A better asymptotic profile of Rosenau-Burgers equation

This paper studies the large-time behavior of the global solutions to the Cauchy problem for the Rosenau-Burgers (R-B) equation u_t + u_xxxxt - αu_xx + (u^p+1/(p + 1))_x = 0. By the variable scaling method, we discover that the solution of the nonlinear parabolic equation u_t - αu_xx + (u^p+1/(p + 1))_x = 0 is a better asymptotic profile of the R-B equation. The convergence rates of the R-B equation to the asymptotic profile have been developed by the Fourier transform method with energy estimates. This result is better than the previous work [1,2] with zero as the asymptotic behavior. Furthermore, the numerical simulations on several test examples are discussed, and the numerical results confirm our theoretical results.