Long Time Behavior For Solution Of Generalized Boussinesq Equations

Boussinesq equations, which describe the nonlinear waves in shallow waters,were frst proposed by Boussinesq, who was a famous French mathematician andphysicist. This equation is based on long waves in shallow water by makingadditional assumptions. First, the seabed is horizontal. Second, the horizontalvelocity does not change with depth. Third, there is a linear relationship betweenthe vertical velocity and the undisturbed depth. The last one, some verticalacceleration is retained. Generalized Boussinesq-type equations were developedfor the case, in which the horizontal velocity of water would change with the waterdepth. These equations are a sort of important nonlinear evolution equations andplay a vital role in the study of wave motion-related physical problems, such asartifcial islands, ofshore airports, water with an ice-cover, etc.In this thesis, we studied the Cauchy problem for two types of generalizedBoussinesq-type equations. We considered the global existence and long timebehavior of solutions. The thesis is arranged as follows.In Chapter1, we introduced the physical background and research statusto the generalized Boussinesq-type equations. We showed our method to studythese equations. We also introduced the organization and the main results of thethesis. Besides, we gave the explanation of symbols.In Chapter2, we considered the global existence of the solutions to theCauchy problem for a sort of generalized Boussinesq-type equation with damp-ing term in multidimensional space. The physical models of these equations arevery varied, like the wave motion with very large foating structure(VLFS). Theenergy method could not solve these equations because of the high order partialderivatives. To overcome this, we divided the solutions into two parts, low fre-quency part and high frequency part. For the low frequency part, we considered the estimates on the L2and L∞by means of the Green’s function combined withconvolution inequalities. Then we employed energy method to estimate the highfrequency part. Finally, we got the global classical solutions to the equations bymeans of constructional iterative equation. We also obtained Lpestimates of thesolutions.The work in Chapter3was based on Chapter2. We considered the point-wise estimates of solutions for the multidimensional generalized Boussinesq-typeequation. The key point of our method was to consider the pointwise estimatesof the Green’s function. For the low frequency part, we divided the low frequencypart of Green function into the convolution of hyperbolic structure and the dis-sipative structure. For the high frequency part, we employed the induction toestimate the solution. Then we obtain the pointwise estimates of solutions bysome technical lemmas.In Chapter4, we considered the well-posedness of the solutions to the Cauchyproblem for another sort of generalized Boussinesq-type equation in multidimen-sional space. Such equations broadened its scope of application for VLFS model,making it suitable for weakly nonlinear wave and long wave with medium length,also fxes the efects of the change of the seabed topography on the pressure onthe water particle. We handled the equation by using frequency decomposition.First, we estimated the low frequency part of the Green function. Second, weestimated the high frequency part of the solutions by means of energy method.By constructing iterative equation, we obtained the global existence and Lpes-timates to the Cauchy problem successfully.