Geodesic
Nonlinear integro-differential equation with a high-degree hyperbolic operator
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 53-64

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In this paper, we examine the solvability of the initial-value problem for a nonlinear integro-differential equation with a hyperbolic operator of arbitrary natural degree and a degenerate kernel. The expression of the high-order partial differential operator on the left-hand side of the equation through the superposition of first-order differential operators allowed us to represent the equation considered as an integral equation for unknown function along the characteristics. Also, we prove the unique solvability of the initial-value problem and the stability of solutions with respect to initial data.
Keywords: initial-value problem, characteristic, superposition of differential operators, high-degree hyperbolic operator, degenerate kernel, unique solvability. 
@article{INTO_2021_201_a4,
     author = {T. K. Yuldashev (Iuldashev) and I. U. Nazarov},
     title = {Nonlinear integro-differential equation with a high-degree hyperbolic operator},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {53--64},
     publisher = {mathdoc},
     volume = {201},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_201_a4/}
}
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T. K. Yuldashev (Iuldashev); I. U. Nazarov. Nonlinear integro-differential equation with a high-degree hyperbolic operator. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 53-64. http://geodesic.mathdoc.fr/item/INTO_2021_201_a4/