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Uniform zeros for beaded strings. (English) Zbl 0607.35055
Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 141-148 (1986).
[For the entire collection see Zbl 0595.00009.]
The author investigates the ”simultaneous crossing” problem for the hyperbolic PDE \[ u_{tt}-u_{xx}+p(x,t)u=0 \] \(p(x,t)>0\) and continuous on \(0\leq x\leq l\), \(t\geq 0\) without involving the separation of variables, and also higher dimensional problems of this type, and establishes some topological criteria for solutions of \[ \underset \sim u''+[\Gamma_ 0+\Pi (t)]\underset \sim u=0 \underset\sim u(0)=\emptyset,\underset\sim u'(0)=\underset\sim g(t), \] \(g\geq \emptyset\) (componentwise).
Here \(\Gamma_ 0\) is a constant matrix with positive eigenvalues. \(\Pi\) (t) is a diagonal matrix regarded as a perturbation of \(\Gamma_ 0\). The author proves that if \(\Pi\) (t)\(\equiv 0\) and the eigenvalues of \(\Gamma_ 0\) satisfy the inequality \[ 1<(\mu_ i/\mu_ 1)^{1/2}<3/2,^ 2\leq i\leq n, \] then if the trajectory first intersects the bounding plane of \({\mathbb{R}}^+_ n\) at some point, it also must exist at that point. He also proves the ”crossover property” and the existence of conjugate points.
Reviewer: V.Komkov
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs