Well-posedness of solutions for the dissipative Boussinesq equation with logarithmic nonlinearity

Abstract

In this paper, we consider a class of Boussinesq-type equations with logarithmic nonlinearity. By employing the classical Faedo–Galerkin method, we first establish the local well-posedness of solutions. Then we investigate the dynamical behaviors of solutions. More precisely, for the solutions with subcritical or critical initial energy, we prove that they exist globally and are uniformly bounded when I(u0)>0, where I(u0) denotes the Nehari functional with the initial value u0. Moreover, under further appropriate assumptions about the initial data, we derive the exponential energy decay estimates of global solutions. In particular, for the solutions with subcritical or critical initial energy, we show that they can be extended over time (the whole half line) and then blow up at infinite time when I(u0)<0. Last but not least, by developing some new methods, we prove the existence of infinite time blow-up solutions with arbitrary high initial energy.