Colombini, Ferruccio; Jannelli, Enrico; Spagnolo, Sergio Well-posedness in the Gevrey classes of the Cauchy problem for a non- strictly hyperbolic equation with coefficients depending on time. (English) Zbl 0543.35056 Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 291-312 (1983). The authors consider the Cauchy problem \((1)\quad u_{tt}- \sum^{n}_{i,j=1}a_{ij}(t)u_{x_ ix_ j}=0, u(x,0)=\phi(x),\quad u_ t(x,0)=\psi(x)\) on \({\mathbb{R}}^ n\times [0,T]\), provided the non- strict hyperbolicity condition \(\sum_{i,j}a_{ij}(t)\xi_ i\xi_ j\geq 0,\) for all \(\xi \in {\mathbb{R}}^ n\) is fulfilled. They prove that if the coefficients \(a_{ij}\in C^{k,\alpha}([0,T])\) with \(k\geq 0\) integer and 0\(\leq \alpha \leq 1\), then the problem (1) is well posed in the Gevrey class \({\mathcal E}^ s\) if \(1\leq s<1+(k+\alpha)/2.\) If the coefficients \(a_{ij}\) are analytic on [0,T], then (1) is \(C^{\infty}\) well-posed. The proof is based on the Fourier-Laplace transform as well as on approximate energy estimates. Some counter-examples are considered showing that the principal result cannot be improved. Reviewer: V.Petkov Cited in 3 ReviewsCited in 74 Documents MSC: 35L15 Initial value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A22 Transform methods (e.g., integral transforms) applied to PDEs 35B45 A priori estimates in context of PDEs Keywords:Cauchy problem; non-strict hyperbolicity; well posed; Gevrey class; Fourier-Laplace transform; approximate energy estimates × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] F. Colombini - E. De Giorgi - S. Spagnolo , Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps , Ann. Scuola Norm. Sup. Pisa , 6 ( 1979 ), pp. 511 - 559 . Numdam | MR 553796 | Zbl 0417.35049 · Zbl 0417.35049 [2] F. Colombini - S. Spagnolo , An example of weakly hyperbolic Cauchy problem not well posed in C\infty , Acta Math. , 148 ( 1982 ), pp. 243 - 253 . Zbl 0517.35053 · Zbl 0517.35053 · doi:10.1007/BF02392730 [3] J. Dieudonné , Sur un théorème de Glaeser , J. Analyse Math. , 23 ( 1970 ), pp. 85 - 88 . MR 269783 | Zbl 0208.07503 · Zbl 0208.07503 · doi:10.1007/BF02795491 [4] G. Glaeser , Racine carrée d’une function differentiable , Ann. Inst. Fourier , 13 ( 1963 ), pp. 203 - 210 . Numdam | MR 163995 | Zbl 0128.27903 · Zbl 0128.27903 · doi:10.5802/aif.146 [5] V. Ya . IVRII - V.M. Petkov , Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed , Uspehi Mat. Nauk , 29 ( 1974 ), pp. 3 - 70 , English Transl. in Russian Math. Surveys. MR 427843 | Zbl 0312.35049 · Zbl 0312.35049 · doi:10.1070/RM1974v029n05ABEH001295 [6] E. Jannelli , Weakly hyperbolic equations of second order with coefficients real analytic in space variables , Comm. in Partial Diff. Equations , 7 ( 1982 ), pp. 537 - 558 . MR 653577 | Zbl 0505.35051 · Zbl 0505.35051 · doi:10.1080/03605308208820231 [7] T. Nishitani , The Cauchy problem for weakly hyperbolic equations of second order , Comm. in Partial Diff. Equations , 5 ( 1980 ), pp. 1273 - 1296 . MR 593968 | Zbl 0497.35053 · Zbl 0497.35053 · doi:10.1080/03605308008820169 [8] O.A. Oleinik , On the Cauchy problem for weakly hyperbolic equations , Comm. Pure Appl. Math. , 23 ( 1970 ), pp. 569 - 586 . MR 264227 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.