Some results on the well-posedness for systems with time dependent coefficients

Marcello D’Abbicco[1]; Giovanni Taglialatela[2]

  • [1] Dipartimento di Matematica, Università di Bari, via E. Orabona 4, 70125 Bari – Italy
  • [2] Dipartimento di Scienze Economiche e Metodi Matematici, Facoltà di Economia, Università di Bari, via C. Rosalba 53, 70124 Bari – Italy

Annales de la faculté des sciences de Toulouse Mathématiques (2009)

  • Volume: 18, Issue: 2, page 247-284
  • ISSN: 0240-2963

Abstract

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We consider hyperbolic systems with time dependent coefficients and size  2 or  3 . We give some sufficient conditions in order the Cauchy Problem to be well-posed in 𝒞 and in Gevrey spaces.

How to cite

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D’Abbicco, Marcello, and Taglialatela, Giovanni. "Some results on the well-posedness for systems with time dependent coefficients." Annales de la faculté des sciences de Toulouse Mathématiques 18.2 (2009): 247-284. <http://eudml.org/doc/10109>.

@article{D2009,
abstract = {We consider hyperbolic systems with time dependent coefficients and size $2$ or $3$. We give some sufficient conditions in order the Cauchy Problem to be well-posed in $\{\cal C\}^\infty $ and in Gevrey spaces.},
affiliation = {Dipartimento di Matematica, Università di Bari, via E. Orabona 4, 70125 Bari – Italy; Dipartimento di Scienze Economiche e Metodi Matematici, Facoltà di Economia, Università di Bari, via C. Rosalba 53, 70124 Bari – Italy},
author = {D’Abbicco, Marcello, Taglialatela, Giovanni},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Gevrey spaces},
language = {eng},
month = {1},
number = {2},
pages = {247-284},
publisher = {Université Paul Sabatier, Toulouse},
title = {Some results on the well-posedness for systems with time dependent coefficients},
url = {http://eudml.org/doc/10109},
volume = {18},
year = {2009},
}

TY - JOUR
AU - D’Abbicco, Marcello
AU - Taglialatela, Giovanni
TI - Some results on the well-posedness for systems with time dependent coefficients
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2009/1//
PB - Université Paul Sabatier, Toulouse
VL - 18
IS - 2
SP - 247
EP - 284
AB - We consider hyperbolic systems with time dependent coefficients and size $2$ or $3$. We give some sufficient conditions in order the Cauchy Problem to be well-posed in ${\cal C}^\infty $ and in Gevrey spaces.
LA - eng
KW - Gevrey spaces
UR - http://eudml.org/doc/10109
ER -

References

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  1. Bronšteĭn (M. D.).— Smoothness of roots of polynomials depending on parameters, Siberian Math. J. 20 (1980) 347-352, translation from Sibirsk. Mat. Zh. 20 p. 493-501 (1979). Zbl0429.30007MR537355
  2. Colombini (F.).— Quelques remarques sur le problème de Cauchy pour des équations faiblement hyperboliques, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1992), École Polytech., Palaiseau, 1992, Exp. No. XIII. Zbl0779.35069MR1191920
  3. Colombini (F.), Ishida (H.), and Orrú (N.).— On the Cauchy problem for finitely degenerate hyperbolic equations of second order, Ark. Mat. 38, p. 223-230 (2000). Zbl1073.35145MR1785400
  4. Colombini (F.), Jannelli (E.) and Spagnolo (S.).— Well posedness in the Gevrey Classes of the Cauchy Problem for a Non Strictly Hyperbolic Equation with Coefficients Depending On Time, Ann. Scu. Norm. Sup. Pisa, 10, p. 291-312 (1983). Zbl0543.35056MR728438
  5. Colombini (F.) and Nishitani (T.).— Two by two strongly hyperbolic systems and Gevrey classes, Ann. Univ. Ferrara Sez. VII (N.S.) 45, suppl. (1999), p. 79-108 (2000). Zbl0993.35061MR1806490
  6. —.— Systèmes 2 fois 2 fortement hyperboliques dans 𝒞 et dans les classes de Gevrey, C. R. Acad. Sci. Paris Sér. I Math. 330, p. 969-972 (2000). Zbl0956.35076MR1779688
  7. Colombini (F.) and Orrú (N.).— Well-posedness in 𝒞 for some weakly hyperbolic equations, J. Math. Kyoto Univ. 39, p. 399-420 (1999). Zbl1007.35045MR1718730
  8. Colombini (F.) and Spagnolo (S.).— An example of a weakly hyperbolic Cauchy problem not well posed in 𝒞 , Acta Math. 148, p. 243-253 (1982). Zbl0517.35053MR666112
  9. Colombini (F.) and Taglialatela (G.).— Levi conditions for higher order operators with finite degeneracy, J. Math. Kyoto Univ. 46 (2006). Zbl1145.35079MR2320353
  10. D’Ancona (P.) and Kinoshita (T.).— On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order, Math. Nach. 278, p. 1147-1162 (2005). Zbl1078.35065MR2155966
  11. D’Ancona (P.), Kinoshita (T.), and Spagnolo (S.).— Weakly hyperbolic systems with Hölder continuous coefficients, J. Differ. Equations 203, p. 64-81 (2004). Zbl1068.35065MR2070386
  12. —.— On the Well-Posedness of the Cauchy Problem for 2 × 2 weakly hyperbolic systems, Osaka J. Math. 45, p. 921-939 (2008). 
  13. D’Ancona (P.) and Racke (R.).— Weakly hyperbolic equations in domains with boundaries, Nonlinear Anal. 33, p. 455-472 (1998). Zbl0933.34067MR1635708
  14. D’Ancona (P.) and Spagnolo (S.).— On pseudosymmetric hyperbolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), p. 397-418 (1998). Zbl1014.35055MR1655523
  15. —.— Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity, Boll. Unione Mat. Ital., Sez. B Artic Ric. Mat. (8) 1, p. 169-185 (1998). Zbl0916.35063MR1618976
  16. —.— A remark on uniformly symmetrizable systems, Adv. in Math. 158, p. 18-25 (2001). Zbl1195.35203MR1814897
  17. Demay (Y.).— Paramètrix pour des systèmes hyperboliques du premier ordre à multiplicité constante, J. Math. Pures Appl., IX. Sér. 56, p. 393-422 (1977). Zbl0379.35068MR466986
  18. Gårding (L.).— Linear hyperbolic partial differential equations with constant coefficients, Acta Math. 85, p. 1-62 (1951). Zbl0045.20202MR41336
  19. Ishida (H.) and Yagdjian (K.).— On a sharp Levi condition in Gevrey classes for some infinitely degenerate hyperbolic equations and its necessity, Publ. Res. Inst. Math. Sci. 38, no. 2, p. 265-287 (2002). Zbl1028.35089MR1903740
  20. Ivrii (V.Ja.).— Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes, Sib. Math. J. 17, p. 921-931 (1977). Zbl0404.35068
  21. Kinoshita (T.) and Spagnolo (S.).— Hyperbolic equations with non analytic coefficients, Math. Ann. 336, p. 551-569 (2006). Zbl1103.35064MR2249759
  22. Larsson (E.).— Generalized hyperbolicity, Arkiv för Matematik 7, p. 11-32 (1967). Zbl0183.41302MR221062
  23. Mandai (T.).— Smoothness of roots of hyperbolic polynomials with respect to one-dimensional parameter, Bull. Fac. Gen. Ed. Gifu Univ. 21, p. 115-118 (1985). MR840968
  24. Matsumoto (W.).— On the condition for the hyperbolicity of systems with double characteristics roots. I and II, J. Math. Kyoto Univ. 21, p. 47-84, p. 251-271 (1981). Zbl0487.35057MR606312
  25. Mencherini (L.).— Il Problema di Cauchy per sistemi lineari di primo ordine debolmente iperbolici, Ph.D. Thesis, University of Pisa. 
  26. Mencherini (L.) and Spagnolo (S.).— Gevrey well-posedness for pseudosymmetric systems with lower order terms, Hyperbolic differential operators and related problems (V. Ancona and J. Vaillant, eds.), Lecture Notes in Pure and Appl. Math., vol. 233, Marcel Dekker, New York, NY, (2003), p. 67-81. Zbl1058.35144MR2004859
  27. —.— Well-posedness of 2 × 2 systems with 𝒞 -coefficients, Hyperbolic problems and related topics (F. Colombini and T. Nishitani, eds.), Grad. Ser. Anal, International Press, Somerville, MA, 2003, Proceedings of the conference, Cortona, Italy, September 10-14, (2002), p. 235-241. Zbl1054.35023MR2056853
  28. —.— Uniformly symmetrizable 3 × 3 matrices, Linear Algebra Appl. 382, p. 25-38 (2004). Zbl1056.15012MR2050097
  29. Nishitani (T.).— Hyperbolicity of two by two systems with two independent variables, Comm. P.D.E. 23, p. 1061-1110 (1998). Zbl0913.35079MR1632796
  30. Orrù (N.).— On a weakly hyperbolic equation with a term of order zero, Annales de la Faculté des Sciences de Toulouse Sér. 6, 6, p. 525-534 (1997). Zbl0895.35057MR1610911
  31. Shinkai (K.) and Taniguchi (K.).— Fundamental solution for a degenerate hyperbolic operator in Gevrey classes, Publ. Res. Inst. Math. Sci. 28, p. 169-205 (1992). Zbl0820.35155MR1152755
  32. Spagnolo (S.).— On the Absolute Continuity of the Roots of Some Algebraic Equation, Ann.Univ.Ferrara, Suppl. XLV, p. 327-337 (1999). Zbl0993.30004MR1806507
  33. Tarama (S.).— On the second order hyperbolic equations degenerating in the infinite order. Example, Math. Japon. 42, p. 523-533 (1995). Zbl0837.35097MR1363842
  34. —.— Note on the Bronshtein theorem concerning hyperbolic polynomials, Sci. Math. Jpn. 63, p. 247-285 (2006). Zbl1106.26016MR2213130
  35. Wakabayashi (S.).— Remarks on hyperbolic polynomials, Tsukuba J. Math. 10, p. 17-28 (1986). Zbl0612.35005MR846411

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