For a hyperbolic second-order differential operator , we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and some algebraic and geometric properties of the principal symbol . If the Hamilton map of (the linearization of the Hamilton field along double characteristics) has nonzero real eigenvalues at every double characteristic (the so-called effectively hyperbolic case), then it is well known that the Cauchy problem for is well posed in any Gevrey class for any lower-order term. In this paper we prove that if is noneffectively hyperbolic and, moreover, such that on a double characteristic manifold of codimension , assuming that there is no null bicharacteristic landing tangentially, then the Cauchy problem for is well posed in the Gevrey class for any lower-order term (strong Gevrey well-posedness with threshold ), extending in particular via energy estimates a previous result of Hörmander in a model case.
Kyoto J. Math.
51(4):
767-810
(Winter 2011).
DOI: 10.1215/21562261-1424857
[1] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45–57. MR435933 10.1215/S0012-7094-77-04402-7 euclid.dmj/1077312091
[1] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45–57. MR435933 10.1215/S0012-7094-77-04402-7 euclid.dmj/1077312091
[2] E. Bernardi, A. Bove, and C. Parenti, Geometric results for a class of hyperbolic operators with double characteristics, II, J. Funct. Anal. 116 (1993), 62–82. MR1237986 0798.35093 10.1006/jfan.1993.1104[2] E. Bernardi, A. Bove, and C. Parenti, Geometric results for a class of hyperbolic operators with double characteristics, II, J. Funct. Anal. 116 (1993), 62–82. MR1237986 0798.35093 10.1006/jfan.1993.1104
[3] E. Bernardi and T. Nishitani, On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness, J. Anal. Math. 105 (2008), 197–240. MR2438425 1209.35009 10.1007/s11854-008-0035-3[3] E. Bernardi and T. Nishitani, On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness, J. Anal. Math. 105 (2008), 197–240. MR2438425 1209.35009 10.1007/s11854-008-0035-3
[4] M. D. Bronšteĭn, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity (in Russian), Trudy Moscov. Mat. Obshch. 41 (1980), 83–99. MR611140 0468.35062[4] M. D. Bronšteĭn, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity (in Russian), Trudy Moscov. Mat. Obshch. 41 (1980), 83–99. MR611140 0468.35062
[6] L. Hörmander, The Cauchy problem for differential equations with double characteristics, J. Anal. Math. 32 (1977), 118–196. MR492751 0367.35054 10.1007/BF02803578[6] L. Hörmander, The Cauchy problem for differential equations with double characteristics, J. Anal. Math. 32 (1977), 118–196. MR492751 0367.35054 10.1007/BF02803578
[7] L. Hörmander, “Quadratic hyperbolic operators” in Microlocal Analysis and Applications (Montecatini Terme, Italy, 1989), Lecture Notes in Math. 1495, Springer, Berlin, 1991, 118–160. MR1178557[7] L. Hörmander, “Quadratic hyperbolic operators” in Microlocal Analysis and Applications (Montecatini Terme, Italy, 1989), Lecture Notes in Math. 1495, Springer, Berlin, 1991, 118–160. MR1178557
[10] V. Ja. Ivrii and V. M. Petkov, Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations (in Russian), Uspehi Mat. Nauk. 29 (1974), no. 5, 3–70. MR427843[10] V. Ja. Ivrii and V. M. Petkov, Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations (in Russian), Uspehi Mat. Nauk. 29 (1974), no. 5, 3–70. MR427843
[13] T. Nishitani, Local energy integrals for effectively hyperbolic operators, I, J. Math. Kyoto Univ. 24 (1984), 623–658; II, 659–666. MR775976 euclid.kjm/1250521222
[13] T. Nishitani, Local energy integrals for effectively hyperbolic operators, I, J. Math. Kyoto Univ. 24 (1984), 623–658; II, 659–666. MR775976 euclid.kjm/1250521222
[14] T. Nishitani, “Microlocal energy estimates for hyperbolic operators with double characteristics” in Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984), Academic Press, Boston, 1986, 235–255. MR925251[14] T. Nishitani, “Microlocal energy estimates for hyperbolic operators with double characteristics” in Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984), Academic Press, Boston, 1986, 235–255. MR925251
[15] T. Nishitani, Non effectively hyperbolic operators, Hamilton map and bicharacteristics, J. Math. Kyoto Univ. 44 (2004), 55–98. MR2062708 1069.35037 euclid.kjm/1250283583
[15] T. Nishitani, Non effectively hyperbolic operators, Hamilton map and bicharacteristics, J. Math. Kyoto Univ. 44 (2004), 55–98. MR2062708 1069.35037 euclid.kjm/1250283583
[16] T. Nishitani, “On Gevrey well-posedness of the Cauchy problem for some noneffectively hyperbolic operators” in Advances in Phase Space Analysis of Partial Differential Equations: In Honor of Ferruccio Colombini’s 60th Birthday, Nonlinear Differential Equations Appl. 78, Birkhäuser, Boston, 2009, 217–233. MR2664613 1203.35159 10.1007/978-0-8176-4861-9_13[16] T. Nishitani, “On Gevrey well-posedness of the Cauchy problem for some noneffectively hyperbolic operators” in Advances in Phase Space Analysis of Partial Differential Equations: In Honor of Ferruccio Colombini’s 60th Birthday, Nonlinear Differential Equations Appl. 78, Birkhäuser, Boston, 2009, 217–233. MR2664613 1203.35159 10.1007/978-0-8176-4861-9_13
[17] T. Nishitani and M. Tamura, A class of Fourier integral operators with complex phase related to the Gevrey classes, J. Pseudo-Differ. Oper. Appl. 1 (2010), 255–292. MR2679894 1204.35183 10.1007/s11868-010-0014-1[17] T. Nishitani and M. Tamura, A class of Fourier integral operators with complex phase related to the Gevrey classes, J. Pseudo-Differ. Oper. Appl. 1 (2010), 255–292. MR2679894 1204.35183 10.1007/s11868-010-0014-1