On the well-posedness and asymptotic behavior of the generalized KdV-Burgers equation

Abstract

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg-de Vries Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval $[1,5)$. We first prove the global well-posedness in $H^s(R)$, for $0\le s\le 3$ and $1\le p<2$, and in $H^3(R)$, when $p\ge 2$. For $2\le p<5$, we prove the existence of global solutions in the $L^2$-setting. Then, by using multiplier techniques combined with interpolation theory, the exponential stabilization is obtained for a indefinite damping term and $1\le p<2$. Under the effect of a localized damping term the result is obtained when $2\le p<5$. Combining multiplier techniques and compactness arguments it is shown that the problem of exponential decay is reduced to prove the unique continuation property of weak solutions

Cited by in corpus (1)

• Controllability Aspects of the Korteweg-de Vries Burgers Equation on Unbounded Domains