Asymptotic profile of L^2-norm of solutions for wave equations with critical log-damping

TL;DR

This paper analyzes the long-term behavior of solutions to wave equations with a specific log-damping, revealing unique asymptotic profiles and decay properties, including a rare case where the L^2-norm neither decays nor blows up.

Contribution

It provides the first detailed asymptotic analysis of wave equations with critical log-damping, highlighting novel phenomena in L^2-norm behavior across different dimensions.

Findings

L^2-norm of solutions in 1D blows up over infinite time.

In 2D, the L^2-norm remains bounded and does not decay or blow up.

Established optimal estimates for solutions as time approaches infinity.

Abstract

We consider wave equations with a special type of log-fractional damping. We study the Cauchy problem for this model in the whole space, and we obtain an asymptotic profile and optimal estimates of solutions as time goes to infinity in L^2-sense. A maximal discovery of this note is that under the effective damping, in case of n = 1 L^2-norm of the solution blows up in infinite time, and in case of n = 2 L^2-norm of the solution never decays and never blows up in infinite time. The latter phenomenon seems to be a rare case.