Global well-posedness for the nonlinear damped wave equation with logarithmic type nonlinearity

The initial boundary value problem for the nonlinear wave equations with damping and logarithmic nonlinearity is investigated in this paper. By making use of modified potential well theory and the technique of Logarithmic-Sobolev inequality, we establish global existence as well as asymptotic behavior of solution, under the assumption that the initial energy is small. Moreover, we obtain an exponential decay which is much faster than the decay in polynomial nonlinear case of Gazzola and Squassina (Ann I H Poincare AN 23:185-207, 2006). These results generalize and extend work in application of potential well theory to wave equations.