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

Abstract

It is shown that if 𝑢 is a sufficiently smooth solution to a two-dimensional nonlinear wave equation such that there exists 𝐿>0 with supp 𝑢(𝑖)⊆[−𝐿,𝐿]×[−𝐿,𝐿], for 𝑖=0,1, then 𝑢≡0.

1. Introduction

In this paper, we consider the following two-dimensional nonlinear wave equation: 𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥𝑥𝑥+𝑐𝑢_𝑦𝑦𝑥+𝑑𝑢_𝑥=0, (1) 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 two-dimensional 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, we mainly use the ideas of [12–15]. The main result is as follows.

Theorem 1. Assume that 𝑏>𝑐 and 𝐿>0, if 𝑢∈𝐶([0,1];𝐻⁴(𝑅²))∩𝐶¹([0,1];𝐿²(𝑅²)) is a solution of (1) such that supp𝑢(0),supp𝑢(1)⊆[−𝐿,𝐿]×[−𝐿,𝐿]; (2) then, 𝑢≡0.

2. Preliminary Estimates

Lemma 2 (see [13]). Assume that 𝑠>0 and 𝜅>0. (i) If 𝑓∈𝐻^𝑠(𝑅²)∩𝐿²(𝑒^2𝜅𝑥𝑑𝑥𝑑𝑦), then 𝐽^𝜃𝑠𝑒^{(1−𝜃)𝜅𝑥}𝑓_𝐿2≤𝐶𝐽^𝑠𝑓^𝜃_𝐿2𝑒^𝜅𝑥𝑓_𝐿^1−𝜃₂, (3) (ii) if 𝑓∈𝐻^𝑠(𝑅²)∩𝐿²(𝑒^{2(𝜅𝑥+𝜅𝑦)}𝑑𝑥𝑑𝑦), then 𝐽^𝜃𝑠𝑒^{(1−𝜃)(𝜅𝑥+𝜅𝑦)}𝑓_𝐿2≤𝐶𝐽^𝑠𝑓^𝜃_𝐿2𝑒^{𝜅𝑥+𝜅𝑦}𝑓_𝐿^1−𝜃₂, (4) where 𝜃∈[0,1], 𝐶=𝐶(𝑠,𝜅), and [𝐽^𝑠𝑓]^∧(𝜉)∶=(1+|𝜉|²)^𝑠/2[𝑓]^∧(𝜉).

Lemma 3. Assume that 𝜅>0, if 𝑢∈𝐶([0,1];𝐻⁴(𝑅²))∩𝐶¹([0,1];𝐿²(𝑅²)) is a solution of (1) such that 𝑢(0)∈𝐿²(𝑒^2𝜅𝑥𝑑𝑥𝑑𝑦); then, 𝑢(𝑡) is bounded in 𝐻³(𝑒^2𝜅𝑥𝑑𝑥𝑑𝑦).

Proof. Assume that 𝜑(𝑥)∈𝐶^∞(𝑅) is a decreasing function with 𝜑(𝑥)=1 if 𝑥<1 and 𝜑(𝑥)=0 if 𝑥>10. Let 𝜃_𝑛(𝑥)=∫^𝑥₀𝜑(𝑡/𝑛)𝑑𝑡 for 𝑛∈𝑍⁺ and 𝜙_𝑛(𝑥)=𝑒^{2𝑘𝜃𝑛(𝑥)}. It is easy to check that 𝜙_𝑛(𝑥)≤𝜙_𝑛+1(𝑥) and 𝜙_𝑛^(𝑗)(𝑥)≤𝐶_𝑗,𝑘𝜙_𝑛(𝑥),∀𝑗∈𝑍⁺,∀𝑥∈𝑅. (5) Multiplying (1) by 𝑢𝜙_𝑛 and integrating by parts in 𝑅²_𝑥𝑦, we obtain 12𝑑𝑢𝑑𝑡²𝜙_𝑛=𝑎3𝑢³𝜙^′−3𝑏2𝑢_𝑥²𝜙^′_𝑛+𝑏2𝑢²𝜙_𝑛^′′′−𝑐2𝑢_𝑦²𝜙^′+𝑑2𝑢²𝜙^′_𝑛≤𝑎3‖𝑢‖_𝐿^∞_(𝑅²₎𝐶_1,𝜅𝑢²𝜙_𝑛+𝑏2𝐶_3,𝜅𝑢²𝜙_𝑛+𝑑2𝐶_1,𝜅𝑢²𝜙_𝑛≤𝑎3𝐶‖𝑢‖_{𝐶([0,1];𝐻}²₎+𝑏2𝐶_3,𝑘+𝑑2𝐶_1,𝑘𝑢²𝜙_𝑛=𝐶_𝑘,𝑢𝑢²𝜙_𝑛. (6) Applying Gronwall Lemma and the Monotone Convergence Theorem, we have 𝑢(𝑡)²𝑒^2𝜅𝑥𝑑𝑥𝑑𝑦≤𝐶𝑢(0)²𝑒^2𝜅𝑥𝑑𝑥𝑑𝑦,∀𝑡∈[0,1]. (7) This proves that 𝑢(𝑡) is bounded in 𝐿²(𝑒^2𝜅𝑥𝑑𝑥𝑑𝑦).
Applying Lemma 2 with 𝑠=4 and 𝜃=3/4, we have that 𝑢(𝑡) is bounded in 𝐻³(𝑒^2𝜅𝑥𝑑𝑥𝑑𝑦). Here, we used the fact that 𝑢∈𝐶([0,1];𝐻⁴). This completes the proof of the lemma.

Lemma 4. Assume that 𝜅>0, 𝑏>𝑐, and 𝑢∈𝐶([0,1];𝐻⁴(𝑅²))∩𝐶¹([0,1];𝐿²(𝑅²)) is a solution of (1). (i) If 𝑢(0)∈𝐿²(𝑒^2𝜅𝑥𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦), then 𝑢(𝑡) is bounded in 𝐻³(𝑒^2𝜅𝑥𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦); (ii) if 𝑢(1)∈𝐿²(𝑒^−2𝑘𝑥𝑒^2𝑘|𝑦|𝑑𝑥𝑑𝑦), then 𝑢(𝑡) is bounded in 𝐻³(𝑒^−2𝜅𝑥𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦).

Proof. Letting 𝑣(𝑡)=𝑒^𝜅𝑥𝑢(𝑡) and 𝑢 a solution to (1), we have 𝑒^𝑘𝑥𝑢^′=𝜅𝑎𝑢+𝑏𝜅²+𝑑𝑣−𝑎𝑢+3𝑏𝜅²+𝑑𝑣_𝑥+3𝑏𝜅𝑣_𝑥𝑥−𝑏𝑣_𝑥𝑥𝑥+𝑐𝜅𝑣_𝑦𝑦−𝑐𝑣_𝑦𝑦𝑥. (8) Multiplying (8) by 𝑣𝜙_𝑛(𝑦) and integrating by parts in 𝑅²_𝑥𝑦, we obtain 𝑒^𝜅𝑥𝑢^′𝑣𝜙_𝑛=𝑎𝜅𝑢𝑣²𝜙_𝑛+𝑎2𝑣²𝑢_𝑥𝜙_𝑛+𝑏𝜅³+𝑑𝜅𝑣²𝜙_𝑛−3𝑏𝜅𝑣_𝑥²𝜙_𝑛−𝑐𝜅𝑣_𝑦²𝜙_𝑛+𝑐2𝜅𝑣²𝜙_𝑛^′′−𝑐𝑣_𝑥𝑣_𝑦𝜙^′_𝑛. (9) Note that |𝜙^′_𝑛(𝑦)|≤2𝜅𝜙_𝑛(𝑦) and 𝑒^𝜅𝑥𝑢^′𝑣𝜙_𝑛=12𝑑𝑣𝑑𝑡²𝜙_𝑛,𝑎.𝑒𝑡∈[0,1]. (10) It follows from (9) that 12𝑑𝑣𝑑𝑡²𝜙_𝑛≤𝑎𝜅𝐶‖𝑢‖_{𝐶([0,1];𝐻}²_(𝑅²₎₎𝑣²𝜙_𝑛+𝑎𝐶𝑢_{𝑥𝐶([0,1];𝐻2(𝑅2))}𝑣²𝜙_𝑛+𝑏𝜅³+𝑑𝜅𝑣²𝜙_𝑛−𝑐𝜅𝑣_𝑥²+2𝑣_𝑥𝑣_𝑦+𝑣_𝑦²𝜙_𝑛+𝑐2𝜅𝐶_2,𝜅𝑣²𝜙_𝑛=𝐶_𝜅,𝑢𝑣²𝜙_𝑛−𝑐𝜅𝑣_𝑥+𝑣_𝑦²𝜙_𝑛≤𝐶_𝜅,𝑢𝑣²𝜙_𝑛,𝑎.𝑒𝑡∈[0,1]. (11) Since 𝑢∈𝐶([0,1];𝐻⁴) and 𝑢(𝑡) is bounded in 𝐿²(𝑒^2𝜅𝑥𝑒^2𝜅𝑦𝑑𝑥𝑑𝑦), applying Lemma 2 with 𝑠=4 and 𝜃=3/4, we have that 𝑢(𝑡) is also bounded in 𝐻³(𝑒^2𝜅𝑥𝑒^2𝜅𝑦𝑑𝑥𝑑𝑦).
Similarly, we can prove that 𝑢(𝑡) is bounded in 𝐻³(𝑒^2𝜅𝑥𝑒^−2𝜅𝑦𝑑𝑥𝑑𝑦). Let 𝑢(𝑥,𝑦,𝑡)=𝑢(𝑥,−𝑦,𝑡); then, 𝑢 is a solution to (1) and satisfies 𝑢(0)∈𝐿²(𝑒^2𝜅𝑥𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦), and therefore 𝑢 is bounded in 𝐻³(𝑒^2𝜅𝑥𝑒^2𝜅𝑦𝑑𝑥𝑑𝑦). This proves (i).
Now, we prove (ii). Let 𝑢(𝑥,𝑦,𝑡)=𝑢(−𝑥,𝑦,1−𝑡); then, 𝑢(𝑥,𝑦,𝑡) is also a solution of (1) and satisfies the hypothesis of (i). This proves (ii) and completes the proof of the lemma.

Remark 5. In particular, if the conditions for 𝑢(0) and 𝑢(1) given in (i) and (ii), respectively, are satisfied, then 𝑢(𝑡) is bounded in 𝐻³(𝑒^2𝜅|𝑥|𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦).

Lemma 6 (see [13]). Let 𝜅>0, 𝑣∈𝐶¹([0,1];𝐿²(𝑅²)) is a function such that 𝑣(𝑡) is bounded in 𝐿²(𝑒^2𝜅|𝑥|𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦), and 𝑣^′∈𝐿¹([0,1];𝐿²(𝑒^2𝜅|𝑥|𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦)). Then, for all 𝜆∈𝑅 and all 𝜉=(𝜉₁,𝜉₂)∈𝑅², the functions 𝑡→[𝑒^𝜆𝑥𝑣(𝑡)]^∧(𝜉) and 𝑡→[𝑒^𝜆𝑦𝑣(𝑡)]^∧(𝜉) are absolutely continuous in [0,1] with derivatives [𝑒^𝜆𝑥𝑣^′(𝑡)]^∧(𝜉) and [𝑒^𝜆𝑦𝑣^′(𝑡)]^∧(𝜉) a.e. 𝑡∈[0,1], respectively.

Lemma 7. Assume that 𝜅>0, 𝑏>𝑐, and 𝜆≠0, if 𝑣∈𝐶([0,1];𝐻³(𝑅²))∩𝐶¹([0,1];𝐿²(𝑅²)) is a function such that 𝑣(𝑡) is bounded in 𝐻³(𝑒^2𝜅|𝑥|𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦) and 𝑣^′(𝑡)∈𝐿¹([0,1];𝐿²(𝑒^2𝜅|𝑥|𝑒^2𝜅|𝑦|𝑑𝑥𝑑𝑦)). Then, 𝑒^𝜆𝑥𝑣_{𝐿2(𝑅2×[0,1])}≤𝑒^𝜆𝑥𝑣(0)_𝐿2(𝑅2)+𝑒^𝜆𝑥𝑣(1)_𝐿2(𝑅2)+𝑒^𝜆𝑥𝑣_𝑡+𝑏𝑣_𝑥𝑥𝑥+𝑐𝑣_𝑦𝑦𝑥+𝑑𝑣_{𝑥𝐿2(𝑅2×[0,1])},𝑒^𝜆𝑦𝑣_{𝐿2(𝑅2×[0,1])}≤𝑒^𝜆𝑦𝑣(0)_𝐿2(𝑅2)+𝑒^𝜆𝑦𝑣(1)_𝐿2(𝑅2)+𝑒^𝜆𝑦𝑣_𝑡+𝑏𝑣_𝑥𝑥𝑥+𝑐𝑣_𝑦𝑦𝑥+𝑑𝑣_{𝑥𝐿2(𝑅2×[0,1])}. (12)(13)

Proof. Let 𝑝(𝑡)=𝑒^𝜆𝑥𝑣(𝑡) and 𝑞(𝑡)=𝑒^𝜆𝑥(𝑣_𝑡+𝑏𝑣_𝑥𝑥𝑥+𝑐𝑣_𝑦𝑦𝑥+𝑑𝑣_𝑥); then, 𝑞(𝑡)=𝑒^𝜆𝑥𝑣^′(𝑡)−𝑏𝜆³+𝑑𝜆𝑝(𝑡)+3𝑏𝜆²+𝑑𝑝_𝑥(𝑡)−3𝑏𝜆𝑝_𝑥𝑥(𝑡)+𝑏𝑝_𝑥𝑥𝑥(𝑡)−𝑐𝜆𝑝_𝑦𝑦(𝑡)+𝑐𝑝_𝑦𝑦𝑥(𝑡). (14) Taking the spatial Fourier transform in (14) and applying Lemma 6, we have 𝑑𝑝(𝑡𝑑𝑡)^∧(𝜉)+−𝑖𝑚(𝜉)−𝑛(𝜉)𝑝(𝑡)^∧(𝜉)=𝑞(𝑡)^∧(𝜉),𝑎.𝑒𝑡∈[0,1], (15) where 𝑚(𝜉)=−3𝑏𝜆²+𝑑𝜉₁+𝑏𝜉³₁+𝑐𝜉₁𝜉²₂,𝑛(𝜉)=𝑏𝜆³+𝑑𝜆−3𝑏𝜆𝜉²₁−𝑐𝜆𝜉²₂. (16) According to (15), when 𝑛(𝜉)≤0, we have 𝑝(𝑡)^∧(𝜉)=𝑒^{𝑖𝑚(𝜉)𝑡}𝑒^{𝑛(𝜉)𝑡}𝑝(0)^∧(𝜉)+^𝑡₀𝑒^{𝑖𝑚(𝜉)(𝑡−𝜏)}𝑒^{𝑛(𝜉)(𝑡−𝜏)}𝑞(𝜏)^∧(𝜉)𝑑𝜏, (17) and when 𝑛(𝜉)>0, we choose to write 𝑝(𝑡)^∧(𝜉)=𝑒^{−𝑖𝑚(𝜉)(1−𝑡)}𝑒^{−𝑛(𝜉)(1−𝑡)}𝑝(1)^∧(𝜉)−¹_𝑡𝑒^{−𝑖𝑚(𝜉)(𝜏−𝑡)}𝑒^{−𝑛(𝜉)(𝜏−𝑡)}𝑞(𝜏)^∧(𝜉)𝑑𝜏. (18) Therefore, we have 𝑝(𝑡)^∧(𝜉)≤𝑝(0)^∧(𝜉)+𝑝(1)^∧(𝜉+₁₀)𝑞(𝜏)^∧(𝜉)𝑑𝜏,𝑡∈[0,1]. (19) Appling Plancherel formula, we have inequality (12).
Similarly, letting 𝑝(𝑡)=𝑒^𝜆𝑦𝑤(𝑡), we can also have (13). This completes the proof of the lemma.

Lemma 8 (see [12, 13]). Assume that 𝐿>0, and 𝑏>𝑐, if 𝑢∈𝐶([0,1];𝐻⁴(𝑅²)) is a solution to (1) such that supp𝑢(𝑡)∈[−𝐿,𝐿]×[−𝐿,𝐿],∀𝑡∈[0,1]; (20) then, 𝑢≡0.

Proof. The proof is similar to that of Theorem  1.1 in [12], and we omit the details.

3. Proof of the Main Result

Assume that 𝜙(𝑥)∶=𝜙(𝑥−𝐴) for 𝐴>𝐿 where 𝜙(𝑥)∈𝐶^∞(𝑅) is a nondecreasing function such that 𝜙(𝑥)=0 for 𝑥<0 and 𝜙(𝑥)=1 for 𝑥>1. Let 𝑔∶=𝜙(𝑥)𝑢; then, 𝑔(0)=𝑔(1)=0. According to Lemma 7, we obtain that 𝑒^𝜆𝑥𝑤≤𝑒^𝜆𝑥𝑔_𝑡+𝑏𝑔_𝑥𝑥𝑥+𝑐𝑔_𝑦𝑦𝑥+𝑑𝑔_𝑥=𝑒^𝜆𝑥𝜙𝑢_𝑡+𝑏𝜙𝑢_𝑥𝑥𝑥+𝑐𝜙𝑢_𝑦𝑦𝑥+𝑏𝜙^′′′+𝑑𝜙^′𝑢+3𝑏𝜙^′′+𝑑𝜙𝑢_𝑥+3𝑏𝜙^′𝑢_𝑥𝑥+𝑐𝜙^′𝑢_𝑦𝑦≤𝑎𝑒^𝜆𝑥𝜙𝑢𝑢_𝑥+𝑒^𝜆𝑥𝐺_𝜙,𝑢, (21) where ‖⋅‖∶=‖⋅‖_𝐿²_(𝑅²_×[0,1]) and 𝐺_𝜙,𝑢=𝑏𝜙^′′′+𝑑𝜙^′𝑢+3𝑏𝜙^′′𝑢_𝑥+3𝑏𝜙^′𝑢_𝑥𝑥+𝑐𝜙^′𝑢_𝑦𝑦. (22) Note that the derivatives of 𝜙(𝑥) are supported in the interval [𝐴,𝐴+1]; then, 𝑒^𝜆𝑥𝐺_𝜙,𝑢=𝐶𝑒^𝜆(𝐴+1), (23) where 𝐶 is dependent on ‖𝑢‖_{𝐶([0,1];𝐻}²₎ and max{𝑏,𝑐,𝑑}.

Combining (21) with (23), we obtain 𝑒^𝜆𝑥𝜙𝑢≤𝑎𝑒^𝜆𝑥𝜙𝑢𝑢_{𝑥𝐿∞([𝐴,+∞]×𝑅×[0,1])}+𝐶𝑒^𝜆(𝐴+1). (24) Applying Lemma 4 with 𝜅=1, we have that 𝑢_𝑥(𝑡)𝑥,𝑦≤𝐶₁𝑒^−𝑥, (25) then 𝑒^𝜆𝑥𝜙𝑢≤𝑎𝐶₁𝑒^−𝐴𝑒^𝜆𝑥𝜙𝑢+𝐶𝑒^𝜆(𝐴+1). (26) Since ‖𝑒^𝜆𝑥𝜙𝑢‖<+∞, taking 𝐴>max{𝐿,1} such that 𝐶₁𝑒^−𝐴<1/2𝑎, we have 𝑒^𝜆𝑥𝜙𝑢≤𝐶𝑒^𝜆(𝐴+1). (27) Note that 𝜙(𝑥)=1 for 𝑥≥2𝐴; we have 𝑒^2𝜆𝐴1₀^∞_−∞^∞_2𝐴𝑢(𝑡)𝑥,𝑦^21/2≤𝑒^𝜆𝑥𝜙𝑢≤𝐶𝑒^𝜆(𝐴+1). (28) Letting 𝜆→+∞, we obtain ¹₀^∞_−∞^∞_2𝐴𝑢(𝑡)𝑥,𝑦^21/2=0, (29) and this proves that 𝑢≡0 in [2𝐴,∞)×𝑅×[0,1].

Next, we will prove that 𝑢≡0 in 𝑅×[2𝐴,∞)×[0,1]. Let ℎ=𝜙(𝑦)𝑢; then, 𝑒^𝜆𝑦ℎ≤𝑒^𝜆𝑦ℎ_𝑡+𝑏ℎ_𝑥𝑥𝑥+𝑐ℎ_𝑦𝑦𝑥+𝑑ℎ_𝑥=𝑒^𝜆𝑦𝜙𝑢_𝑡+𝑏𝜙𝑢_𝑥𝑥𝑥+𝑐𝜙𝑢_𝑦𝑦𝑥+𝑑𝜙𝑢_𝑥+2𝑐𝜙^′𝑢_𝑥𝑦+𝑐𝜙^′′𝑢_𝑥≤𝑎𝑒^𝜆𝑦𝜙𝑢𝑢_𝑥+𝑒^𝜆𝑦𝐻_𝜙,𝑢, (30) where 𝐻_𝜙,𝑢∶=2𝑐𝜙^′𝑢_𝑥𝑦+𝑐𝜙^′′𝑢_𝑥.

In fact, Let 𝑢(𝑥,𝑦,𝑡)=𝑢(−𝑥,−𝑦,1−𝑡); it is easy to check that 𝑢 also satisfies the hypotheses of this theorem, and then we find that 𝑢≡0 in (−∞,−2𝐴]×𝑅×[0,1]∪𝑅×(−∞,−2𝐴]×[0,1]. (31) Thus, there exists 𝐴>0 such that supp𝑢(𝑡)∈[−𝐴,𝐴]×[−𝐴,𝐴] for all 𝑡∈[0,1]. Applying Lemma 8, we complete the proof of Theorem 1.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11171135), the Natural Science Foundation of Jiangsu (no. BK 2010329), the Project of Excellent Discipline Construction of Jiangsu Province of China, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), as well as the Taizhou Social Development Project (no. 2011213).