Stable pencils of hyperbolic polynomials and the Cauchy problem for hyperbolic equations with a small parameter at the highest derivatives

Authors:
L. R. Volevich and E. V. Radkevich

Translated by:
O. Khleborodova

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **65** (2004).

Journal:
Trans. Moscow Math. Soc. **2004**, 63-104

MSC (2000):
Primary 35B25, 35L25.

DOI:
https://doi.org/10.1090/S0077-1554-04-00147-5

Published electronically:
November 4, 2004

MathSciNet review:
2193437

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study pencils of hyperbolic polynomials of the form , where is a real homogeneous polynomials of degree resolved with respect to the highest power of and ; the numbers are positive. In the first part of the paper we find necessary and close to sufficient conditions of stability of the polynomial (i.e., the condition that its roots lie in the open upper half-plane of the complex plane). This problem is closely related to the problem on uniform (with respect to a small parameter) estimates for the solution of the Cauchy problem for hyperbolic equations with a small parameter. The latter problem (both for constant and variable coefficients) is the topic of the second part of the paper.

**1.**Harold Grad,*On the kinetic theory of rarefied gases*, Comm. Pure Appl. Math.**2**(1949), 331–407. MR**0033674**, https://doi.org/10.1002/cpa.3160020403**2.**E. V. Radkevich,*Well-posedness of mathematical models in continuum mechanics and thermodynamics*. Contemporary Mathematics. Fundamental Directions. 2003. Vol. 3. pp. 5-32.**3.**M. M. Postnikov,*Stable polynomials*, URSS, Moscow, 2004. (Russian)**4.**L. R. Volevich and M. G. Dzhavadov,*Uniform estimates for solutions of hyperbolic equations with a small parameter multiplying the highest derivatives*, Differentsial'nye Uravneniya**19**(1983), no. 12, 2082-2090; English transl. in Differential Equations**19**(1983). MR**0729566 (85j:35020)****5.**L. R. Volevich and E. V. Radkevich,*Uniform estimates for solutions of the Cauchy problem for hyperbolic equations with a small parameter at the highest derivative*, Differentsial'nye Uravneniya**39**(2003), no. 4, 486-499; English transl. in Differential Equations**39**(2003).**6.**Ch. Hermite,*Oeuvres*I. Paris, 1905, pp. 397-414.**7.**Lars Hörmander,*Linear partial differential operators*, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR**0161012****8.**M. I. Višik and L. A. Lyusternik,*Regular degeneration and boundary layer for linear differential equations with small parameter*, Uspehi Mat. Nauk (N.S.)**12**(1957), no. 5(77), 3–122 (Russian). MR**0096041****9.**S. G. Gindikin and L. R. Volevich,*Mixed problem for partial differential equations with quasihomogeneous principal part*, Translations of Mathematical Monographs, vol. 147, American Mathematical Society, Providence, RI, 1996. Translated from the Russian manuscript by V. M. Volosov. MR**1357662****10.**L. R. Volevich,*The Vishik-Lyusternik method in general elliptic boundary value problems with small parameter*, Keldysh Inst. Appl. Math. RAS, 2002, Preprint no. 26.**11.**S. Gindikin and L. R. Volevich,*The method of Newton’s polyhedron in the theory of partial differential equations*, Mathematics and its Applications (Soviet Series), vol. 86, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian manuscript by V. M. Volosov. MR**1256484**

Retrieve articles in *Transactions of the Moscow Mathematical Society*
with MSC (2000):
35B25,
35L25.

Retrieve articles in all journals with MSC (2000): 35B25, 35L25.

Additional Information

**L. R. Volevich**

Affiliation:
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow 125047, Russia

Email:
volevich@spp.keldysh.ru

**E. V. Radkevich**

Affiliation:
Moscow State University, Mechanics and Mathematics Department, Moscow 119899, Russia

Email:
evrad@land.ru

DOI:
https://doi.org/10.1090/S0077-1554-04-00147-5

Published electronically:
November 4, 2004

Additional Notes:
The first author was supported by the Russian Foundation for Basic Research (Grant 03–01–00189) and INTAS (Project no. 899). The second author was supported by the Russian Foundation for Basic Research (Grant 03–01–00189).

Article copyright:
© Copyright 2004
American Mathematical Society