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Long-time behavior of solution for Rosenau-Burgers equation. II. (English) Zbl 0909.35122
This paper establishes results on the decay of solutions for the Rosenau-Burgers equation (R-B E) $$u_t+ u_{xxxxt}- \alpha u_{xx}+\beta u_x+ \phi(u)_x= 0,\quad x\in\bbfR^1,\quad t>0,$$ with initial data $u|_{t=0}= u_0(x)\to u_{\pm}$, as $x\to\pm\infty$. When $u_+= u_-=\overline u\in\bbfR$, the author proves that the solution of this Cauchy problem tends toward the constant $\overline u$ in the forms $\|(u-\overline u)(t)\|_{L^2}= O(t^{-3/4})$ and $\|(u-\overline u)(t)\|_{L^\infty}= O(t^{-1})$ for any nonlinearity $\phi(u)\in C^2$, which improves the previous work [{\it M. Mei}, Appl. Anal. 63, 315-330 (1996; Zbl 0865.35113)]. Moreover, when $u_+\ne u_-$, under some restrictions on the state constants $u_{\pm}$, there exists a stationary travelling wave solution of (R-B E) in the form $u(x- st)= U(x)$, $U(\pm\infty)= u_{\pm}$, with the zero speed $s= 0$. In this case, the author shows that it is nonlinearly stable for both the nondegenerate shock condition $\phi'(u_+)< -\beta< \phi'(u_-)$ and the degenerate shock condition $\phi'(u_+)= -\beta< \phi'(u_-)$ or $\phi'(u_+)< -\beta= \phi'(u_-)$ or $\phi'(u_{\pm})= -\beta$. Especially, in the nondegenerate case $\phi'(u_+)< -\beta< \phi'(u_-)$, the author shows that the solution $u(t,x)$ of (R-B E) asymptotically converges to the travelling wave $U(x)$ at the decay rate $O(e^{-\theta t})$ for some $\theta>0$, if the initial perturbation is suitably small and decays like $O(e^{-\eta| x|})$ for some $\eta>0$.
Reviewer: M.Mei (Kanazawa)

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35B40 Asymptotic behavior of solutions of PDE 35L65 Conservation laws
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