Decay of solution of generalized Benjamin—Bona—Mahony—Burgers equations in n-space dimensions

We consider the large time behavior of the solutions to the Cauchy problem for the BBM–Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers equation called the nonlinear diffusion wave. Moreover, we construct the appropriate second asymptotic profiles of the solutions depending on the initial data. Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. Especially, the important point of this study is that the second asymptotic profiles of the solutions with slowly decaying data, whose case has not been studied, are obtained.

This paper studies the asymptotic behavior of a solution of dispersive shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation. The energy decay rates of the solution to the model are examined through the Fourier transform method. Moreover, we develop a pseudo-compact finite difference scheme for solving the model, study solution behavior, and confirm our theoretical results. The fundamental energy-decreasing property, which is obtained from the model of coupling the Rosenau-RLW equation with the Rosenau-Burgers equation, is derived and preserved by the present numerical scheme. The existence, uniqueness, convergence, and stability of the numerical solution are theoretically analyzed. Some numerical experiments are also conducted to demonstrate the accuracy and robustness of the present method. Finally, to verify the optimal decay rates, the numerical results are carried out at variant time scales by applying the moving boundary technique. The simulations are successfully constructed to support the theoretical results.

In this paper, we investigate the initial value problem for the nonlinear pseudo-parabolic equation. Global existence and optimal decay estimate of solution are established, provided that the initial value is suitably small. Moreover, when $n⩾2$ and the nonlinear term $f(u)$ disappears, we prove that the global solutions can be approximated by the linear solution as time tends to infinity. When $n=1$ and the nonlinear term $f(u)$ disappears, we show that as time tends to infinity, the global solution approaches the nonlinear diffusion wave described by the self-similar solution of the viscous Burgers equation.

In this paper, we consider the nonlinear Sobolev equation with an initial condition and Dirichlet boundary condition. In different regions divided by $p$, $q$, and $m$, we obtain the sufficient conditions for finite-time blow-up of the solutions, and get the upper and lower bounds of the blow-up time.

This paper is concerned with nonlinear stability of planar viscous shock profiles of the generalized Benjamin–Bona–Mahony–Burgers equations in two dimensions. Our analysis is motivated by J. Goodman’s work (1989) [4] on the nonlinear stability of scalar viscous shock profiles in two dimensions and is based on some new decay estimates on the planar viscous shock profiles of the generalized Benjamin–Bona–Mahony–Burgers equations.