# On the Stability of Wave Equation

Authors: Soon-Mo Jung
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We prove the generalized Hyers-Ulam stability of the wave equation,Δu=(1/c2)utt, in a class of twice continuously differentiable functions under some conditions.


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## Body

## 1. Introduction

In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms.LetG1 be a group and let G2 be a metric group with the metric d(·,·). Given ε>0, does there exist a δ>0 such that if a function h:G1→G2 satisfies the inequality d(h(xy),h(x)h(y))<δ for all x,y∈G1, then there exists a homomorphism H:G1→G2 with d(h(x),H(x))<ε for all x∈G1?The case of approximately additive functions was solved by Hyers [2] under the assumption that G1 and G2 are Banach spaces. Indeed, he proved that each solution of the inequality ∥f(x+y)-f(x)-f(y)∥≤ε, for all x and y, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, f(x+y)=f(x)+f(y), is said to have the Hyers-Ulam stability.Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows:
(1)∥f(x+y)-f(x)-f(y)∥≤ε(∥x∥p+∥y∥p)
and proved the Hyers’ theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4–10].The terminologies, the generalized Hyers-Ulam stability and the Hyers-Ulam stability, can also be applied to the case of other functional equations, of differential equations, and of various integral equations.Given a real numberc>0, the partial differential equation
(2)Δu(x,t)-1c2utt(x,t)=0
is called the wave equation, where utt(x,t) and Δu(x,t) denote the second time derivative and the Laplacian of u(x,t), respectively.For an integern≥2, assume that U and T are open (connected) subsets of ℝn and ℝ, respectively. Let φ:U×T→[0,∞) be a function. If, for each twice continuously differentiable function u:U×T→ℝ satisfying
(3)|Δu(x,t)-1c2utt(x,t)|≤φ(x,t)(x∈U,t∈T),
there exist a solution u0:U×T→ℝ of the wave equation (2) and a function Φ:U×T→[0,∞) such that
(4)|u(x,t)-u0(x,t)|≤Φ(x,t)(x∈U,t∈T),
where Φ(x,t) is independent of u(x,t) and u0(x,t), then we say that the wave equation (2) has the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability).In this paper, using ideas from [11, 12], we prove the generalized Hyers-Ulam stability of the wave equation (2).


## 2. Main Results

For a given integern≥2, xi denotes the ith coordinate of any point x in ℝn; that is x=(x1,…,xi,…,xn), and |x| denotes the Euclidean distance between x and the origin; that is,
(5)|x|=x12+x22+⋯+xn2.Given a real numberc>0, assume that real numbers a and t2 satisfy a>c and 0<t2<∞, and define
(6)T:=(0,t2),U:={x∈ℝn:|x|>at2},R:=(a,∞).
We remark that (x,t)∈U×T if and only if |x|/t∈R. Using an idea from [11], we define a class W of all twice continuously differentiable functions u:U×T→ℝ with the properties (i)
u(x,t)=tv(|x|/t) for all x∈U and t∈T and for some v:R→ℝ;(ii)
lim|x|→at2limt→t2u(x,t)=0.If we define(7)(u1+u2)(x,t)=u1(x,t)+u2(x,t),(λu1)(x,t)=λu1(x,t),
for all u1,u2∈W and λ∈ℝ, then W is a vector space over real numbers. That is, W is a large class such that it is a vector space.Theorem 1.
Let a functionφ:U×T→[0,∞) be given such that there exists a positive real number s with
(8)s:=supx∈U,t∈Ttφ(x,t).
If a u∈W satisfies the inequality
(9)|Δu(x,t)-1c2utt(x,t)|≤φ(x,t),
for all x∈U and t∈T, then there exists a solution u0:U×T→ℝ of the wave equation (2) which belongs to W and satisfies
(10)|u(x,t)-u0(x,t)|≤t∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2∫z∞‍c2sq2-c2dqdz,
for all x∈U and t∈T.Proof.
Letv:ℝ→ℝ be a function which satisfies
(11)u(x,t)=tv(|x|t),
for all x∈U and t∈T. For any i∈{1,2,…,n}, we differentiate u(x,t) with respect to xi to get
(12)uxi(x,t)=xi|x|v′(|x|t).
Similarly, we obtain the second partial derivative of u(x,t) with respect to xi as follows:
(13)uxixi(x,t)=(1|x|-xi2|x|3)v′(|x|t)+1txi2|x|2v′′(|x|t).
Hence, we have
(14)Δu(x,t)=∑i=1nuxixi(x,t)=n-1tt|x|v′(|x|t)+1tv′′(|x|t).
By a similar way, we further get the second derivative of u(x,t) with respect to t as follows:
(15)utt(x,t)=1t|x|2t2v′′(|x|t).
Therefore, it follows from (14) and (15) that
(16)Δu(x,t)-1c2utt(x,t)=n-1tt|x|v′(|x|t)+1tv′′(|x|t)-1c2t|x|2t2v′′(|x|t)=n-1t1rv′(r)+(1t-1c2tr2)v′′(r)=1t(1-r2c2)(v′′(r)+n-1rc2c2-r2v′(r)),
for any x∈U, t∈T, and r∶=|x|/t∈R, and it follows from (8) and (9) that
(17)|v′′(r)+n-1rc2c2-r2v′(r)|≤c2r2-c2tφ(x,t)≤c2sr2-c2
or
(18)|w′(r)+n-1rc2c2-r2w(r)|≤c2sr2-c2,
for all r∈R, where we set w(r)∶=v′(r).
Set(19)g(r)∶=n-1rc2c2-r2,h(r):=0,ϕ(r)∶=c2sr2-c2,
for each r∈R. Then we have
(20)∫arg(p)dp=ln(r2a2·a2-c2r2-c2)(n-1)/2,∫a∞‍ϕ(r)exp{ℜ(∫ar‍g(p)dp)}dr=∫a∞(r2a2·a2-c2r2-c2)(n-1)/2c2sr2-c2dr<∫a∞‍c2sr2-c2dr<∞.
According to (18) and [13, Theorem 1], there exists a unique real number α such that
(21)|w(r)-α(a2r2·r2-c2a2-c2)(n-1)/2|≤(a2r2·r2-c2a2-c2)(n-1)/2×∫r∞‍c2sq2-c2(q2a2·a2-c2q2-c2)(n-1)/2dq≤(a2r2·r2-c2a2-c2)(n-1)/2∫r∞‍c2sq2-c2dq
or
(22)(a2r2·r2-c2a2-c2)(n-1)/2(α-∫r∞‍c2sq2-c2dq)≤v′(r)≤(a2r2·r2-c2a2-c2)(n-1)/2(α+∫r∞‍c2sq2-c2dq),
for all r∈R.
Hence, it follows from the last inequalities that(23)∫ar‍(a2z2·z2-c2a2-c2)(n-1)/2(α-∫z∞‍c2sq2-c2dq)dz≤v(r)-limz→a+v(z)≤∫ar‍(a2z2·z2-c2a2-c2)(n-1)/2(α+∫z∞‍c2sq2-c2dq)dz,
for any r∈R.
Due to(ii), it holds that limz→a+v(z)=0. Replacing r with |x|/t in the last inequalities, we get
(24)|u(x,t)-αt∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2dz|≤t∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2∫z∞‍c2sq2-c2dqdz,
for all x∈U and t∈T.
If we define a functionu0:U×T→ℝ by
(25)u0(x,t):=αt∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2dz,
then we have
(26)∂∂xiu0(x,t)=αxi|x|(a2r2·r2-c2a2-c2)(n-1)/2,∂2∂xi2u0(x,t)=α(1|x|-xi2|x|3)(a2r2·r2-c2a2-c2)(n-1)/2+(n-1)αa2c2xi2(a2-c2)tr3|x|2(a2r2·r2-c2a2-c2)(n-3)/2,Δu0(x,t)=(n-1)αa2(a2-c2)|x|(a2r2·r2-c2a2-c2)(n-3)/2,∂∂tu0(x,t)=α∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2dz-α|x|t(a2r2·r2-c2a2-c2)(n-1)/2,∂2∂t2u0(x,t)=(n-1)αa2c2(a2-c2)|x|(a2r2·r2-c2a2-c2)(n-3)/2,
for all x∈U and t∈T, which implies that u0(x,t) is a solution of the wave equation (2).
It is now to show thatu0∈W. Let F:R→ℝ be a function with the property
(27)F(r)∶=∫‍(a2r2·r2-c2a2-c2)(n-1)/2dr.
Then we have
(28)u0(x,t)=αt(F(|x|t)-F(a)),
which implies that u0(x,t) can be expressed as tv(|x|/t), where v(r)=αF(r)-αF(a). Moreover, we get
(29)lim|x|→at2limt→t2u0(x,t)=lim|x|/t→a+αt∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2dz=0,
which verifies that u0∈W. Finally, by (24), the inequality (10) holds true.Assume now thatb and t1 are given real numbers satisfying 0<b<c and 0<t1<∞. We then set
(30)T′:=(t1,∞),U′:={x∈ℝn:0<|x|<bt1},R′:=(0,b)
and define a class W′ of all twice continuously differentiable functions u:U′×T′→ℝ with the properties (iii)
u(x,t)=tv(|x|/t) for all x∈U′ and t∈T′ and for some v:R′→ℝ;(iv)
lim|x|→bt1limt→t1u(x,t)=0.It might be remarked that(x,t)∈U′×T′ if and only if |x|/t∈R′. If we define
(31)(u1+u2)(x,t)=u1(x,t)+u2(x,t),(λu1)(x,t)=λu1(x,t),
for all u1,u2∈W′and λ∈ℝ, then W′ is a vector space over real numbers.Theorem 2.
Let a functionφ:U′×T′→[0,∞) be given such that there exists a positive real number s′ with
(32)s′:=supx∈U′,t∈T′tφ(x,t).
If a u∈W′ satisfies the inequality
(33)|Δu(x,t)-1c2utt(x,t)|≤φ(x,t),
for all x∈U′ and t∈T′, then there exists a solution u0:U′×T′→ℝ of the wave equation (2) which belongs to W′ and satisfies
(34)|u(x,t)-u0(x,t)|≤t∫|x|/tb‍(b2z2·c2-z2c2-b2)(n-1)/2∫0z‍c2s′c2-q2dqdz
for all x∈U′ and t∈T′.Proof.
Ifv:ℝ→ℝ is given by (11), then we can simply follow the lines in the first part of the proof of Theorem 1 to obtain
(35)|w′(r)+n-1rc2c2-r2w(r)|≤c2s′c2-r2,
for all r∈R′, where w(r)∶=v′(r).
Set(36)g(r):=n-1rc2c2-r2,h(r):=0,ϕ(r):=c2s′c2-r2,
for any r∈R′. Then we get
(37)∫rb‍g(p)dp=ln(b2r2·c2-r2c2-b2)(n-1)/2,∫0b‍ϕ(r)exp{ℜ(∫br‍g(p)dp)}dr=∫0b‍(r2b2·c2-b2c2-r2)(n-1)/2c2s′c2-r2dr<∫0b‍c2s′c2-r2dr<∞.
According to (35) and [13, Corollary 2], there exists a unique real number α such that
(38)|w(r)-α(b2r2·c2-r2c2-b2)(n-1)/2|≤(b2r2·c2-r2c2-b2)(n-1)/2∫0r‍c2s′c2-q2dq
or
(39)(b2r2·c2-r2c2-b2)(n-1)/2(α-∫0r‍c2s′c2-q2dq)≤v′(r)≤(b2r2·c2-r2c2-b2)(n-1)/2(α+∫0r‍c2s′c2-q2dq),
for all r∈R′.
From the last inequalities, it follows that(40)∫rb‍(b2z2·c2-z2c2-b2)(n-1)/2(α-∫0z‍c2s′c2-q2dq)dz≤limz→b-v(z)-v(r)≤∫rb‍(b2z2·c2-z2c2-b2)(n-1)/2(α+∫0z‍c2s′c2-q2dq)dz,
for each r∈R′.
On account of(iv), we have limz→b-v(z)=0. Replacing r with |x|/t in the last inequalities, we obtain
(41)|u(x,t)-αt∫b|x|/t‍(b2z2·c2-z2c2-b2)(n-1)/2dz|≤t∫|x|/tb‍(b2z2·c2-z2c2-b2)(n-1)/2∫0z‍c2s′c2-q2dqdz,
for allx∈U′and t∈T′.
Let us define a functionu0:U′×T′→ℝ by
(42)u0(x,t):=αt∫b|x|/t‍(b2z2·c2-z2c2-b2)(n-1)/2dz.
Then, a similar argument to the last part of the proof of Theorem 1 shows that u0(x,t) is a solution of the wave equation (2) and it belongs to W′. Finally, the validity of (34) immediately follows from (41).


## 3. Remarks

Remark 1.
The inequality (10) in Theorem 1 can be rewritten as
(43)|u(x,t)-u0(x,t)|≤t∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2∫z∞‍c2sq2-c2dqdz≤t∫a|x|/t‍(a2z2·z2-c2a2-c2)(n-1)/2cs2(lna+ca-c)dz≤cst2(a2a2-c2)(n-1)/2(lna+ca-c)×∫a|x|/t‍(1-c2z2)(n-1)/2dz,
for all x∈U and t∈T. If we further substitute sinθ for c/z in the previous inequality, then we obtain
(44)|u(x,t)-u0(x,t)|≤cst2(a2a2-c2)(n-1)/2(lna+ca-c)×∫sin-1(c/a)sin-1(ct/|x|)‍cosn-1θ(-ccosθsin2θ)dθ=-c2st2(a2a2-c2)(n-1)/2(lna+ca-c)×[-cosn-1θsinθ]sin-1(c/a)sin-1(ct/|x|)+(n-1)c2st2×(a2a2-c2)(n-1)/2(lna+ca-c)×∫sin-1(c/a)sin-1(ct/|x|)‍cosn-2θdθ=-c2st2(a2a2-c2)(n-1)/2(lna+ca-c)×[ac(1-(ca)2)(n-1)/2-|x|ct(1-(ct|x|)2)(n-1)/2]+(n-1)c2st2(a2a2-c2)(n-1)/2×(lna+ca-c)∫sin-1(c/a)sin-1(ct/|x|)‍cosn-2θdθ,
for any x∈U and t∈T.
For the case ofn=3, the inequality (10) can be rewritten as
(45)|u(x,t)-u0(x,t)|≤c2st2a2a2-c2(lna+ca-c)×(ct|x|+|x|ct-ca-ac),
for all x∈U and t∈T.Remark 2.
As in Remark1, the inequality (34) in Theorem 2 can be rewritten as
(46)|u(x,t)-u0(x,t)|≤cs′t2(b2c2-b2)(n-1)/2×(lnc+bc-b)∫|x|/tb‍(c2z2-1)(n-1)/2dz,
for all x∈U′ and t∈T′. If we substitute ccosθ for z in the previous inequality, then we get
(47)|u(x,t)-u0(x,t)|≤-cs′t2(b2c2-b2)(n-1)/2(lnc+bc-b)×[1t(c2t2-|x|2)(n-1)/2|x|n-2-(c2-b2)(n-1)/2bn-2]-(n-1)c2s′t2(b2c2-b2)(n-1)/2×(lnc+bc-b)∫cos-1(|x|/ct)cos-1(b/c)sinn-2θcosn-1θdθ,
for any x∈U′ and t∈T′.
For the case ofn=3, the inequality (34) can be rewritten as
(48)|u(x,t)-u0(x,t)|≤c2s′t2b2c2-b2(lnc+bc-b)(ct|x|+|x|ct-bc-cb),
for all x∈U′ and t∈T′.Remark 3.
It is an open problem whether the wave equation (2) has the generalized Hyers-Ulam stability for the case of either T=(0,t2) and U={x∈ℝn:0<|x|<at2} or T=(t1,∞) and U={x∈ℝn:|x|>bt1} or T=(0,∞) and U={x∈ℝn:|x|>0}.



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