On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–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 ≤ s ≤ 3 and 1 ≤ p < 2, and in H³(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L²-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.