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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

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