Geodesic
Ill-posed boundary-value problem for a system of partial differential equations with two degenerate lines
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 3, pp. 392-401

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This paper is devoted to the investigation of ill-posed boundary-value problem for system of parabolic type equations with changing time direction with two degenerate lines. The problem under consideration is ill-posed in the sense of J. Hadamard, namely, there is no continuous dependence of the solution on the initial data. Such equations have many different applications, for example, describe the processes of heat propagation in inhomogeneous media, the interaction of filtration flows, mass transfer near the surface of an aircraft, and the description of complex viscous fluid flows. As possible applications should also indicate the task of calculating heat exchangers, in which the counter flow principle is used. Theorems on the uniqueness and conditional stability of a solution on a set of well-posedness are proved. We construct a sequence of approximate (regularized) solutions that are stable on the set of well-posedness.
Keywords: parabolic equation with changing time direction, ill-posed problem, a priori estimate, estimation of conditional stability, uniqueness of solution, approximate solution. 
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     author = {Kudratillo S. Fayazov and Yashin K. Khudayberganov},
     title = {Ill-posed boundary-value problem for a system of partial differential equations with two degenerate lines},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {392--401},
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     number = {3},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2019_12_3_a13/}
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Kudratillo S. Fayazov; Yashin K. Khudayberganov. Ill-posed boundary-value problem for a system of partial differential equations with two degenerate lines. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 3, pp. 392-401. http://geodesic.mathdoc.fr/item/JSFU_2019_12_3_a13/