The Long-Time Behavior Of Solutions For The Boussinesq Equation With Nonlocal Damping

We study in this master's thesis the existence of finite dimensional global at-tractor and exponential attractor for the singularly perturbed Boussinesq equation with nonlocal weak damping as well as the Boussinesq equation with strong nonlinear damping respectively,where u=u(x,t)is an unknow function,??(0,1)is the small pertur-bation coefficient for(0.1),and ?>0 is a constant for(0.2),? is a bounded domain in R3 with smooth boundary(?)?,v is the unit outward normal on(?)?,?ut?ut(r?0)is the nonlocal weak damping,f?H-1(?)is the external force term.In the first part of the thesis,when the nonlinearity g(u)has no any growth restrictions,and ?>0 is only a constant,we firstly prove the well-posedness of so-lution for problem(0.1)by means of monotone operator theory.Secondly,we obtain the dissipativity of the dynamical system(E,S(t))associated with the problem(0.1)in the space H02(?)× L2(?)and D(A3/4)× H01(?).Subsequently,the asymptotic s-moothness of(E,S(t))is demonstrated by the energy reconstruction method,and further we acquire the quasi-stability of(E,S(t)).Finally,we obtain the main results based on the above conclusions.In addition,an estimate of the difference between the solutions of the problem(0.1)and the corresponding pseudo-parabolic equation is verified by combining with the small perturbation of ?.Utilizing the key estimate,we also obtain the existence of global solution for problem(0.1).In the second part of the thesis,under the condition that the growth of nonlin-earity is invariable,we prove the well-posedness of solution for problem(0.2)by using the method of Galerkin appropriation.Consequently,combining with the asymptotic smoothness and the quasi-stability estimates of the dynamical system(E,S(t))and the contractive function method,we establish the existence of the finite-dimensional global attractor and the generalized exponential attractor.