Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (omega = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.