×

Global solvability for the degenerate Kirchhoff equation. (English) Zbl 0914.35085

From the introduction: We consider the Cauchy problem \[ u_{tt} (t,x)+\Phi \biggl(\bigl(Au (t,\cdot),u(t, \cdot)\bigr)\biggr) Au(t,x)=f(t,x)\quad (t>0,x\in \mathbb{R}^n), \]
\[ u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x), \] where \(Au(t,x)=\sum_{hk} D_{x_h}(a_{hk} (x)D_{x_k} u(t,x))\), \((Au(t,\cdot), u(t, \cdot))\) is the inner product of \(Au(x)\) and \(u(x)\) in \(L^2(\mathbb{R}^n_x)\), and \(\Phi(\eta)\) is a nonnegative function in \(C^1([0,\infty))\). When \(A= \sum^n_{h=1} D^2_{x_h}=-\Delta\) (i.e. \(a_{hk}(x)= \delta_{hk}\), Kronecker’s \(\delta)\) the equation is called the Kirchhoff equation, which has been studied by many authors. The problem which we treat in this paper is a generalization of \(-\Delta\) to a degenerate elliptic operator \(A\), where \([a_{hk}(x)]_{hk}\) is a real symmetric matrix which satisfies \[ \sum_{hk}a_{hk}(x) \xi_h\xi_k\geq 0\quad \bigl(\forall x\in \mathbb{R}^n_x,\;\forall\xi= (\xi_1,\dots, \xi_n)\in \mathbb{R}^n_\xi\bigr). \]
Reviewer: V.Mustonen (Oulu)

MSC:

35L80 Degenerate hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] A. Arosio - S. Spagnolo , Global existence for abstract evolution equations of weakly hyperbolic type , J. Math. Pures Appl. 65 ( 1986 ), 263 - 305 . MR 875159 | Zbl 0616.35049 · Zbl 0616.35049
[2] A. Arosio - S. Spagnolo , Global solution to the Cauchy problem for a nonlinear equation , Pitman Res. Notes Math. 109 ( 1984 ), 1 - 26 . MR 772234 | Zbl 0598.35062 · Zbl 0598.35062
[3] S. Bernstein , Sur une class d’équations fonctionnelles aux dérivées partielles , Izv. Akad. Nauk SSSR, Ser. Mat. 4 ( 1940 ), 17 - 26 . MR 2699 | Zbl 0026.01901 | JFM 66.0471.01 · Zbl 0026.01901
[4] P. D’Ancona , Gevrey well posedness of an abstract Cauchy problem of weakly hyperbolic type , Publ. Res. Inst. Math. Sci , Kyoto Univ. 24 ( 1988 ), 433 - 449 . Article | MR 966182 | Zbl 0706.35077 · Zbl 0706.35077 · doi:10.2977/prims/1195175035
[5] P. D’Ancona - S. Spagnolo , Global solvability for the degenerate Kirchhoff equation with real analytic data , Invent. Math. 108 ( 1992 ), 247 - 262 . MR 1161092 | Zbl 0785.35067 · Zbl 0785.35067 · doi:10.1007/BF02100605
[6] G.H. Hardy - J.E. Littlewood - G. Polya , ” Inequalities ”, Cambridge UP , Cambridge , 1952 . MR 46395 | Zbl 0047.05302 · Zbl 0047.05302
[7] F. Hirosawa , Global solvability for the generalized degenerate Kirchhoff equation with real-analytic data in Rn , Tsukuba J. Math. 21 ( 1997 ), 483 - 503 . MR 1473934 | Zbl 0896.35073 · Zbl 0896.35073
[8] T. Kinoshita , On the wellposedness in the ultradifferentiable classes of the Cauchy ploblem for a weakly hyperbolic equation of second order , Tsukuba J. Math. (to appear). MR 1637629 | Zbl 0942.35104 · Zbl 0942.35104
[9] K. Kajitani - K. Yamaguti , On global real analytic solutions of the degenerate Kirchhoff equation , Ann. Scuola Norm. Sup. Pisa. Cl. Sci. ( 4 ) 21 ( 1994 ), 279 - 297 . Numdam | MR 1288368 | Zbl 0819.35099 · Zbl 0819.35099
[10] G. Krantz - H.R. Parks , ” A Primer of Real Analytic Functions ”, Birkhäuser Verlag , Basel - Boston - Berlin , 1992 . MR 1182792 | Zbl 0767.26001 · Zbl 0767.26001
[11] W. Matsumoto , Theory of pseudo-differential operators of ultradifferentiable class , J. Math. Kyoto Univ. 27 ( 1987 ), 453 - 500 . Article | MR 910230 | Zbl 0651.35088 · Zbl 0651.35088
[12] T. Nishida , A note on the nonlinear vibrations of the elastic string , Mem. Fac. Engrg. Kyoto Univ. 33 ( 1971 ), 329 - 34 1 . MR 279387
[13] K. Nishihara , On a global solution of some quasilinear hyperbolic equation , Tokyo J. Math. 7 ( 1984 ), 437 - 459 . MR 776949 | Zbl 0586.35059 · Zbl 0586.35059 · doi:10.3836/tjm/1270151737
[14] G. Perla , On classical solutions of a quasilinear hyperbolic equation , Nonlinear Anal. 3 ( 1979 ), 613 - 627 . MR 541872 | Zbl 0419.35062 · Zbl 0419.35062 · doi:10.1016/0362-546X(79)90090-7
[15] O.A. Oleinik , On linear equations of second order with non-negative characteristic form , Mat. Sb. N.S. 69 ( 1966 ), 111 - 140 (English translation: Transl. Amer. Mat. Soc. 65 , 167 - 199 ). MR 193383
[16] S.I. Pohozaev , On a class ofquasilinear hyperbolic equations , Math. USSR-Sb. 96 ( 1975 ), 152 - 166 . MR 369938 | Zbl 0328.35060 · Zbl 0328.35060 · doi:10.1070/SM1975v025n01ABEH002203
[17] K. Yamaguti , On global quasi-analytic solution of the degenerate Kirchhoff equation , Tsukuba J. Math. (to appear). MR 1603843 | Zbl 1028.35090 · Zbl 1028.35090
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.