Remarks on the Decay Rate of the Energy for Damped Modified Boussinesq-Beam Equations on the 1-D Half Line

Abstract

We consider the mixed problem for weakly damped modified Boussinesq-Beam equations on the one dimensional half line (0, + ∞). We shall derive fast decay results of the total energy and L²-norm of solutions based on the idea due to [7], which is an essential modification of that developed by Morawetz [15]. In order to apply that idea due to [7] to the one dimensional exterior mixed problem, one also constructs an important Hardy-Sobolev type inequality, which holds only in the 1-D half line case.