Factorization of second-order strictly hyperbolic operators with logarithmic slow scale coefficients and generalized microlocal approximations

Abstract

We give a factorization procedure for a strictly hyperbolic partial differential operator of second order with logarithmic slow scale coefficients. From this we can microlocally diagonalize the full wave operator which results in a coupled system of two first-order pseudodifferential equations in a microlocal sense. Under the assumption that the full wave equation is microlocal regular in a fixed domain of the phase space, we can approximate the problem by two one-way wave equations where a dissipative term is added to suppress singularities outside the given domain. We obtain well-posedness of the corresponding Cauchy problem for the approximated one-way wave equation with a dissipative term. Contents 1. Basic notions 1 2. Pseudodifferential calculus 3 3. The governing equation in the Colombeau setting 8 4. Ellipticity of pseudodifferential operators 11 5. A factorization procedure for the generalized strictly hyperbolic wave operator 21 6. The generalized wave front set 29 7. Microlocal decomposition of the wave equation 33 8. Approximated first-order wave equations 40 9. Well-posedness of the approximated first-order equations 43 References 49