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Nonsimultaneous resonances in porous media. (English. Russian original) Zbl 0827.35104

Differ. Equations 29, No. 12, 1878-1887 (1993); translation from Differ. Uravn. 29, No. 12, 2149-2159 (1993).
A mathematical model is considered for a three-dimensional two- temperature porous medium filled by a viscous liquid partly being in the form of a film. The dynamic process of propagation of nonlinear waves in such a deformable medium is described by the Cauchy problem for a system of partial differential equations. In order to determine conditions for the existence of resonance modes the authors carry out an asymptotic analysis based on Maslov’s nonlinear WKB-method and the technique of multiscale expansions. For this purpose the original model is transformed to nonlinear evolution equations describing the propagation of the high- frequency elastic waves as well as the low-frequency filtrational and thermal ones. The main theorem proves the existence of an asymptotic solution for the reduced system. In conclusion the essential influence of dispersion and nonlinearities is shown in generating modulated waves.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
76S05 Flows in porous media; filtration; seepage
35G25 Initial value problems for nonlinear higher-order PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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