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Global solutions to the Cauchy problem for a nonlinear hyperbolic equation. (English) Zbl 0598.35062
Nonlinear partial differential equations and their applications, Coll. de France Semin., Paris 1982-83, Vol. VI, Res. Notes Math. 109, 1-26 (1984).
[For the entire collection see Zbl 0543.00005.]
This paper deals with the global existence of a solution to the abstract Cauchy problem in a complex Hilbert space V: \[ (*)\quad u''+m(<Au,u>)Au=0\quad (t\geq 0);\quad u(0)=u_ 0,\quad u'(0)=u_ 1, \] where \(<\cdot,\cdot >\) denotes the antiduality between \(V^*\) (the antidual space of V) and V, and \(A: V\to V^*\) is a symmetric positive definite isomorphism. This problem has such important applications as the IBVP to the equations \[ (**)\quad u_{tt}-m(\int_{\Omega} | u_ x|^ 2 dx)\Delta u=0. \] The authors show that the problem (*) has a solution \(u\in C^ 2([0,+),V)\) for any A-analytic initial data \(u_ 0\) and \(u_ 1\) under the assumptions
(i) m(r) is continuous and \(\geq 0\) on [0,\(\infty)\), and \(\int^{\infty}_{0}m(r)dr=+\infty\) or \(\sup_{r\geq 0} m(r)<+\infty,\)
(ii) the embedding of V into \(V^*\) is a compact operator, and they also show that the above u(t) and u’(t) are A-analytic for each time \(t>0\). The authors used the Fourier expansion method to show the above results. Their results weakened conditions on m(r) assumed by several papers to the IBVP to (**).
Reviewer: M.Yamaguchi

35L10 Second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A22 Transform methods (e.g., integral transforms) applied to PDEs