Exponential stabilization of a coupled system of Korteweg-de Vries equations with localized damping

Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equations as a singular limit of the Kuramoto-Sivashinsky system

“…For instance, one can take the damping term 1 ω (− d 2 dx 2 ) −1 . The results described above, except [15] and [23], were extended with similar statements for system (1.4)-(1.6) in [19,3,17]. The analysis developed in these works was first inspired by the work [19] due to Micu and Ortega where they showed that the decay of solutions of the linearized system fails for some critical values of the length L of the interval (0, L).…”

“…It has the structure of a pair of KdV equations with both linear and nonlinear coupling terms. This somewhat complicated system has been the object of intensive research in recent years (see, for instance, [3], [4] and [14]).…”

“…Clearly, in order to avoid ambiguities in the limiting process, it is necessary to guarantee the existence and uniqueness of solutions for both models. In this regard we mention the works [3] and [16].…”

Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

“…Moreover, after addition to the equation of the damping term id(x)u, where the non-negative function d was strictly positive on a certain sub-interval, large-time decay of solutions was established. Certain preceding results for the truncated HNLS equation (β = γ = 0) can be found in [2,3,4,5,12,13].…”

“…2,+ ) for certain admissible weight function ψ(x), such that ψ ′ (x) is also an admissible weight function. Let either p ∈ [1, 3) or p ∈ [1,4), γ = 0. Then there exists a weak solution of problem…”