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Stability and instability zones for equations of continuous medium vibrations. (English. Russian original) Zbl 0855.35082

Differ. Equations 30, No. 7, 1178-1181 (1994); translation from Differ. Uravn. 30, No. 7, 1270-1273 (1994).
In the short paper the continuous medium vibrations are considered in the form of the equation \[ {\partial^2 u\over \partial t^2}+ \sum^n_{i, j= 1} a_{ij} {\partial^2 u\over \partial x_i \partial x_j}+ \Biggl[ \overline\varphi\Biggl( t+ \sum^n_{j= 1} a_j x_j\Biggr)+ \overline\lambda\Biggr] u= 0 \] with constant real coefficients. For this equation the author studies the behavior of self-similar motions of the form \(u= u(x)\), where \(x= t+ \sum^n_{j= 1} a_j x_j\). Such motions are called plane waves. Substituting \(u(x)\) in the above equation allows one to reduce it to \(d^2 u/dx^2+ [\varphi(x)+ \lambda] u= 0\).
The main result concerning the last equation consists in establishing the necessary and sufficient conditions for it to belong to stability or instability zones as well as for its solutions to be oscillatory or nonoscillatory.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B35 Stability in context of PDEs
74J10 Bulk waves in solid mechanics
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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