DE Seminar: Haoran Qi (Penn State)

Title: Microlocal Analysis of Hyperbolic Equations with Memory

Abstract: Hyperbolic equations with memory arise in models of wave propagation in viscoelastic and other history-dependent media. In this talk, I will discuss a microlocal analysis of such equations for two classes of memory kernels: continuous kernels and fractional kernels. The main question is how memory affects the propagation of singularities. In the classical memoryless setting, singularities propagate along bicharacteristics of the principal symbol. I will explain how this picture changes once a time-nonlocal memory term is introduced.

For continuous memory kernels, I will show that the usual hyperbolic propagation persists, but an additional stationary singularity appears at the initial location. For fractional memory kernels, the behavior is genuinely different: after a microlocal diagonalization, the branch that would normally propagate becomes infinitely regular, while a stationary singularity remains. The analysis combines wave-packet methods, Volterra structure, and microlocal reduction to first-order equations with memory. I will also discuss connections to viscoelasticity, wave attenuation, and possible directions toward inverse problems in media with memory.