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

Abstract

Initial-boundary value problems for nonlinear dispersive equations of evolution of order 2l + 1, l ∈ N with a convective term of the form u k u x , k ∈ N have been considered on intervals (0, L), L ∈ (0, +∞). 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]. Although dispersive equations were deduced for the whole real line, necessity to calculate numerically the Cauchy problem, approximating the real line by finite intervals [4], implies to study initial-boundary value problems posed on bounded and unbounded intervals, [5,7,19,21,23]. What concerns (1.1) with k > 1, l = 1, called generalized Korteweg-de Vries equations, the Cauchy problem for (1.1) has been studied in [8,9,10,18,25,26], where was proved that for k = 4, called the critical case, the initial

Global solutions for the generalized Benney-Lin equation posed on bounded intervals and on a half-line

“…See also [8,9]. Mathematical results on initial and initial boundary value problems for various variants of (1.1) are presented in [2,4,6,7,10,12,13,20,21,23,24,25,26,32,33,36,38], see references there for more information. In [4,13,21,38], Kuramoto-Sivashinsky type equations have been considered which included u xxx (KdV) term.…”

Formulation of Problems for Stationary Dispersive Equations of Higher Orders on Bounded Intervals with General Boundary Conditions

“…making use of the semigroup theory (see [1], Theorem 4.1) or a semi-discrete approach [2] . Here, we propose (1) as a stationary analog of (2) because the last equation includes as special cases classical dispersive equations: when l = 1, we have the Korteweg-de Vries (KdV) equation [3,4] and for l = 2 the Kawahara equation [5][6][7] .…”

“…Therefore, precise mathematical analysis of mixed problems in bounded domains for dispersive equations is welcome and attracts attention of specialists in this area [2,[11][12][13][15][16][17][18] . Last years, publications on stationary and evolution dispersive equations of higher orders appeared [1,15,[19][20][21][22] . Usually, simple boundary conditions at x = 0 and x = L such as D i u(0) = D i u(L) = D l u(L) = 0, i = 0,..., l − 1 for (1) were imposed, see [1,21,22].…”

“…Last years, publications on stationary and evolution dispersive equations of higher orders appeared [1,15,[19][20][21][22] . Usually, simple boundary conditions at x = 0 and x = L such as D i u(0) = D i u(L) = D l u(L) = 0, i = 0,..., l − 1 for (1) were imposed, see [1,21,22]. Different kind of boundary conditions for KdV and Kawahara equations was considered in [16,[23][24][25].…”

“…The goal of our work is to formulate correct general boundary value problems for (1) and to prove the existence and uniqueness of regular solutions. Obviously, boundary conditions for (1) are the same as for (2).…”