Infinite Propagation Speed For Wave Solutions on Some P.C.F. Fractals

TL;DR

This paper investigates wave solutions on p.c.f. fractals, demonstrating that under certain heat kernel bounds, wave propagation can occur at infinite speed, which has implications for understanding wave behavior on fractal structures.

Contribution

The paper proves that wave solutions on p.c.f. fractals exhibit infinite propagation speed when the heat kernel satisfies sub-Gaussian bounds, and establishes bounds for the wave kernel based on heat kernel estimates.

Findings

Wave solutions on p.c.f. fractals can propagate infinitely fast.

Heat kernel bounds influence wave kernel behavior.

Sub-Gaussian bounds are key to understanding wave propagation on fractals.

Abstract

From the finite difference method for wave equation on p.c.f. fractals, we would expect that infinite prorogation speed property for wave solutions on a large class of p.c.f. fractals. We prove that is true if the heat kernel satisfies the sub-Gaussian lower bound. Furthermore, we provide a sub-Gaussian upper bound for wave kernel given the heat kernel sub-Gaussian upper bound.