  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.
