Geodesic
On the solvability of the boundary value~problem for one class of nonlinear systems of high-order partial differential equations
Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 841-860

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For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.
Keywords: high-order nonlinear systems; fixed point principles; existence, uniqueness and nonexistence of a solution. 
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     title = {On the solvability of the boundary value~problem for one class of nonlinear systems of high-order partial differential equations},
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S. S. Kharibegashvili; B. G. Midodashvili. On the solvability of the boundary value~problem for one class of nonlinear systems of high-order partial differential equations. Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 841-860. http://geodesic.mathdoc.fr/item/SM_2024_215_6_a6/