Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity

Abstract

To understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity, we aim to study the global well-posedness of nonlinear fourth order wave equations with logarithmic source term, where the dispersive, the nonlinear weak damping and linear strong damping are taken into account. Based on the potential well method, the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow up with subcritical initial energy. Moreover, we extend the results to the critical initial energy, which is realized by introducing scaling technique. Finally, surrounding the blow up at arbitrarily high initial energy, we compare and discuss the research program stemming from two different strategies. In the first one, it is assumed that the blow up result is bound to the original logarithmic source by weakening the dispersive–dissipative structure, while the second one is based on the nonlinear wave equation with complete dispersive–dissipative structure, but the logarithmic source is replaced by an enhanced version. Through these two strategies, we explain the mechanism of these two structures respectively and establish two kinds of sufficient conditions for initial data leading to high energy blow up.