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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 $$u\sb{tt}-\varphi(\int\sb P\vert\nabla u(t,x)\vert\sp 2dx)\Delta u(t,x)=0,\ u(0,x)=u\sb 0(x),u\sb t(0,x)=u\sb 0(x), \tag P$$ where $u(t,x)$ is $2\pi$-periodic in $x\sb i$ $(i=1,\dots,n)$, $t\in\bbfR\sp 1$, $x\in\bbfR\sp n$, $P=[0,2\pi]\sp n$ and $\varphi$ is nonnegative and continuous in $[0,\infty)$. They prove that if $u\sb 0$ and $u\sb 1$ are real analytic and $2\pi$-periodic in $x\sb i$ $(i=1,\ldots,n)$, then the problem (P) has a $C\sp 2$-solution $u$ in $(0,\infty)\times\bbfR\sp n$ which is real analytic in $x$ and $2\pi$-periodic. The result that the global solution in $t$ is obtained only under the condition $\varphi\ge 0$ is very interesting.

##### MSC:
 35L80 Hyperbolic equations of degenerate type 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B10 Periodic solutions of PDE
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##### References:
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