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

D’Abbicco, Marcello; Taglialatela, Giovanni

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 18 (2009) no. 2, pp. 247-284.

On considère des systèmes hyperboliques dont les coefficients ne dépendent que du temps. On donne des conditions suffisantes pour que le problème de Cauchy soit bien posé en ${𝒞}^{\infty }$ et dans les espaces de Gevrey

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 ${𝒞}^{\infty }$ and in Gevrey spaces.

[B] 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). | MR | Zbl

[C] 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. | EuDML | Numdam | MR | Zbl

[CIO] 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). | MR | Zbl

[CJS] 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). | EuDML | Numdam | MR | Zbl

[CN1] 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). | MR | Zbl

[CN2] —.— Systèmes 2 fois 2 fortement hyperboliques dans ${𝒞}^{\infty }$ et dans les classes de Gevrey, C. R. Acad. Sci. Paris Sér. I Math. 330, p. 969-972 (2000). | MR | Zbl

[CO] Colombini (F.) and Orrú (N.).— Well-posedness in ${𝒞}^{\infty }$ for some weakly hyperbolic equations, J. Math. Kyoto Univ. 39, p. 399-420 (1999). | MR | Zbl

[CS] Colombini (F.) and Spagnolo (S.).— An example of a weakly hyperbolic Cauchy problem not well posed in ${𝒞}^{\infty }$, Acta Math. 148, p. 243-253 (1982). | MR | Zbl

[CT] Colombini (F.) and Taglialatela (G.).— Levi conditions for higher order operators with finite degeneracy, J. Math. Kyoto Univ. 46 (2006). | MR | Zbl

[DAK] 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). | MR | Zbl

[DAKS1] D’Ancona (P.), Kinoshita (T.), and Spagnolo (S.).— Weakly hyperbolic systems with Hölder continuous coefficients, J. Differ. Equations 203, p. 64-81 (2004). | MR | Zbl

[DAKS2] —.— On the Well-Posedness of the Cauchy Problem for $2\times 2$ weakly hyperbolic systems, Osaka J. Math. 45, p. 921-939 (2008).

[DAR] D’Ancona (P.) and Racke (R.).— Weakly hyperbolic equations in domains with boundaries, Nonlinear Anal. 33, p. 455-472 (1998). | MR | Zbl

[DAS1] D’Ancona (P.) and Spagnolo (S.).— On pseudosymmetric hyperbolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), p. 397-418 (1998). | EuDML | Numdam | MR | Zbl

[DAS2] —.— 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). | EuDML | MR | Zbl

[DAS3] —.— A remark on uniformly symmetrizable systems, Adv. in Math. 158, p. 18-25 (2001). | MR | Zbl

[D] Demay (Y.).— Paramètrix pour des systèmes hyperboliques du premier ordre à multiplicité constante, J. Math. Pures Appl., IX. Sér. 56, p. 393-422 (1977). | MR | Zbl

[G] Gårding (L.).— Linear hyperbolic partial differential equations with constant coefficients, Acta Math. 85, p. 1-62 (1951). | MR | Zbl

[IY] 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). | MR | Zbl

[I] 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). | Zbl

[KS] Kinoshita (T.) and Spagnolo (S.).— Hyperbolic equations with non analytic coefficients, Math. Ann. 336, p. 551-569 (2006). | MR | Zbl

[L] Larsson (E.).— Generalized hyperbolicity, Arkiv för Matematik 7, p. 11-32 (1967). | MR | Zbl

[M] 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). | MR

[Ma] 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). | MR | Zbl

[Me] Mencherini (L.).— Il Problema di Cauchy per sistemi lineari di primo ordine debolmente iperbolici, Ph.D. Thesis, University of Pisa.

[MS1] 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. | MR | Zbl

[MS2] —.— Well-posedness of $2\times 2$ systems with ${𝒞}^{\infty }$-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. | MR | Zbl

[MS3] —.— Uniformly symmetrizable $3\times 3$ matrices, Linear Algebra Appl. 382, p. 25-38 (2004). | MR | Zbl

[N] Nishitani (T.).— Hyperbolicity of two by two systems with two independent variables, Comm. P.D.E. 23, p. 1061-1110 (1998). | MR | Zbl

[O] 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). | Numdam | MR | Zbl

[ST] 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). | MR | Zbl

[S] Spagnolo (S.).— On the Absolute Continuity of the Roots of Some Algebraic Equation, Ann.Univ.Ferrara, Suppl. XLV, p. 327-337 (1999). | MR | Zbl

[T1] Tarama (S.).— On the second order hyperbolic equations degenerating in the infinite order. Example, Math. Japon. 42, p. 523-533 (1995). | MR | Zbl

[T2] —.— Note on the Bronshtein theorem concerning hyperbolic polynomials, Sci. Math. Jpn. 63, p. 247-285 (2006). | MR | Zbl

[W] Wakabayashi (S.).— Remarks on hyperbolic polynomials, Tsukuba J. Math. 10, p. 17-28 (1986). | MR | Zbl