×

Time deformations of powers of the wave operator. (Russian, English) Zbl 1100.35002

Zap. Nauchn. Semin. POMI 324, 213-228 (2005); translation in J. Math. Sci., New York 138, No. 2, 5603-5612 (2006).
There exists a complete characterization under which conditions the wave operator satisfies the Huygens’ principle.
The author is interested in the question which other operators possess the property of a forward and a backward wave front. First he recalls a definition of equivalence of operators (they can be transformed into each other by elementary transformations) which guarantees that both operators satisfy Huygens’ principle. An integrability condition for the coefficients of the drift and dissipation term helps on the one hand to describe the equivalence to a fixed class of second order wave operators and on the other hand to describe exact elementary transformations yielding the equivalence property. Some “deformations” of the wave operator are given.
Finally “deformations” of powers of the wave operator with lower order terms having time-dependent coefficients are considered. The special structure of these operators allows to introduce an iteration scheme which gives an explicit representation of the fundamental solution of powers of the considered operators with time dependent coefficients. In the case of odd spatial dimensions some conditions for parameters imply the Huygens’ principle.

MSC:

35A08 Fundamental solutions to PDEs
35G05 Linear higher-order PDEs
35L05 Wave equation
PDFBibTeX XMLCite
Full Text: EuDML Link