A better asymptotic profile of Rosenau–Burgers equation

Abstract

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.