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A better asymptotic profile of Rosenau-Burgers equation. (English) Zbl 1020.35097
Summary: This paper studies the large-time behavior of the global solutions to the Cauchy problem for the Rosenau-Burgers (R-B) equation \[ u_t+ u_{xxxxt}-\alpha u_{xx}+ (u^{p+1}/(p+ 1))_x= 0. \] By the variable scaling method, we discover that the solution of the nonlinear parabolic equation \[ u_t-\alpha u_{xx}+ (u^{p+1}/(p+ 1))_x= 0 \] 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 with zero as the asymptotic behavior. Furthermore, the numerical simulations on several test examples are discussed, and the numerical results confirm our theoretical results.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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