KdV equation in domains with moving boundaries

Abstract

The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for the Korteweg-de Vries equation is considered. Existence and uniqueness of global strong solutions are proved as well as the exponential decay of small solutions in asymptotically cylindrical domains.

Quasilinear evolution equations of the third order

“…This approach implies, that the function ϕ, which is an extension of the values of the solution itself at the boundary into the corresponding domain, can be chosen such, that its properties ensure a possibility of derivation of a relevant estimate on the solution in L 2 (I) from (24). In particular, it seems natural, that ϕ must satisfy the following condition…”

“…The boundary potential J was also used in [37], [39] to construct the auxiliary function ϕ and use it in (23), (24) to obtain a global estimate in L 2 . More precisely,…”

“…Note, that the rate of growth of the nonlinearity in KdV, which does not exceed quadratic, is essential for a global estimate in L 2 in the case of non-homogeneous boundary data (see (24)). Certain global results for the considered problems in the case of the greater rate of growth can be found in [58], [5], [34], [21], [11], but they are beyond the scope of this survey.…”

“…Intial-boundary value problems for the KdV equation with different from (19)-(21) boundary conditions or in other domains (for example, with moving boundaries) are considered in [12], [13], [18], [80], [24].…”

Odd-order quasilinear evolution equations posed on a bounded interval

“…In this occasion some boundary conditions are needed to specify the solution. Therefore, precise mathematical analysis of boundary value problems in bounded domains for general dispersive equations is welcome and attracts attention of specialists in the area of dispersive equations, especially KdV and BBM equations, [3,5,6,7,8,9,12,15,16,17,18,21,25,26,27,28,31,37,38]. Cauchy problem for dispersive equations of high orders was successfully explored by various authors, [2,10,11,15,23,33,36].…”