Kawahara-Burgers equation on a strip

An initial-boundary value problem for the 2D Kawahara-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in the $L^2$-norm.


Introduction
We are concerned with an initial-boundary value problem (IBVP) for the two-dimensional Kawahara-Burgers (KB) equation u t + u x − u xx + uu x + u xxx + u xyy − ∂ 5 x u = 0 (1.1) posed on a strip modeling an infinite channel {(x, y) ∈ R 2 : x ∈ R, y ∈ (0, B), B > 0}. This equation is a two-dimensional analog of the Kawahara type equation which includes dissipation and dispersion and has been studied intensively last years due to its applications in Mechanics and Physics [1,3,4,5,7,6,8,18,25]. Equations (1.1) and (1.2) are typical examples of so-called dispersive equations which attract considerable attention of both pure and applied mathematicians in the past decades. 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 MSC2010 35Q53;35B35 keywords: Kawahara-Burgers equation , Dispersive equations, Exponential Decay. 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. Later, this effect has been proven for a wide class of dispersive equations of any odd order with one space variable [16].
On the other hand, it has been shown in [39] that control of the linear KdV equation with the transport term u x may fail for critical domains, but it is possible to eliminate the term u x by simple scaling when the KdV and Kawahara equations are posed on the whole line. The same is true also for (1.1) posed on a strip (y ∈ (0, B), x ∈ R, t > 0) [31].
Recently, interest on dispersive equations became to be extended to multi-dimensional models such as Kadomtsev-Petviashvili (KP), Zakharov-Kuznetsov (ZK) equations [42] and dispersive equations of higher orders [12]. As far as the ZK equation and its generalizations are concerned, the results on IVPs can be found in [13,17,35,36,38] and IBVPs were studied in [2,14,15,29,34,41]. It was shown that IBVP for the ZK equation posed on a half-strip unbounded in x direction with the Dirichlet conditions on the boundaries possesses regular solutions which decay exponentially as t → ∞ provided initial data are sufficiently small and the width of a half-strip is not too large [30,34]. The similar result was established for the 2D Kawahara equation posed on a half-strip [29]. This means that multi-dimensional dispersive equation may create an internal dissipative mechanism for some types of IBVPs.
The goal of our note is to prove that the KB equation on a strip also may create a dissipative effect without adding any artificial damping. We must mention that IBVP for the ZK equation on a strip (x ∈ (0, 1), y ∈ R) has been studied in [11,41] and IBVPs on a strip (y ∈ (0, L), x ∈ R) for the ZK equation and Zakharov-Kuzetsov-Burgers equation were considered in [2,31] and for the ZK equation with some internal damping in [15]. In the domain (y ∈ (0, B), x ∈ R, t > 0), the term u x in (1.1) can be scaled out by a simple change of variables. Nevertheless, it can not be safely ignored for problems posed both on finite and semi-infinite intervals as well as on infinite in y direction bands without changes in the original domain [11,39].
The main results of our paper are the existence and uniqueness of regular and weak global-in-time solutions for (1.1) posed on a strip with the Dirichlet boundary conditions and the exponential decay rate of these solutions as well as continuous dependence on initial data. To explore dissipativity of the term u xyy , we used exponential weight e 2bx which implied to define solutions of (1.1) as the product e bx [u t − u xx + uu x + u xxx + u xyy − ∂ 5 x u] = 0 in L 2 (S). We must mention that this idea has been proposed yearlier in [19].
The paper has the following structure. Section 1 is Introduction. Section 2 contains formulation of the problem. In Section 3, we prove global existence and uniqueness theorems for regular solutions in some weighted spaces and continuous dependence on initial data. Surprisingly, we did not succeed to prove global existence for all positive weights e 2bx as in [30,34] and imposed a restriction 6 − 40b 2 ≥ 0. In Section 4, we prove exponential decay of small regular solutions in an elevated norm. In Section 5, we prove the existence, uniqueness and continuous dependence on initial data for weak solutions as well as the exponential decay rate of the L 2 (S)-norm for small solutions without limitations on the width of the strip.

Problem and preliminaries
Let B, T, r be finite positive numbers.
Hereafter subscripts u x , u xy , etc. denote the partial derivatives, as well as ∂ x or ∂ 2 xy when it is convenient. Operators ∇ and ∆ are the gradient and Laplacian acting over S. By (·, ·) and · we denote the inner product and the norm in L 2 (S), and · H k stands for norms in the L 2 -based Sobolev spaces. We will use also the spaces H s ∩ L 2 b , where L 2 b = L 2 (e 2bx dx), see [19]. Consider the following IBVP:

Existence of regular solutions
Approximate solutions. We will construct solutions to (2.1)-(2.3) by the Faedo-Galerkin method: let w j (y) be orthonormal in L 2 (S) eigenfunctions of the following Dirichlet problem: w jyy + λ j w j = 0, y ∈ (0, B); (3.1) Define approximate solutions of (2.1)-(2.3) as follows: where g j (x, t) are solutions to the following Cauchy problem for the system of N generalized Kawahara equations: It can be shown that for g j (x, 0) ∈ H s , s ≥ 5, the Cauchy problem (3.4)-(3.5) has a unique regular solution [1,18,19,37]. To prove the existence of global solutions for (2.1)-(2.3), we need uniform in N global in t estimates of approximate solutions u N (x, y, t). Estimate I. Multiply the j-th equation of (3.4) by g j , sum up over j = 1, ..., N and integrate the result with respect to x over R to obtain It follows from here that for N sufficiently large and ∀t > 0 In our calculations we will drop the index N where it is not ambiguous. Estimate II. For some positive b, multiply the j-th equation of (3.4) by e 2bx g j , sum up over j = 1, ..., N and integrate the result with respect to x over R. Dropping the index N, we get The proof is obvious. In our calculations, we will frequently use the following multiplicative inequalities [26]: where the constant C D depends on a way of continuation of Extending u N (x, y, t) for a fixed t into the exterior of S by 0 and exploiting (3.10), we find Substituting this into (3.8), we come to the inequality By the Gronwall lemma, It follows from this estimate and (3.6) that uniformly in N and for any r > 0 and t ∈ (0, T ) where C does not depend on N.
Estimates (3.13), (3.14) make it possible to prove the existence of a weak solution to (2.1)-(2.3) passing to the limit in (3.4) as N → ∞. For details of passing to the limit in the nonlinear term see [19].
We will need the following lemma : where δ, δ 1 are arbitrary positive numbers.
Proof. Denote v = e bx u. Then simple calculations give Returning to the function u(x, y, t), we prove Lemma 3.3 Estimate III. Multiplying the j-th equation of (3.4) by −(e 2bx g jx ) x , and dropping the index N, we come to the equality Making use of Proposition 3.2, we estimate Similarly, Substituting I 1 , I 2 into (3.16) with 2δ = b, we obtain for ∀t ∈ (0, T ) : Estimate IV. Multiplying the j-th equation of (3.4) by −2(e 2bx λ j g j ), and dropping the index N, we come to the equality Making use of Proposition 3.2, we estimate Taking δ = b, we transform (3.18) into the inequality . Making use of (3.7) and the Gronwall lemma, we get ∀t ∈ (0, T ) : This and (3.17) give for ∀t ∈ (0, T ): which imply that for all finite r > 0 and all t ∈ (0, T ) Estimate V. Multiplying the j-th equation of (3.4) by (e 2bx g jxx ) xx , and dropping the index N, we come to the equality Using (3.10), we estimate Taking 2δ = b and substituting I into (3.22), we obtain . Making use of (3.7), we find Estimate VI. Differentiate (3.4) by t and multiply the result by e 2bx g jt to obtain Making use of (3.10), we estimate Taking δ = b and substituting I into (3.24), we get . This implies ∀t ∈ 0, T ): where Multiplying the j-th equation of (3.4) by −e 2bx g jx and dropping the index N, we come to the equality Estimate VIII.

Proof. Rewrite (3.4) in the form
where Φ N (y) is an arbitrary function from the set of linear combinations N i=1 α i w i (y) and Ψ(x) is an arbitrary function from H 1 (R). Taking into account estimates (3.7), (3.37) and fixing Φ N , we can easily pass to the limit as N → ∞ in linear terms of (3.39). To pass to the limit in the nonlinear term, we must use (3.21) and repeat arguments of [19]. Since linear combinations [ N i=1 α i w i (y)]Ψ(x) are dense in L 2 (S), we come to (3.38). This proves the existence of regular solutions to (2.1)-(2.3). Remark 1. Estimates (3.7),(3.37) are valid also for the limit function u(x, y, t) and (3.7) obtains its sharp form: Uniqueness of a regular solution.

Decay of regular solutions
In this section we will prove exponential decay of regular solutions in an elevated weighted norm. We start with Theorem ?? which is crucial for the main result.  ∈ (0, b 0 ), u 0 ≤ 3π 8B and u(x, y, t) be a regular solution of (2.1)-(2.3). Then for all finite B > 0 the following inequalities are true: Proof. Multiplying (2.1) by 2e 2bx u, we get the equality d dt Taking into account (3.10), we estimate The following proposition is principal for our proof. Proof. Since u(x, 0, t) = u(x, B, t) = 0, fixing (x, t), we can use with respect to y the following Steklov inequality: if f (y) ∈ H 1 0 (0, π) then π 0 f 2 (y) dy ≤ π 0 |f y (y)| 2 dy.
After a corresponding process of scaling we prove Proposition 4.2.
Observe that differently from [29,30,34], we do not have any restrictions on the width of a strip B.
The main result of this section is the following assertion.
Proof. We start with the following lemma.

Weak solutions
Here we will prove the existence, uniqueness and continuous dependence on initial data as well as exponential decay results for weak solutions of (2.1)-(2.3) when the initial function u 0 ∈ L 2 (S).
Then for all finite positive T and B there exists at least one function u(x, y, t): such that e bx u ∈ L ∞ (0, T ; L 2 (S)) ∩ L 2 (0, T ; H 1 (S)), e bx u xx ∈ L 2 (0, T ; L 2 (S)) and the following integral identity takes a place: where v ∈ C ∞ (S T ) is an arbitrary function.
Proof. In order to justify our calculations, we must operate with sufficiently smooth solutions u m (x, y, t). With this purpose, we consider first initial functions u 0m (x, y), which satisfy conditions of Theorem 3.4, and obtain estimates (3.7), (3.21) for functions u m (x, y, t). This allows us to pass to the limit as m → ∞ in the following identity: y , bv y + v xy )(t)} dt = (e bx u m 0 , v(x, y, 0)), (5.2) and come to (5.1).
Uniqueness of a weak solution. Proof. Actually, this proof is provided by Theorem 3.5. It is sufficient to approximate the initial function u 0 ∈ L 2 (S) by regular functions u 0m in the form: lim m→∞ u 0m − u 0 = 0, where u om satisfies the conditions of Theorem 3.4. This guarantees the existence of the unique regular solution to (2.1)-(2.3) and allows us to repeat all the calculations which have been done during the proof of Theorem 3.5 and to come to the following inequality: d dt (e 2bx , z 2 m )(t) + (e 2bx , |∇z m | 2 )(t) ≤ C(b) 1 + u 1m 2 (t) + u 2m 2 (t) + u 1xm 2 (t) + u 2xm 2 (t) (e 2bx , z 2 m )(t).
Hence, Since e bx z(x, y, t) is a weak limit of regular solutions {e bx z m (x, y, t)}, then (e 2bx , z 2 )(t) ≤ (e 2bx , z 2 m )(t) = 0. This implies u 1 ≡ u 2 a.e. in S T . The proof of Theorem 5.2 is complete.
We have in this Theorem a more strict condition u 0 ≤ 3π 16B instead of u 0 ≤ 3π 8B in the case of decay for regular solution because for weak solutions we do not have the sharp estimate (3.40), but only (3.7).