Finding an unknown rapidly oscillating right-hand side in multidimensional first-order hyperbolic system

P. V. Babich

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 15-23

Résumé

The paper considers the Cauchy problem with a zero initial condition for a multidimensional linear hyperbolic system of first-order differential equations with constant coefficients and a right-hand side rapidly oscillating in time. Each component of the latter is a product of two functions, one of which depends only on the spatial variable, and the second only on the temporal and “fast temporal” variables. The the multiplier functions that depend on the spatial variable are known, but the time-dependent, rapidly oscillating multiplier are unknown. The inverse coefficient problem of recovering the latter from some additional information on the partial asymptotics of the solution of the Cauchy problem in the case when the right-hand side of the system is known (direct problem) is posed and solved. This additional information consists in setting the values of the first few asymptotic coefficients calculated at a certain point of the space. This type of overdetermination condition (additional condition) distinguishes the statement of the inverse problem from the one used in the classical theory of inverse coefficient problems, where the overdetermination conditions are imposed on the exact solution. Thus, the formulation and solution of the inverse problem is preceded by the solution of the problem, which consists in constructing and justifying the partial asymptotics of the solution. At this stage, in particular, it is determined how many first coefficients of the asymptotic expansion of the solution will be used in the condition of redefining the inverse problem. We also note that evolutionary problems with rapidly oscillating data play an important role in mathematics and its applications, already because they simulate many physical processes; for example, associated with high-frequency mechanical, electromagnetic or other vibrations. Moreover, the question of constructing for such problems the first few terms of the asymptotics of the solution is often much simpler than constructing the solution itself (and also calculating its values at the required points). Therefore, the development of the theory of inverse coefficient problems for rapidly oscillating problems seems to be undoubtedly relevant.

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