×

zbMATH — the first resource for mathematics

On the wellposedness in the Gevrey classes of the Cauchy problem for weakly hyperbolic equations whose coefficients are Hölder continuous in \(t\) and degenerate in \(t=T\). (English) Zbl 0927.35055
Let us consider in \([0,T]\times \mathbb{R}^n_x\) the weakly hyperbolic Cauchy problem \[ u_{tt}- \sum^n_{i,j=1} a_{ij}(t) u_{x_ix_j}+ \sum^n_{i=1} b_i(t) u_{x_i}= 0,\quad u(0,x)= u_0(x),\quad u_t(0,x)= u_1(x), \] where \(A(t)\equiv (a_{ij})\) is a real symmetric matrix with \(a_{ij}(t)\) belonging to \(C^{k+\alpha}[0,T)\) (\(k\in\mathbb{N}\), \(\alpha\in[0,1]\)). The real vector \(B(t)\equiv (b_1(t))\) has components belonging to \(C^0[0,T)\). On the line \(t= T\) the matrix \(A(t)\) and the vector \(B(t)\) degenerate. There exist results [F. Colombini, E. Jannelli and S. Spagnolo, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 291-312 (1983; Zbl 0543.35056)] about a relation between Hölder continuity of the coefficients and Gevrey well-posedness of the above Cauchy problem, namely, if \(A(t)\in C^{k+\alpha}[0,T)\), \(B(t)\in C^0[0,T)\), then the Cauchy problem is well posed in \(G^s\), \(1\leq s<1+{k+\alpha\over 2}\).
The author followed a nice idea to include in the assumptions the order of degeneration at \(t= T\). He assumes for \(a(t,\xi)\equiv (\sum a_{ij}\xi_i, \xi_j)\) and \(b(t,\xi)\equiv (\sum b_i(t)\xi_i)\) the conditions \[ C_1\Biggl(1- {t\over T}\Biggr)^\beta|\xi|^2\leq a(t,\xi)\leq C^{-1}_1\Biggl(1-{t\over T}\Biggr)^\beta|\xi|^2,\quad \beta\in [0,\infty]; \] \[ | b(t,\xi)|\leq C_2\Biggl(1- {t\over T}\Biggr)^\gamma|\xi|,\quad \gamma\in (-1,\infty]. \] The orders of degeneration are \(\beta\), \(\gamma\), respectively, some Gevrey type Levi conditions are satisfied. The author obtains results about the connection between \(s\), \(\alpha\), \(\beta\) and \(\gamma\).

MSC:
35L15 Initial value problems for second-order hyperbolic equations
35L80 Degenerate hyperbolic equations
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] M. Cicognani , On the strictly hyperbolic equations which are Hölder continuous with respect to time , preprint. MR 1695477 | Zbl 0967.35087 · Zbl 0967.35087
[2] F. Colombini - E. De Giorgi - S. Spagnolo , Sur les equations hyperboliques avec des coefficients qui ne d6pendent que du temps , Ann. Scuola Norm. Sup. Pisa , 6 ( 1979 ), pp. 511 - 559 . Numdam | MR 553796 | Zbl 0417.35049 · Zbl 0417.35049 · numdam:ASNSP_1979_4_6_3_511_0 · eudml:83819
[3] F. Colombini - E. Jannelli - S. Spagnolo , Wellposedness in the Gevrey classes of the Cauchy problem for a non strictly hyperbolic equation with coefficients depending on time , Ann. Scuola Norm. Sup. Pisa , 10 ( 1983 ), pp. 291 - 312 . Numdam | MR 728438 | Zbl 0543.35056 · Zbl 0543.35056 · numdam:ASNSP_1983_4_10_2_291_0 · eudml:83908
[4] P. D’Ancona , Gevrey well posedness of an abstract Cauchy problem of weakly hyperbolic type , Publ. RIMS Kyoto Univ. , 24 ( 1988 ), pp. 433 - 449 . Article | MR 966182 | Zbl 0706.35077 · Zbl 0706.35077 · doi:10.2977/prims/1195175035 · minidml.mathdoc.fr
[5] P. D’Ancona , Local existence for semilinear weakly hyperbolic equations with time dependent coefficients, Nonlinear Analysis . Theory, Methods and Applications , Vol 21 , No. 9 ( 1993 ), pp. 685 - 696 . MR 1246287 | Zbl 0830.35089 · Zbl 0830.35089 · doi:10.1016/0362-546X(93)90064-Y
[6] V. Ya . IVRII, Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes , Siberian. Math. , 17 ( 1976 ), pp. 921 - 931 . Zbl 0404.35068 · Zbl 0404.35068 · doi:10.1007/BF00968018
[7] K. Kajitani , The well posed Cauchy problem for hyperbolic operators , Exposé au Séminaire de Vaillant du 8 février ( 1989 ).
[8] T. Kinoshita , On the wellposedness in the Gevrey classes of the Cauchy problem for weakly hyperbolic systems with Hölder continuous coefficients in t , preprint. Zbl 0933.35119 · Zbl 0933.35119
[9] M. Reissig - K. Yagdjian , Levi conditions and global Gevrey regularity for the solutions of quasilinear weakly hyperbolic equations , Mathematische Nachrichten , 178 ( 1996 ), pp. 285 - 307 . MR 1380714 | Zbl 0848.35078 · Zbl 0848.35078 · doi:10.1002/mana.19961780114
[10] T. Nishitani , Sur les équations hyperboliues à coefficients hölderiens en t et de classes de Gevrey en x , Bull. Sci. Math. , 107 ( 1983 ), pp. 739 - 773 . MR 704720 | Zbl 0552.35051 · Zbl 0552.35051
[11] H. Odai , On the Cauchy problem for a hyperbolic equation of second order , Doctoral thesis ( 1994 ). · Zbl 0802.11043
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.