Dynamic Properties Of Solutions To A Class Of Sixth-order Boussinesq Models

This dissertation mainly investigates the dynamic properties of the solutions to a class of the sixth-order Boussinesq models,such as the well-posedness,global existence and finite time blow-up of solutions.In the first part,we study a class of the sixth-order Boussinesq equation with linear strong damping and nonlinear source.Firstly,by using potential-well method,we establish some suitable conditions for the global existence and finite time blow-up of solutions at low initial energy level.Secondly,by scaling technique,we extend the results to the critical initial energy level.Finally,under the further study of the Boussinesq-type equations,we establish some appropriate assumptions independent of the depth of the potential well,which will help us obtain the global existence,finite time blow-up of the solutions and the upper bound of blow-up time for blowing-up solutions with subcritical initial energy level,critical initial energy level and supercritical initial energy level.In the second part,we study a class of sixth-order Boussinesq equation with logarithmic nonlinearity.Based on the Faedo-Gal¨erkin method,the logarithmic Sobolev inequality,and the potential well method,we get the conditions on global existence with subcritical initial energy level and critical initial energy level,infinite time blow-up with subcritical initial energy level,the polynomial decay and the exponential decay of the energy of the system with subcritical initial energy level and critical initial energy level.More specifically,this dissertations is mainly divided into the following four chapters:In the first chapter,we first make a brief introduction to the Boussinesq-type problems.Then we give the research background and purpose of this dissertation.In the second chapter,we investigate a class of the sixth-order Boussinesq equation with linear strong damping and nonlinear source.First of all,by means of appropriate Faedo-Gal¨erkin method,we study the local well-posedness.Secondly,we establish the existence of global solutions and obtain the finite time blow-up solutions with subcritical and critical initial energy by potential well method and concavity method.Moreover,the decay rate of energy functional for global solutions and upper bound for finite time blow-up solutions are also studied.Thirdly,we study the global existence and finite time blow-up results for general positive initial energy.Especially,by establishing a new finite time blow-up condition,which is independent of the mountain-pass level,we show the solution can blow up in finite time at arbitrary initial energy level.Finally,under a further assumption for the parameters,we establish the sufficient and necessary conditions of existing finite time blow-up solutions.In the third chapter,we study a sixth-order Boussinesq equation with logarithmic nonlinearity.Based on the Faedo-Gal¨erkin method,the logarithmic Sobolev inequality,and the potential well method,we get the conditions on global existence with subcritical initial energy and critical initial energy,infinite time blow-up with subcritical initial energy,and the polynomial decay and the exponential decay of the energy of the system of the solutions with subcritical initial energy and critical initial energy.