Properties Of Solutions To The Generalized Benjamin-Bona-Mahony-Burgers Equations

This thesis is concerned with some properties of solutions to the generalized Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equations. The main results obtained in this thesis contain the following two parts:The first part is concerned with the nonlinear stability of traveling wave solutions to the Cauchy problem of the generalized Benjamin-Bona-Mahony-Burgers equationsHere v(>0),βare constants, u±are two given constants satisfying u_+≠u_- and the nonlinear smooth function f(u)∈C~2(R) is assumed to be either convex or concave for all u under our consideration. Firstly, by exploiting the phase-plane analysis method, a sufficient condition to guarantee the existence of monotonic traveling waves to the generalized BBM-Burgers equation is given and then based on the elementary energy method together with the continuation argument, we obtain the nonlinear stability of traveling wave solutionsφ(x-st) to the Cauchy problem (E). Secondly, decay rates (both algebraic and exponential) of the solutions of the Cauchy problem (E) to the traveling waves are obtained by employing the space-time weighted energy method developed by Kawashima and Matsumura in [12] to discuss the asymptotic behavior of traveling wave solutions to the Burgers equation.The second part is devoted to discussing the global nonlinear stability of the boundary layer solutions to the initial-boundary value problem of the generalized Benjamin-Bona-Mahony-Burgers equation in the half space R_+Here u(t, x) is an unknown function of t > 0 and x∈R_+, u_b≠u_+ are two given constant states and the nonlinear smooth function f(u)∈C~2 (R) is assumed to be a strictly convex function of u. Based on the elementary energy method and the continuation argument, we first show that the corresponding boundary layer solutionφ(x) is globally nonlinear stable and then, by employing the space-time weighted energy method which was initiated by Kawashima and Matsumura [12], the convergence rates (both algebraic and exponential) of the global solution u(t, x) to the problem (Ⅰ) toward the boundary layer solutionφ(x) are also obtained for both the non-degenerate case f′(u_+) < 0 and the degenerate case f′(u_+) = 0.