On the decay of step for the Korteweg-de Vries-Burgers equation
We study the behavior, as \(t\to\infty\), of the solution \(u(x,t)\) of the Cauchy problem for \[ u_ t+ 2uu_ x+ au_{xxx}- u_{xx}=0 \] in the case \(| a|>1/8\). We prove that the solution \(u(x,t)\) converges to the stationary wave \(\varphi(x)\) as \(t\to\infty\) uniformly with respect to \(x\in \mathbb{R}\) in the case \(| a|>1/8\), and we obtain an estimate for the rate of convergence. Thus, the stationary wave is also asymptotically stable in the case \(| a|>1/8\).
scientific article; zbMATH DE number 525657