Global Well-posedness And Long Time Behavior Of Boussinesq-Burgers Equation

This thesis is mainly concerned with a kind of nonlinear evolution equations de-scribing the motion of water waves:the Boussinesq-Burgers equation and generalized Boussinesq-Burgers equation.By utilizing energy methods and applying some important inequalities,we study the global well-posedness and long time behavior of the large data solution to the equation on a finite interval or the whole space.Furthermore,the global existence and asympotic behavior of the solution are established.Moreover,the explicit decay rate and the diffusion limit of the solution are obtained.The main works of this dissertation are consist of the following four chapters.In Chapter 1,we mainly introduce the background,research developments and recent works,and give a brief overview of the main content of this paper.In Chapter 2,we study the initial-boundary value problem for the Boussinesq-Burgers equation on a finite interval.Based on the Dirichlet type dynamic boundary conditions,and H1 ×H2 initial data which are compatible with boundary conditions,we construct the delicate energy estimates and explore appropriate conditions on the dy-namic boundary data,then there exists the unique global-in-time solution to the initial-boundary value problem,and the solution converges to the boundary data as time goes to infinity,regardless of the magnitude of the initial data.This result has been published by“Journal of Differential Equations".In Chapter 3,we study the Cauchy problem of the Boussinesq-Burgers equation in one space dimension.By using the LP(p>2)estimates,we obtain the lower order esti-mates of the large data solution,then utilizing the standard energy methods,we establish the global existence and asymptotic behavior of the large data solution.Furthermore,by defining the anti-derivatives and using the time-weighted energy estimate method,we prove that the global solution converges to the constant equilibrium state with an alge-braic decay rate as time approaches infinity.This result,has been published by "Journal of Mathematical Analysis and Applications".In Chapter 4,we study the initial problem of a generalized Boussinesq-Burgers e-quation in one space dimension.The equation has a nonlinear term including high order power exponents.By exploring a series of importantly technical inequalities,and building the entropy estimate of the equation,we obtain the lower order and high order estimates under varies of power exponents,then prove the unique global-in-time solution to the Cauchy problem of the equation.In addition,the solution of the equation converges to the constant equilibrium state with a algebraic decay rate as time goes to infinity,regard-less of the magnitude of the initial data.Moreover,the diffusion limit of the equation is obtained.