Abstract
In this paper, we establish a global Carleman estimate for the Kawahara equation. Based on this estimate, we obtain the Unique Continuation Property (UCP) for this equation and the global exponential stability for the Kawahara equation with a very weak localized dissipation.
Highlights
In this paper, we consider the Kawahara equation yt + yx + yxxx − yxxxxx + yyx = 0 in Q, y(0, t) = 0=y(1, t) in (0, T ), yx(0, t) = 0 = yx(1, t) (1) yxx(1, t) = 0y(x, 0) = y0(x) in (0, T ), in I, where I = (0, 1), T > 0 and Q = I × (0, T )
The Kawahara equation is a dispersive partial differential equation (PDE) describing numerous wave phenomena such as magneto-acoustic waves in a cold plasma [19], the propagation of long waves in a shallow liquid beneath an ice sheet [16], gravity waves on the surface of a heavy liquid [7], etc. In this model the conservative dispersive effect is represented by the term yxxx − yxxxxx
In the literature this equation is referred as the fifth-order KdV equation [4], or singularly perturbed KdV equation [23], or the special version of the Benney-Lin equation [2, 3]
Summary
The Kawahara equation, global Carleman estimate, unique continuation property, exponential decay, stabilization, asymptotic analysis. This paper is devoted to establish a global Carleman estimate for the Kawahara equation. Based on the global Carleman estimate established in this paper, a Unique Continuation Property (UCP) for the Kawahara equation and the global exponential stability for the Kawahara equation with a very weak localized dissipation can be obtained. 1. The Carleman estimate for the Kawahara equation considered in [8] is local, namely the function considered in [8] should satisfy supp y ⊂ Q, while the Carleman estimate in Theorem 1.1 is global. We consider the global exponential stability for the Kawahara equation with weak damping yt + yx + yxxx − yxxxxx + yyx + By. KdV equation has been considered in [20].
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