Kirchhoff-type system with linear weak damping and logarithmic nonlinearities

Abstract

For the nonlinear Kirchhoff-type wave system with logarithmic nonlinearities and weak dissipation the global well-posedness of initial boundary value problem is analyzed. Focusing on the interplay between Kirchhoff terms and logarithmic sources, we investigate the Kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in blow up analysis which is the primary scenario of interest. By employing potential well method and concavity method, we obtain several results related to the sufficient conditions posed on subcritical initial energy and critical initial energy, which is used to classify initial data for global existence and finite time blow up. Finally, via careful analysis involving the unstable invariant set under supercritical initial energy, we are able to show an affirmative result that the solution blows up in finite time when initial data satisfy some suitable assumptions.