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

Authors: Wenbin Zhang; Jiangbo Zhou; Lixin Tian; Sunil Kumar
Journal: Journal of Mathematics
 (2013)

---

## Abstract

It is shown that ifu 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.


---

## Body

## 1. Introduction

In this paper, we consider the following two-dimensional nonlinear wave equation:(1)ut+auux+buxxx+cuyyx+dux=0,
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.Fora=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 thatb>c and L>0, if u∈C([0,1];H4(R2))∩C1([0,1];L2(R2)) is a solution of (1) such that
(2)suppu(0),suppu(1)⊆[-L,L]×[-L,L];
then, u≡0.


## 2. Preliminary Estimates

Lemma 2 (see [13]).
Assume thats>0 and κ>0. (i) If f∈Hs(R2)∩L2(e2κxdxdy), then
(3)∥Jθs(e(1-θ)κxf)∥L2≤C∥Jsf∥L2θ∥eκxf∥L21-θ,
(ii) if f∈Hs(R2)∩L2(e2(κx+κy)dxdy), then
(4)∥Jθs(e(1-θ)(κx+κy)f)∥L2≤C∥Jsf∥L2θ∥eκx+κyf∥L21-θ,
where θ∈[0,1], C=C(s,κ), and [Jsf]∧(ξ):=(1+|ξ|2)s/2[f]∧(ξ).Lemma 3.
Assume thatκ>0, if u∈C([0,1];H4(R2))∩C1([0,1];L2(R2)) is a solution of (1) such that u(0)∈L2(e2κxdxdy); then, u(t) is bounded in H3(e2κxdxdy).Proof.
Assume thatφ(x)∈C∞(R) is a decreasing function with φ(x)=1 if x<1 and φ(x)=0 if x>10. Let θn(x)=∫0x‍φ(t/n)dt for n∈Z+ and ϕn(x)=e2kθn(x). It is easy to check that ϕn(x)≤ϕn+1(x) and
(5)|ϕn(j)(x)|≤Cj,kϕn(x),∀j∈Z+,∀x∈R.
Multiplying (1) by uϕn and integrating by parts in Rxy2, we obtain
(6)12ddt∫‍u2ϕn=a3∫‍u3ϕ′-3b2∫‍(ux)2ϕn′+b2∫‍u2ϕn′′′-c2∫‍(uy)2ϕ′+d2∫‍u2ϕn′≤a3∥u∥L∞(R2)C1,κ∫‍u2ϕn+b2C3,κ∫‍u2ϕn+d2C1,κ∫‍u2ϕn≤(a3C∥u∥C([0,1];H2)+b2C3,k+d2C1,k)∫‍u2ϕn=Ck,u∫‍u2ϕn.
Applying Gronwall Lemma and the Monotone Convergence Theorem, we have
(7)∫‍u(t)2e2κxdxdy≤C∫‍u(0)2e2κxdxdy,∀t∈[0,1].
This proves that u(t) is bounded in L2(e2κxdxdy).
Applying Lemma2 with s=4 and θ=3/4, we have that u(t) is bounded in H3(e2κxdxdy). Here, we used the fact that u∈C([0,1];H4). This completes the proof of the lemma.Lemma 4.
Assume thatκ>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(e2κxe2κ|y|dxdy), then u(t) is bounded in H3(e2κxe2κ|y|dxdy); (ii) if  u(1)∈L2(e-2kxe2k|y|dxdy), then u(t) is bounded in H3(e-2κxe2κ|y|dxdy).Proof.
Lettingv(t)=eκxu(t) and u a solution to  (1), we have
(8)ekxu′=κ(au+bκ2+d)v-(au+3bκ2+d)vx+3bκvxx-bvxxx+cκvyy-cvyyx.
Multiplying (8) by vϕn(y) and integrating by parts in Rxy2, we obtain
(9)∫‍eκxu′vϕn=aκ∫‍uv2ϕn+a2∫‍v2uxϕn+(bκ3+dκ)∫‍v2ϕn-3bκ∫‍(vx)2ϕn-cκ∫‍(vy)2ϕn+c2κ∫‍v2ϕn′′-c∫‍vxvyϕn′.
Note that |ϕn′(y)|≤2κϕn(y) and
(10)∫‍eκxu′vϕn=12ddt∫‍v2ϕn,a.et∈[0,1].
It follows from (9) that
(11)12ddt∫‍v2ϕn≤aκC∥u∥C([0,1];H2(R2))∫‍v2ϕn+aC∥ux∥C([0,1];H2(R2))∫‍v2ϕn+(bκ3+dκ)∫‍v2ϕn-cκ∫‍[(vx)2+2|vx||vy|+(vy)2]ϕn+c2κC2,κ∫‍v2ϕn=Cκ,u∫‍v2ϕn-cκ∫‍(|vx|+|vy|)2ϕn≤Cκ,u∫‍v2ϕn,a.et∈[0,1].
Since u∈C([0,1];H4) and u(t) is bounded in L2(e2κxe2κydxdy), applying Lemma 2 with s=4 and θ=3/4, we have that u(t) is also bounded in H3(e2κxe2κydxdy).
Similarly, we can prove thatu(t) is bounded in H3(e2κxe-2κydxdy). Let u~(x,y,t)=u(x,-y,t); then, u~ is a solution to (1) and satisfies u~(0)∈L2(e2κxe2κ|y|dxdy), and therefore u~ is bounded in H3(e2κxe2κydxdy). This proves (i).
Now, we prove (ii). Letu-(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 foru(0) and u(1) given in (i) and (ii), respectively, are satisfied, then u(t) is bounded in H3(e2κ|x|e2κ|y|dxdy).Lemma 6 (see [13]).
Letκ>0, v∈C1([0,1];L2(R2)) is a function such that v(t) is bounded in L2(e2κ|x|e2κ|y|dxdy), and v′∈L1([0,1];L2(e2κ|x|e2κ|y|dxdy)). Then, for all λ∈R and all ξ=(ξ1,ξ2)∈R2, the functions t→[eλxv(t)]∧(ξ) and t→[eλyv(t)]∧(ξ) are absolutely continuous in [0,1] with derivatives [eλxv′(t)]∧(ξ) and [eλyv′(t)]∧(ξ) a.e. t∈[0,1], respectively.Lemma 7.
Assume thatκ>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(e2κ|x|e2κ|y|dxdy) and v′(t)∈L1([0,1];L2(e2κ|x|e2κ|y|dxdy)). Then,
(12)∥eλxv∥L2(R2×[0,1])≤∥eλxv(0)∥L2(R2)+∥eλxv(1)∥L2(R2)+∥eλx(vt+bvxxx+cvyyx+dvx)∥L2(R2×[0,1]),(13)∥eλyv∥L2(R2×[0,1])≤∥eλyv(0)∥L2(R2)+∥eλyv(1)∥L2(R2)+∥eλy(vt+bvxxx+cvyyx+dvx)∥L2(R2×[0,1]).Proof.
Letp(t)=eλxv(t) and q(t)=eλx(vt+bvxxx+cvyyx+dvx); then,
(14)q(t)=eλxv′(t)-(bλ3+dλ)p(t)+(3bλ2+d)px(t)-3bλpxx(t)+bpxxx(t)-cλpyy(t)+cpyyx(t).
Taking the spatial Fourier transform in (14) and applying Lemma 6, we have
(15)ddt[p(t)]∧(ξ)+[-im(ξ)-n(ξ)][p(t)]∧(ξ)=[q(t)]∧(ξ),a.et∈[0,1],
where
(16)m(ξ)=-(3bλ2+d)ξ1+bξ13+cξ1ξ22,n(ξ)=(bλ3+dλ)-3bλξ12-cλξ22.
According to (15), when n(ξ)≤0, we have
(17)[p(t)]∧(ξ)=eim(ξ)ten(ξ)t[p(0)]∧(ξ)+∫0t‍eim(ξ)(t-τ)en(ξ)(t-τ)[q(τ)]∧(ξ)dτ,
and when n(ξ)>0, we choose to write
(18)[p(t)]∧(ξ)=e-im(ξ)(1-t)e-n(ξ)(1-t)[p(1)]∧(ξ)-∫t1‍e-im(ξ)(τ-t)e-n(ξ)(τ-t)[q(τ)]∧(ξ)dτ.
Therefore, we have
(19)|[p(t)]∧(ξ)|≤|[p(0)]∧(ξ)|+|[p(1)]∧(ξ)|+∫01‍|[q(τ)]∧(ξ)|dτ,t∈[0,1].
Appling Plancherel formula, we have inequality (12).
Similarly, lettingp(t)=eλyw(t), we can also have (13). This completes the proof of the lemma.Lemma 8 (see [12, 13]).
Assume thatL>0, and b>c, if u∈C([0,1];H4(R2)) is a solution to (1) such that
(20)suppu(t)∈[-L,L]×[-L,L],∀t∈[0,1];
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 thatϕ(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
(21)∥eλxw∥≤∥eλx(gt+bgxxx+cgyyx+dgx)∥=∥eλx[ϕut+bϕuxxx+cϕuyyx+(bϕ′′′+dϕ′)u=eλx+(3bϕ′′+dϕ)ux+3bϕ′uxx+cϕ′uyy]∥≤∥aeλxϕuux∥+∥eλxGϕ,u∥,
where ∥·∥:=∥·∥L2(R2×[0,1]) and
(22)Gϕ,u=(bϕ′′′+dϕ′)u+3bϕ′′ux+3bϕ′uxx+cϕ′uyy.
Note that the derivatives of  ϕ(x) are supported in the interval [A,A+1]; then,
(23)∥eλxGϕ,u∥=Ceλ(A+1),
where C is dependent on ∥u∥C([0,1];H2) and max{b,c,d}.Combining (21) with (23), we obtain
(24)∥eλxϕu∥≤a∥eλxϕu∥∥ux∥L∞([A,+∞]×R×[0,1])+Ceλ(A+1).
Applying Lemma 4 with κ=1, we have that
(25)|ux(t)(x,y)|≤C1e-x,
then
(26)∥eλxϕu∥≤aC1e-A∥eλxϕu∥+Ceλ(A+1).
Since ∥eλxϕu∥<+∞, taking A>max{L,1} such that C1e-A<1/2a, we have
(27)∥eλxϕu∥≤Ceλ(A+1).
Note that ϕ(x)=1 for x≥2A; we have
(28)e2λA(∫01‍∫-∞∞‍∫2A∞‍|u(t)(x,y)|2)1/2≤∥eλxϕu∥≤Ceλ(A+1).
Letting λ→+∞, we obtain
(29)(∫01‍∫-∞∞‍∫2A∞‍|u(t)(x,y)|2)1/2=0,
and this proves that u≡0 in [2A,∞)×R×[0,1].Next, we will prove thatu≡0 in R×[2A,∞)×[0,1]. Let h=ϕ(y)u; then,
(30)∥eλyh∥≤∥eλy(ht+bhxxx+chyyx+dhx)∥=∥eλy(ϕut+bϕuxxx+cϕuyyx+dϕux+2cϕ′uxy+cϕ′′ux)∥≤∥aeλyϕuux∥+∥eλyHϕ,u∥,
where Hϕ,u:=2cϕ′uxy+cϕ′′ux.In fact, Letu~(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
(31)(-∞,-2A]×R×[0,1]∪R×(-∞,-2A]×[0,1].
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.



---

*Source: 578094-2013-03-04.xml*