The Goursat problem for hyperbolic linear third order equations

• Main Author: V. I. Korzyuk
• Format: Article
• Language: English
• Published: Vilnius Gediminas Technical University 2001-12-01
• Series: Mathematical Modelling and Analysis
• Subjects: -
• Online Access: https://journals.vgtu.lt/index.php/MMA/article/view/9913

The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality. Trečios eilės tiesinių hiperbolinių lygčių Goursat uždavinys Santrauka Daugiamate Euklido erdves necilindrineje srityje nagrinejama trečios eiles tiesine hiper‐boline lygtis. Lygties operatorius yra pirmos eiles diferencialinio operatoriaus ir antros eiles operatoriaus, kuris yra hiperbolinis apibrežto vektorinio lauko atžvilgiu, kompozicija. Srities kontūra sudaro charakteristinis hiperpaviršius (jame formuojama Goursat salyga), hiperpaviršiaus, kuriame formuluojama Caushy salyga, ir laisvas nuo bet kokiu salygu hiperpaviršius. Naudojantis kintamojo žingsnio suvidurkinto operatoriaus bei funkcines analizes metodais, paremtais energetine nelygybe, irodytas šio uždavinio stipriojo sprendinio egzistavimas ir vienatis. First Published online: 14 Oct 2010