On the quadratic wave equation with randomized oscillating coefficients

Abstract

In this paper, we consider the asymptotic behavior of the solutions of the quadratic wave equations with perturbed oscillating coefficients which are very important mathematical models in describing the oscillation of particles imposed with time‐dependent electromagnetic fields. By the application of microlocal analysis and stochastic analysis, this paper makes a delicate classification of various types of time‐dependent oscillating coefficients on the principal quantum harmonic oscillator part and explores the corresponding regularity behavior and L2$$ {L}^2 $$‐estimates of the solution of the equation. It is interesting to find out that weak or mild oscillations will not cause any energy growth, while regular and strong oscillations will lead to polynomial or exponential growth of the energy. This is meaningful to help us explore the energy conversion mechanism in the electromagnetic field. Furthermore, in order to demonstrate the optimality of the L2$$ {L}^2 $$‐estimates, typical counterexamples with periodic coefficients will be constructed to show the lower bound of growth rate by the application of harmonic analysis and instability arguments. Compared with formal literatures focusing on the constant coefficients, current results in this paper reveal the essential relation between oscillation types and energy growth.