Logarithmic wave equations in non-cylindrical domains

TL;DR

This paper investigates logarithmic wave equations in non-cylindrical domains, proving existence of solutions, addressing energy positivity issues, and establishing exponential decay of energy.

Contribution

It introduces a method for proving existence of solutions and analyzes energy decay in a novel class of logarithmic wave equations in non-cylindrical domains.

Findings

Existence of weak solutions established.

Energy positivity depends on initial data.

Exponential decay of positive energy proven.

Abstract

This paper is devoted to studying a type of logarithmic wave equation in non-cylindrical domains. Firstly, by the penalty method, we prove the existence of weak solutions to such kind of equations. Secondly, different from the dissipative wave equation, the energy defined in this problem is not always positive. Thus, some suitable initial data are selected to let the energy be positive. Finally, by a difference inequality, we derive a exponential decay estimate for the positive energy.