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

## TL;DR

This paper develops a factorization method for second-order hyperbolic PDEs with logarithmic slow scale coefficients, enabling microlocal diagonalization and approximation by dissipative one-way wave equations.

## Contribution

It introduces a novel factorization and microlocal diagonalization technique for hyperbolic operators with slow scale coefficients, leading to effective approximation by one-way wave equations.

## Key findings

- Successfully diagonalizes the wave operator microlocally.
- Provides well-posedness results for the approximated equations.
- Demonstrates suppression of singularities outside the domain.

## 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.