Geodesic
Boundary value problem with a nonlocal boundary condition of integral form for a multidimensional equation of IV order
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 1, pp. 15-28

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The aim of this paper is to study the solvability of solution of non-local problem with integral condition in spatial variables for high-order linear equation in the classe of regular solutions (which have all the squared derivatives generalized by S.L. Sobolev that are included in the corresponding equation). It is indicated that at first similar problems were studied for high-order equations either in the one-dimensional case, or under certain conditions of smallness by the value of $T$. A list of new works for the multidimensional case is also given. In this paper, we present new results on the solvability of non-local problem with integral spatial variables for high-order equation a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions by the value $T$; however, this condition exists for the kernel $K(x,y,t)$. The research method is based on obtaining a priori estimates of the solution of the problem, which implies its existence and uniqueness in a given space.
Keywords: boundary value problem, high-order equation, priori estimates, integral boundary conditions, regular solutions, uniqueness
Mots-clés : non-local condition, Boussinesq equation, existence. 
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     author = {V. B. Dmitriev},
     title = {Boundary value problem with a nonlocal boundary condition of integral form for a multidimensional equation of {IV} order},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {15--28},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2021_27_1_a1/}
}
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V. B. Dmitriev. Boundary value problem with a nonlocal boundary condition of integral form for a multidimensional equation of IV order. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 1, pp. 15-28. http://geodesic.mathdoc.fr/item/VSGU_2021_27_1_a1/