The Well-posedness Of The Definite Solution Problem For The Dissipative Boussinesq Equation

This thesis is devoted to the initial value problem and the initial-boundary value problem of the dissipative Boussinesq equation,where ? and ? are constants satisfying ? + ?>0.We study the global well-posedness,the decay properties,the long time behavior of solutions,the blow-up of solutions in finite time and the effect of the dissipative terms-?ut and ?2ut upon the decay rate and the regularity of small solutions.Firstly,we investigate the initial value problem of Equation(1),proving the global existence and uniqueness of solutions,showing the decay estimates of global solutions,and studying the finite time blow-up of solutions.We establish the pointwise estimates of fundamental solutions by the energy method in the Fourier space,consequently giving the corresponding space-time estimates by the technology of high and low frequency decomposition.Moreover,by means of the contracting mapping principle,we obtain the existence and uniqueness of local solutions in the energy space C([0,T];H1).The global existence and nonexistence of solutions are discussed under the following three different initial energy(E(0))conditions,respectively:(i)the subcritical initial energy E(0)<d,(i)the critical initial energy E(0)= d,(i)the supercritical initial energy E(0)>d,(where d is the potential well depth).Under the subcritical initial energy condition,we provide the necessary and sufficient conditions of global existence and nonexistence of solutions respectively,by the standard potential well theory and concavity method.For the critical initial energy condition,we present the sufficient conditions of global existence and nonexistence of solutions by the approximation method.It is complex and interesting to study the supercritical initial energy condition,we also give the sufficient conditions of global existence and nonexistence of solutions by selecting suitable functionals.At last,with the help of the pointwise estimates of the fundamental solutions and the technology of high and low frequency decomposition we establish the estimates of the solution operators in the spaces H? and L? to study the linearized problem of Equation(1)to get the asymptotic profile and the optimal estimates of solutions.Under the smallness of initial data,we study the global existence solutions in C([0,T];Hs),where s>n/2-2,by using the contracting mapping principle,moreover get the decay estimates of solutions.Secondly,we are interested in the initial-boundary value problem contain-ing Hinged boundary conditions and Dirichlet boundary conditions,studying the global existence,uniqueness,the decay estimates,the asymptotic behavior of so-lutions and the finite time blow-up of solutions.In the first part,we mainly discuss Equation(1)with the source nonlinear terms and the Hinged boundary conditions.We establish local well-posedness of solutions by the compactness method,and we provide the necessary and sufficient conditions of' existence and nonexistence of global solutions respectively,by the potential well method under the assumption E(0)<d and show the exponential decay of the solutions.For E(0)= d and E(0)>d,we investigate the global existence and nonexistence of solutions and the asymptotic behavior of the global solutions.In the second part,we study Equation(1)with the sink nonlinear functions and the Hinged boundary conditions and we prove the existence of the global compact attractor and exponential attractor by the quasi-stable method.At last,we discuss the local existence of solutions of Equation(1)with the Dirichlet conditions.