Abstract
An initial-boundary value problem for the 2D Zakharov-Kuznetsov-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.
Highlights
Quite recently, the interest on dispersive equations became to be extended to multi-dimensional models such as Kadomtsev-Petviashvili (KP) and ZakharovKuznetsov (ZK) equations [24]
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
We will prove exponential decay of regular solutions in an elevated weighted norm corresponding to the H1(S) norm
Summary
Let B, T, r be finite positive numbers. Define S = {(x, y) ∈ R2 : x ∈ R, y ∈ (0, B)}; Sr = {(x, y) ∈ R2 : x ∈ (−r, +∞), y ∈ (0, B)} and ST = S × (0, T ). Denote the partial derivatives, as well as ∂x or ∂x2y 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 L2(S), and · Hk stands for norms in the L2-based Sobolev spaces. We will use the spaces Hs ∩ L2b , where L2b = L2(e2bxdx), see [11].
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