Kawahara equation in a bounded domain

Initial-Boundary Value Problems for Generalized Dispersive Equations of Higher Orders Posed on Bounded Intervals

“…The existence and uniqueness of local regular solutions have been established.where x ∈ (0, L), Q T = (0, T ) × (0, L); l, k ∈ N; T, L are real positive numbers. This equation includes as special cases classical dispersive equations: when l = k = 1, we have the well-known Kortewegde Vries (KdV) equation, see [15,20,31], and when k = 1, l = 2, we have the Kawahara equation [5,16,23]. For k = 1, the Cauchy problem for dispersive equations of higher orders has been studied in [3,6,11,12,17,28,32] and initial boundary value problems have been studied in [4,5,7,23,30].…”

Kawahara-Burgers Equation on a Strip

“…The theory of the Cauchy problem for (1.2) and other dispersive equations like the KdV equation has been extensively studied and is considerably advanced today [1,4,5,6,7,8,19,20,18,22,23,25,37,40]. Results on IBVPs for one-dimensional dispersive equations both in bounded and unbounded domains may be found in [5,6,9,10,24,28,32]. It was shown in [9,10,27,29,30,33] that the KdV and Kawahara equations have an implicit internal dissipation.…”

“…Results on IBVPs for one-dimensional dispersive equations both in bounded and unbounded domains may be found in [5,6,9,10,24,28,32]. It was shown in [9,10,27,29,30,33] that the KdV and Kawahara equations have an implicit internal dissipation. This allowed the proof of exponential decay of small solutions in bounded domains without adding any artificial damping term.…”

On the controllability of the fifth-order Korteweg–de Vries equation

“…concerning the Cauchy problem in the real line, see for instance [8,9,17,16,18,21] and references therein. For what concerns the boundary value problem, the Kawahara equation with homogeneous boundary conditions was investigated by Doronin and Larkin [10]. Note that the initial boundary value problem for the (third-order) Korteweg-de Vries equation has drained much attention (see in particular [1,4,5,11,14]).…”