Global solvability for the degenerate Kirchhoff equation with real analytic data. (English) Zbl 0785.35067

D’Ancona, Piero; Spagnolo, Sergio

Invent. Math. 108, No. 2, 247-262 (1992).

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.

Reviewer: K.Kajitani (Ibaraki)

MSC:

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

Keywords:

Kirchhoff equation; periodic solution; quasilinear hyperbolic Cauchy problem

References:

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