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Klassische Lösungen nichtlinearer Wellengleichungen im Großen. (German) Zbl 0177.36602

This paper deals with the question of classical solvability of the Cauchy problem for nonlinear wave equations \[ \partial^2u/\partial t^2 + A(t)u + F'(\vert u\vert^2)u + \zeta u =0 \] over the whole of \(\mathbb R^+\times \mathbb R^n\), \(n\ge 3\), \(\zeta\) a complex constant. \(F'\) is the derivative of a nonnegative function \(F\) satisfying certain growth conditions depending on the space dimension \(n\). \(\{A(t),\ t\ge 0\}\) is a family of positive formally self-adjoint elliptic operators of order \(2m\). For \(A(t) = - \Delta\) and \(n=3\) various results are known, especially those of K. Jörgens [ibid. 77, 295–308 (1961, this Zbl. 111, 91)]. For every \(n\) and constant \(A(t)\) F. E. B r o w d e r [ibid. 80, 249–264 (1962, this Zbl. 109, 321)] constructed solutions in a Hilbert-space-theoretical sense. Similar to Browder the wave equation is understood as an equation of evolution in \(L^2(\mathbb R^n)\). At first a solution for small \(t\) is constructed which lies – for sufficiently regular initial data – in the domain of definition of a certain power \(A^{(k+1)/2}(t)\), \(k\) being a positive integer. If an a priori estimate of \(\Vert A^{(k+1)/2}(t)u(t)\Vert\) is available, the solution can be continued to the whole positive real axis and is regular over \(\mathbb R^+\times \mathbb R^n\). This problem can be reduced to the question whether an a priori estimate of a “critical power” \(A^{(k_0+1)/2}(t)u(t)\) can be found where \(k_0\) is a positive integer less than \(k\) and depends on \(n\) and \(m\). The desired a priori estimate is derived for space dimensions not exceeding 6 .
Reviewer: W. von Wahl

MSC:

35Lxx Hyperbolic equations and hyperbolic systems
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References:

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