# Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients

Vladyslav Litovchenko; Iryna Dovzhytska

Open Mathematics (2012)

- Volume: 10, Issue: 3, page 1084-1102
- ISSN: 2391-5455

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topVladyslav Litovchenko, and Iryna Dovzhytska. "Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients." Open Mathematics 10.3 (2012): 1084-1102. <http://eudml.org/doc/269354>.

@article{VladyslavLitovchenko2012,

abstract = {In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.},

author = {Vladyslav Litovchenko, Iryna Dovzhytska},

journal = {Open Mathematics},

keywords = {Cauchy problem; Shilov parabolic systems with variable coefficients; Generalized initial data; Correct solvability; Principle of localization; generalized initial data; correct solvability; principle of localization},

language = {eng},

number = {3},

pages = {1084-1102},

title = {Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients},

url = {http://eudml.org/doc/269354},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Vladyslav Litovchenko

AU - Iryna Dovzhytska

TI - Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients

JO - Open Mathematics

PY - 2012

VL - 10

IS - 3

SP - 1084

EP - 1102

AB - In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.

LA - eng

KW - Cauchy problem; Shilov parabolic systems with variable coefficients; Generalized initial data; Correct solvability; Principle of localization; generalized initial data; correct solvability; principle of localization

UR - http://eudml.org/doc/269354

ER -

## References

top- [1] Gel’fand I.M., Shilov G.E., Generalized Functions. 3, Academic Press, New York-London, 1967
- [2] Gel’fand I.M., Shilov G.E. Generalized Functions. 2, Academic Press, New York-London, 1968
- [3] Gorodetskii V.V., Localization principle for solutions of the Cauchy problem for parabolic systems in the class of generalized functions of infinite order, Differ. Uravn., 1985, 21(6), 1077–1079 (in Russian) Zbl0609.35046
- [4] Khou-sin’ U., On the definition of parabolic systems of partial differential equations, Uspekhi Mat. Nauk, 1960, 15(6), 157–161 (in Russian)
- [5] Litovchenko V.A., Dovzhitska I.M., The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients, J. Math. Sci., 2011, 175(4), 450–476 http://dx.doi.org/10.1007/s10958-011-0356-0 Zbl1282.35185
- [6] Zhitomirskii Ya.I., The Cauchy problem for certain types of systems, parabolic in the sense of G.E. Shilov, of linear partial differential equations with variable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 1959, 23, 925–932 (in Russian)

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