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Global solvability for the degenerate Kirchhoff equation with real analytic data. (English) Zbl 0785.35067
The authors consider the following quasilinear hyperbolic Cauchy problem $u_{tt}-\varphi(\int_ P|\nabla u(t,x)|^ 2dx)\Delta u(t,x)=0,\;u(0,x)=u_ 0(x),u_ t(0,x)=u_ 0(x), \tag{P}$ where $$u(t,x)$$ is $$2\pi$$-periodic in $$x_ i$$ $$(i=1,\dots,n)$$, $$t\in\mathbb{R}^ 1$$, $$x\in\mathbb{R}^ n$$, $$P=[0,2\pi]^ n$$ and $$\varphi$$ is nonnegative and continuous in $$[0,\infty)$$. They prove that if $$u_ 0$$ and $$u_ 1$$ are real analytic and $$2\pi$$-periodic in $$x_ i$$ $$(i=1,\ldots,n)$$, then the problem (P) has a $$C^ 2$$-solution $$u$$ in $$(0,\infty)\times\mathbb{R}^ n$$ which is real analytic in $$x$$ and $$2\pi$$-periodic. The result that the global solution in $$t$$ is obtained only under the condition $$\varphi\geq 0$$ is very interesting.

##### MSC:
 35L80 Degenerate hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B10 Periodic solutions to PDEs
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##### References:
 [1] [A] Arosio, A.: Global (in time) solution of the approximated non-linear string equation of G.F. Carrier and R. Narashima. Comment. Math. Univ. Carol.26, 169–171 (1985) · Zbl 0563.73020 [2] [AS1] Arosio, A., Spagnolo, S.: Global existence for abstract evolution equations of weakly hyperbolic type. J. Math. Pures Appl.65, 263–305 (1986) · Zbl 0616.35049 [3] [AS2] Arosio, A., Spagnolo, S.: Global solution to the Cauchy problem for a nonlinear equation, In: Brezis, H. Lions, J.L. (eds.) Nonlinear PDE’s and their applications. Collège de France seminar, vol. VI. (Res. Notes Math., vol. 109, pp. 1–26) Boston: Pitman 1984 [4] [B] Bernstein, S.: Sur une classe d’équations fonctionnelles aux dérivées partielles. Izv. Akad. Nauk SSSR, Sér. Mat.4, 17–26 (1940) · Zbl 0026.01901 [5] [D] Dickey, R.W.: Infinite systems of nonlinear oscillation equations, related to the string. Proc. Am. Math. Soc.23, 459–468 (1969); Infinite systems of nonlinear oscillation equations. J. Differ. Equations8, 16–26 (1970) · Zbl 0218.34015 · doi:10.1090/S0002-9939-1969-0247189-8 [6] [G] Garabedian, P.R.: Partial differential equations. New York: J. Wiley & Sons 1964 · Zbl 0124.30501 [7] [K] Kirchhoff, G.: Vorlesungen über Mechanik. Leipzig: Teubner 1883 · JFM 08.0542.01 [8] [L] Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: De La Penha, G.M. Medeiros, L.A. (eds.) Contemporary developments in continuum mechanics and partial differential equations. London: North-Holland 1978 [9] [M] Medeiros, L.A.: On a new class of nonlinear wave equations. J. Math. Anal. Appl.69, 252–262 (1979) · Zbl 0407.35051 · doi:10.1016/0022-247X(79)90192-6 [10] [P] Perla Menzala, G.: On classical solutions of a quasilinear hyperbolic equation. Nonlinear Anal.3, 613–627 (1979) · Zbl 0419.35062 · doi:10.1016/0362-546X(79)90090-7 [11] [Po] Pohožaev, S.I.: On a class of quasilinear hyperbolic equations. Mat. Sb. 96 (138) (1975) N.1, 152–166; Translation: Math. USSR Sb.25 (N.1) (1975) [12] [R] Rivera Rodriguez, P.H.: On local strong solution of a non-linear partial differential equation. Appl. Anal.8, 93–104 (1980) · Zbl 0451.35042 · doi:10.1080/00036818008839291 [13] [W] Wakabayashi, S.: Generalized flows and their applications. In: Garnir, H.G. (ed.), Advances in microlocal analysis, pp. 363–384. Dordrecht: Reidel 1986
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