Geodesic
An analogue of the Tricomi problem for a mixed type of~quasilinear equation with two lines of degeneracy
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 4, pp. 630-649

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The paper proves the unique solvability of an analog of the Tricomi problem for a quasilinear equation of mixed type with two lines of degeneracy. The class $R_1$ of generalized solutions in the hyperbolic part of the domain is introduced. The uniqueness of the solution is proved by the method of energy integrals. The existence of a solution is proved by the method of integral equations. The boundary value problem is reduced to an equivalent system of integral equations, the solvability of which is proved using the Schauder principle. As a result, the application of the Schauder principle resulted in the global solvability of the problem under study without any restrictions on the size of the area of the region under consideration and on the value of the given functions.
Keywords: generalized solution, normal curve, method of integrals energy, integral equation of normal type, index of integral equation, regularization, equicontinuity, Schauder's principle. 
@article{VSGTU_2022_26_4_a1,
     author = {X. R. Rasulov},
     title = {An analogue of the {Tricomi} problem for a mixed type of~quasilinear equation with two lines of degeneracy},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {630--649},
     publisher = {mathdoc},
     volume = {26},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2022_26_4_a1/}
}
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X. R. Rasulov. An analogue of the Tricomi problem for a mixed type of~quasilinear equation with two lines of degeneracy. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 4, pp. 630-649. http://geodesic.mathdoc.fr/item/VSGTU_2022_26_4_a1/