On the Support of Solutions to a Two-Dimensional Nonlinear Wave Equation

where a, b, c, d are arbitrary positive constants. Equation (1) was recently derived by Gottwald [1] for large scale motion from the barotropic quasigeostrophic equation as a twodimensional model for Rossby waves. He [2] showed that (1) has traveling wave solutions via the homotopy perturbation method. Using a subequation method, the traveling wave solutions are also studied by Fu et al. [3]. Aslan [4] constructed solitary wave solutions and periodic wave solutions to (1) by the Exp-function method. For a = b = c = 1 and d = 0 in (1), one obtains the classical Zakharov-Kuznetsov (ZK) equation [5], which is a mathematical model to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma. Solitary wave solutions and the Cauchy problem to ZK equation have extensively been studied in the literature ([6–11]). Panthee [12] proved that if a sufficiently smooth solution to the initial value problem associated with the ZK equation is supported compactly in a nontrivial time interval, then it vanishes identically. Recently, Bustamante et al. [13] showed that sufficiently smooth solutions of the ZK equation that have compact support for two different times are identically zero. The purpose of this paper is to investigate the support of solutions to (1). To solve the problem, wemainly use the ideas of [12–15]. The main result is as follows.


Introduction
In this paper, we consider the following two-dimensional nonlinear wave equation: where , , ,  are arbitrary positive constants.Equation (1) was recently derived by Gottwald [1] for large scale motion from the barotropic quasigeostrophic equation as a twodimensional model for Rossby waves.He [2] showed that (1) has traveling wave solutions via the homotopy perturbation method.Using a subequation method, the traveling wave solutions are also studied by Fu et al. [3].Aslan [4] constructed solitary wave solutions and periodic wave solutions to (1) by the Exp-function method.
For  =  =  = 1 and  = 0 in (1), one obtains the classical Zakharov-Kuznetsov (ZK) equation [5], which is a mathematical model to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma.Solitary wave solutions and the Cauchy problem to ZK equation have extensively been studied in the literature ( [6][7][8][9][10][11]).Panthee [12] proved that if a sufficiently smooth solution to the initial value problem associated with the ZK equation is supported compactly in a nontrivial time interval, then it vanishes identically.Recently, Bustamante et al. [13] showed that sufficiently smooth solutions of the ZK equation that have compact support for two different times are identically zero.
The purpose of this paper is to investigate the support of solutions to (1).To solve the problem, we mainly use the ideas of [12][13][14][15].The main result is as follows.
Multiplying (1) by   and integrating by parts in  2  , we obtain Applying Gronwall Lemma and the Monotone Convergence Theorem, we have This proves that () is bounded in  2 ( 2  ).
Proof.The proof is similar to that of Theorem 1.1 in [12], and we omit the details.