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"x"x"x"x"[email protected]"x"x+(u^p^+^1/(p+1))"x=0. By the variable scaling method, we discover that the solution of the nonlinear parabolic equation u"[email protected]"x"x+(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.