federation, Samara State Technical University, 244 Molodogvardeyskaya Str., Samara, 443100, Russian Federation, tel.: (383)333-28-92, email: kozhanov@math.nsc.ru

Alexandra Dyuzheva, Cand. Sci. (Phys.–Math.), Assoc. Prof., Samara State Technical University, 244 Molodogvardeyskaya Str., Samara, 443100, Russian Federation, tel.: (846)278-43-53, email: aduzheva@rambler.ru

Kozhanov A.I., Dyuzheva A.V. Non-local Problems with Integral Displacement for High-order Parabolic Equations. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 36, pp. 14-28. (in Russian) https://doi.org/10.26516/1997-7670.2021.36.14

high-order parabolic equations, non-local problems, integral boundary conditions, regular solutions, uniqueness, existence.

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