Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order

  • Alessandro Oliaro
  • Petar Popivanov
Part of the Operator Theory: Advances and Applications book series (OT, volume 172)


In this paper we study the local solvability in Gevrey classes for degenerate parabolic operators of order ≥ 2. We assume that the lower order term vanishes at a suitably smaller rate with respect to the principal part; we then analyze its influence on the behavior of the operator, proving local solvability in Gevrey spaces G s for small s, and local nonsolvability in G s for large s.


Degenerate parabolic operators Gevrey classes local solvability non local solvability 


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  1. [1]
    A. Bove and D. Tartakoff, Propagation of Gevrey regularity for a class of hypoelliptic equations, Trans. Amer. Math. Soc. 348 (1996), 2533–2575.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Bove and D. Tartakoff, Optimal non-isotropic Gevrey exponents for sum of squares of vector fields, Comm. Partial Differential Equations 22 (1997), 1263–1282.MATHMathSciNetGoogle Scholar
  3. [3]
    D. Calvo and P. Popivanov, Solvability in Gevrey classes for second powers of the Mizohata operator, C. R. Acad. Bulg. Sci. 57 (2004), 11–18.MATHMathSciNetGoogle Scholar
  4. [4]
    F. Colombini, L Pernazza, and F Treves, Solvability and nonsolvability of second-order evolution equations, in Hyperbolic Problems and Related Topics, Grad. Ser. Anal., Int. Press, Somerville, MA, 2003, 111–120.Google Scholar
  5. [5]
    A. Corli, On local solvability in Gevrey classes of linear partial differential operators with multiple characteristics, Comm. Partial Differential Equations 14 (1989), 1–25.MATHMathSciNetGoogle Scholar
  6. [6]
    T. Gramchev, P. Popivanov and M. Yoshino, Critical Gevrey index for hypoellipticity of parabolic equations and Newton polygons, Ann. Mat. Pura Appl. 170 (1996), 103–131.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    B. Helffer, Sur l’hypoellipticité d’une classe d’opérateurs paraboliques dégénérés, in Sur quelques équations aux dérivées partielles singulières, Astérisque, 19, Soc. Math. France, Paris, 1974, 79–105.Google Scholar
  8. [8]
    Y. Kannai, An unsolvable hypoelliptic differential operator, Israel J. Math. 9 (1971), 306–315.MATHMathSciNetGoogle Scholar
  9. [9]
    M. Mascarello and L. Rodino, Partial differential equations with multiple characteristics, Wiley-Akademie Verlag, Berlin, 1997.MATHGoogle Scholar
  10. [10]
    T. Matsusawa, On some degenerate parabolic equations I, Nagoya Math. J. 51 (1973), 57–77.MathSciNetGoogle Scholar
  11. [11]
    T. Matsusawa, On some degenerate parabolic equations II, Nagoya Math. J. 52 (1973), 61–84.MathSciNetGoogle Scholar
  12. [12]
    O.A. Oleinik and E.V. Radkevič, The method of introducing a parameter for the investigation of evolution equations, Uspekhi Mat. Nauk 33,5(203) (1978), 7–76, 237.Google Scholar
  13. [13]
    A. Oliaro, On a Gevrey non-solvable partial differential operator, in Recent Advances in Operator Theory and its Applications, Editors: M.A. Kaashoek, C Van der Mee and S. Seatzu, Birkhäuser, 337–356.Google Scholar
  14. [14]
    P. Popivanov, Local properties of linear pseudodifferential operators with multiple characteristics, C. R. Acad. Bulg. Sci. 29, (1976), 461–464.MATHMathSciNetGoogle Scholar
  15. [15]
    P. Popivanov, Local solvability of several classes of Partial Differential Equations, C. R. Acad. Bulg. Sci. 48, (1995), 15–18.MATHMathSciNetGoogle Scholar
  16. [16]
    P. Popivanov, On a nonsolvable partial differential operator, Ann. Univ. Ferrara VII, Sc. Mat. 49 (2003), 197–208.MathSciNetGoogle Scholar
  17. [17]
    L. Rodino, Gevrey hypoellipticity for a class of operators with multiple characteristics, in Analytic Solutions of Partial Differential Equations (Trento), Astérisque, 89–90, Soc. Math. France, Paris, 1981, 249–262.Google Scholar
  18. [18]
    R. Rubinstein, Examples of nonsolvable partial differential equations, Trans. Amer. Math. Soc. 199 (1974), 123–129.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    F. Trèves, On the existence and regularity of solutions of linear partial differential equations, in Proceedings of Symposia in Pure Mathematics, Vol. XXIII, 33–60.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Alessandro Oliaro
    • 1
  • Petar Popivanov
    • 2
  1. 1.Department of MathematicsUniversity of TorinoTorinoItaly
  2. 2.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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