Decay of Small Solutions for the Zakharov-Kuznetsov Equation posed on a half-strip

Dispersive equations attract attention of many mathematicians. More popular are Korteweg-de Vries and Schrödinger equations. The theory of the Cauchy problem for them nowadays is well developed and presented in papers of Bona and his colleagues [1], Kruzhkov and Faminskii [11], Kato [9], Bourgain [3], Saut [21], Temam [23], Ponce and his colleages [10,17], etc. Last years appeared papers on initial boundary value problems for dispersive equations in bounded and non-bounded domains. Here we can mention again Bona and his colleagues [2], Bubnov [4], Faminskii [6], Faminskii and Larkin [7], Larkin [13,14]. Quite recently was discovered that the KdV equation has an implicit internal dissipation. This property allowed to prove exponential decay of small solutions in bounded domains without adding any artificial damping. Later, this effect was proved for a wide class of dispersive equations of any odd order in the space variable. We can mention here papers of Larkin [13,14], Faminskii and Larkin [7,8]. In [20] Rosier showed that control of the linear KdV equation with a "drift term", ux, is impossible for the critical domains. It means that there is not decay of solutions with time for a set of critical domains. Hence, there is not also decay of solutions in a quarter-plane. By the way, without the "drift term" it is possible to prove exponential decay of small solutions for the KdV equation posed on any bounded interval (0, L). Recently appeared papers of Faminskii [6], Pyatkov [19], Linares and Pastor [15], Linares and Saut [18] on initial boundary value problems for the ZakharovKuznetsov (ZK) equation which may be considered as multi-dimensional analogue

Quite recently was discovered that the KdV equation has an implicit internal dissipation.This property allowed to prove exponential decay of small solutions in bounded domains without adding any artificial damping.Later, this effect was proved for a wide class of dispersive equations of any odd order in the space variable.We can mention here papers of Larkin [13,14], Faminskii and Larkin [7,8].In [20] Rosier showed that control of the linear KdV equation with a "drift term", u x , is impossible for the critical domains.It means that there is not decay of solutions with time for a set of critical domains.Hence, there is not also decay of solutions in a quarter-plane.By the way, without the "drift term" it is possible to prove exponential decay of small solutions for the KdV equation posed on any bounded interval (0, L).
Recently appeared papers of Faminskii [6], Pyatkov [19], Linares and Pastor [15], Linares and Saut [18] on initial boundary value problems for the Zakharov-Kuznetsov (ZK) equation which may be considered as multi-dimensional analogue 58 N. A. Larkin and E. Tronco of the KdV equation , see [24].Our work was motivated by the paper of Saut and Temam [22] on initial boundary value problem in a domain bounded in x variable and non-bounded in y variable.Studying this paper, we discovered that the term u xyy in ZK equation delivers additional "dissipation" which can help to prove decay of small solutions in non-bounded domains of a channel type non-bounded in x direction and we consider the following initial boundary value problem.

Formulation of the problem and main results
Let T, L be real positive numbers; Consider in Q t the following initial boundary value problem: (2.1) where 3) by a parabolic problem as in [22], one can prove the following result.
Theorem 2.1.Let T and L be arbitrary finite real positive numbers.Let and there is a real positive k such that Then there exists a unique regular solution of (2.1)-( 2.3) such that

Decay of Solutions
The main result of this article is the following theorem.
Then regular solutions of (2.1)-(2.3)satisfy the inequality Decay of Small Solutions for the C Equation 59where Proof: Transforming the integral to the equality Next, consider for some k > 0 the equality which can be reduced to the form x )(t)
With this choice of k > 0, (3.11) becomes Now we assume u 0 2 such that which gives d dt (e kx , u 2 )(t) + χ(e kx , u 2 )(t) ≤ 0, where The function With this, χ becomes Solving (3.13), we prove Theorem. 2 . Then regular solutions of (2.Remark.The presence in (2.1) of a linear term u x (α = 1) implies a restriction for value of L : (L < 2 √ 2), which means that a channel D has a limitation in width.On the other hand , absence of that term (α = 0) allows to L bo be any finite positive number; it means that a channel may be of any finite width.