Geodesic
Differential properties of a~generalized solution to a~hyperbolic system of first-order differential equations
Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 2, pp. 26-39

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We study some questions of the qualitative theory of solutions to differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations. The right-hand sides of these equations contain discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. It is in particular established that its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts a common approach of investigation which uses the reduction of a system of two first-order equations to a single second-order equation.
Keywords: hyperbolic equations, discontinuous functions, generalized solutions, differential properties. 
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     author = {D. S. Anikonov and S. G. Kazantsev and D. S. Konovalova},
     title = {Differential properties of a~generalized solution to a~hyperbolic system of first-order differential equations},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2013_16_2_a2/}
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D. S. Anikonov; S. G. Kazantsev; D. S. Konovalova. Differential properties of a~generalized solution to a~hyperbolic system of first-order differential equations. Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 2, pp. 26-39. http://geodesic.mathdoc.fr/item/SJIM_2013_16_2_a2/