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.


35L80 Degenerate hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B10 Periodic solutions to PDEs
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