Logarithmic wave equations in non-cylindrical domains

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.