×

A simple construction of a parametrix for a regular hyperbolic operator. (English) Zbl 0714.35042

Let \(P=D^ 2_ t-\phi (t,x)A(t,x,D_ x)+B(t,x,D_ t,D_ x)\) with \(\phi \in C^{\infty}({\mathbb{R}}\times {\mathbb{R}}^ N)\), \(\phi\geq 0\) for \(t\geq 0\), \(\partial \phi /\partial t\neq 0\) where \(\phi =0\). \(A(t,x,D_ x)=\sum^{N}_{i,j=1}a_{ij}(t,x)D_ iD_ j\) is an elliptic differential operator and B is a differential operator of order one.
The author describes the construction of a local parametrix for the Cauchy problem for P on \(t\geq 0\) and x near to \(x_ 0\). Moreover he gives the tools to have a local parametrix for \(Pu=f.\)
The techniques used here are introduced in two earlier papers of the author [Ann. Mat. Pura Appl., IV. Ser. 140, 285-299 (1985; Zbl 0592.35097) and 146, 311-336 (1987; Zbl 0637.35020)].
Reviewer: K.H.Jansen

MSC:

35L10 Second-order hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35A08 Fundamental solutions to PDEs
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] M. Imai , On a parametrix for a weakly hyperbolic operator, Hokk . Math. J. , 9 ( 1980 ), pp. 190 - 216 . MR 597970 | Zbl 0471.35048 · Zbl 0471.35048
[2] M. Imai , Cauchy problems for operators of Tricomi’s type, Hokk . Math. J. , 8 ( 1979 ), pp. 126 - 143 . MR 533096 | Zbl 0415.35058 · Zbl 0415.35058
[3] L. Hörmander , Fourier integral operators I , Acta Math. , 127 ( 1971 ), pp. 79 - 183 . MR 388463 | Zbl 0212.46601 · Zbl 0212.46601 · doi:10.1007/BF02392052
[4] F. Segala , Parametrices for operators of Tricomi type , Ann. Mat. Pura e Appl. , 140 ( 1985 ), pp. 285 - 299 . MR 807641 | Zbl 0592.35097 · Zbl 0592.35097 · doi:10.1007/BF01776853
[5] F. Segala , Parametrices for a class of differential operators with multiple characteristics , Ann. Mat. Pura e Appl. (to appear). MR 916697 | Zbl 0637.35020 · Zbl 0637.35020 · doi:10.1007/BF01762369
[6] W. Wasow , Asymptotic expansions for ordinary differential equations , Interscience , 1965 . MR 203188 · Zbl 0133.35301
[7] A. Yoshikawa , A parametric for a completely regularly hyperbolic operator, Hokk . Math. J. , 10 ( 1981 ), pp. 723 - 742 . MR 662334 | Zbl 0516.35046 · Zbl 0516.35046
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.