On the Stability of Wave Equation

Abstract

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

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. Let 𝐺₁ 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: 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝+𝑦^𝑝 (1) 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 number 𝑐>0, the partial differential equation ()−1Δ𝑢𝑥,𝑡𝑐²𝑢_𝑡𝑡(𝑥,𝑡)=0 (2) is called the wave equation, where 𝑢_𝑡𝑡(𝑥,𝑡) and Δ𝑢(𝑥,𝑡) denote the second time derivative and the Laplacian of 𝑢(𝑥,𝑡), respectively.

For an integer 𝑛≥2, assume that 𝑈 and 𝑇 are open (connected) subsets of ℝ^𝑛 and ℝ, respectively. Let 𝜑∶𝑈×𝑇→[0,∞) be a function. If, for each twice continuously differentiable function 𝑢∶𝑈×𝑇→ℝ satisfying ()−1Δ𝑢𝑥,𝑡𝑐²𝑢_𝑡𝑡(𝑥,𝑡)≤𝜑(𝑥,𝑡)(𝑥∈𝑈,𝑡∈𝑇), (3) there exist a solution 𝑢₀∶𝑈×𝑇→ℝ of the wave equation (2) and a function Φ∶𝑈×𝑇→[0,∞) such that 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤Φ(𝑥,𝑡)(𝑥∈𝑈,𝑡∈𝑇), (4) 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 integer 𝑛≥2, 𝑥_𝑖 denotes the 𝑖th coordinate of any point 𝑥 in ℝ^𝑛; that is 𝑥=(𝑥₁,…,𝑥_𝑖,…,𝑥_𝑛), and |𝑥| denotes the Euclidean distance between 𝑥 and the origin; that is, |𝑥|=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛. (5)

Given a real number 𝑐>0, assume that real numbers 𝑎 and 𝑡₂ satisfy 𝑎>𝑐 and 0<𝑡₂<∞, and define 𝑇∶=0,𝑡₂,𝑈∶=𝑥∈ℝ^𝑛∶|𝑥|>𝑎𝑡₂,𝑅∶=(𝑎,∞). (6) 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|→𝑎𝑡2}lim_𝑡→𝑡2𝑢(𝑥,𝑡)=0.

If we define 𝑢₁+𝑢₂(𝑥,𝑡)=𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡),𝜆𝑢₁(𝑥,𝑡)=𝜆𝑢₁(𝑥,𝑡), (7) 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 function 𝜑∶𝑈×𝑇→[0,∞) be given such that there exists a positive real number 𝑠 with 𝑠∶=sup_{𝑥∈𝑈,𝑡∈𝑇}𝑡𝜑(𝑥,𝑡). (8) If a 𝑢∈𝑊 satisfies the inequality ()−1Δ𝑢𝑥,𝑡𝑐²𝑢_𝑡𝑡(𝑥,𝑡)≤𝜑(𝑥,𝑡), (9) 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 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤𝑡_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2∞}_𝑧𝑐²𝑠𝑞²−𝑐²𝑑𝑞𝑑𝑧, (10) for all 𝑥∈𝑈 and 𝑡∈𝑇.

Proof. Let 𝑣∶ℝ→ℝ be a function which satisfies 𝑢()|𝑥|𝑥,𝑡=𝑡𝑣𝑡, (11) for all 𝑥∈𝑈 and 𝑡∈𝑇. For any 𝑖∈{1,2,…,𝑛}, we differentiate 𝑢(𝑥,𝑡) with respect to 𝑥_𝑖 to get 𝑢_𝑥𝑖()=𝑥𝑥,𝑡^𝑖|𝑥|𝑣^′|𝑥|𝑡. (12) Similarly, we obtain the second partial derivative of 𝑢(𝑥,𝑡) with respect to 𝑥_𝑖 as follows: 𝑢_{𝑥𝑖𝑥𝑖}()=1𝑥,𝑡|𝑥|−𝑥²_𝑖|𝑥|³𝑣^′|𝑥|𝑡+1𝑡𝑥²_𝑖|𝑥|²𝑣^′′|𝑥|𝑡. (13) Hence, we have Δ𝑢(𝑥,𝑡)=^𝑛_𝑖=1𝑢_{𝑥𝑖𝑥𝑖}(𝑥,𝑡)=𝑛−1𝑡𝑡|𝑥|𝑣^′|𝑥|𝑡+1𝑡𝑣^′′|𝑥|𝑡. (14) By a similar way, we further get the second derivative of 𝑢(𝑥,𝑡) with respect to 𝑡 as follows: 𝑢_𝑡𝑡()=1𝑥,𝑡𝑡|𝑥|²𝑡²𝑣^′′|𝑥|𝑡. (15) Therefore, it follows from (14) and (15) that ()−1Δ𝑢𝑥,𝑡𝑐²𝑢_𝑡𝑡()=𝑥,𝑡𝑛−1𝑡𝑡|𝑥|𝑣^′|𝑥|𝑡+1𝑡𝑣^′′|𝑥|𝑡−1𝑐²𝑡|𝑥|²𝑡²𝑣^′′|𝑥|𝑡=𝑛−1𝑡1𝑟𝑣^′(𝑟)+1𝑡−1𝑐²𝑡𝑟²𝑣^′′(𝑟)=1𝑡𝑟1−²𝑐²𝑣^′′(𝑟)+𝑛−1𝑟𝑐²𝑐²−𝑟²𝑣^′(𝑟), (16) for any 𝑥∈𝑈, 𝑡∈𝑇, and 𝑟∶=|𝑥|/𝑡∈𝑅, and it follows from (8) and (9) that 𝑣^′′(𝑟)+𝑛−1𝑟𝑐²𝑐²−𝑟²𝑣^′(𝑟≤𝑐)²𝑟²−𝑐²()≤𝑐𝑡𝜑𝑥,𝑡²𝑠𝑟²−𝑐² (17) or 𝑤^′(𝑟)+𝑛−1𝑟𝑐²𝑐²−𝑟²𝑤(𝑟≤𝑐)²𝑠𝑟²−𝑐², (18) for all 𝑟∈𝑅, where we set 𝑤(𝑟)∶=𝑣^′(𝑟).
Set 𝑔(𝑟)∶=𝑛−1𝑟𝑐²𝑐²−𝑟²(𝑟)𝜙(𝑟)𝑐,ℎ∶=0,∶=²𝑠𝑟²−𝑐², (19) for each 𝑟∈𝑅. Then we have ^𝑟_𝑎𝑔𝑝𝑟𝑑𝑝=ln²𝑎²⋅𝑎²−𝑐²𝑟²−𝑐^{2(𝑛−1)/2}_∞𝑎,𝜙(𝑟)expℜ^𝑟_𝑎𝑔𝑝=𝑑𝑝𝑑𝑟_∞𝑎𝑟²𝑎²⋅𝑎²−𝑐²𝑟²−𝑐^{2(𝑛−1)/2}𝑐²𝑠𝑟²−𝑐²<𝑑𝑟_∞𝑎𝑐²𝑠𝑟²−𝑐²𝑑𝑟<∞. (20) According to (18) and [13, Theorem 1], there exists a unique real number 𝛼 such that 𝑤(𝑟)𝑎−𝛼²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}≤𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}×^∞_𝑟𝑐²𝑠𝑞²−𝑐²𝑞²𝑎²⋅𝑎²−𝑐²𝑞²−𝑐^{2(𝑛−1)/2}≤𝑎𝑑𝑞₂𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2∞}_𝑟𝑐²𝑠𝑞²−𝑐²𝑑𝑞 (21) or 𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝛼−^∞_𝑟𝑐²𝑠𝑞²−𝑐²𝑑𝑞≤𝑣^′(𝑟)≤𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝛼+^∞_𝑟𝑐²𝑠𝑞²−𝑐²𝑑𝑞, (22) for all 𝑟∈𝑅.
Hence, it follows from the last inequalities that ^𝑟_𝑎𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝛼−^∞_𝑧𝑐²𝑠𝑞²−𝑐²(𝑟)𝑑𝑞𝑑𝑧≤𝑣−lim_𝑧→𝑎+𝑣(𝑧)≤^𝑟_𝑎𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝛼+^∞_𝑧𝑐²𝑠𝑞²−𝑐²𝑑𝑞𝑑𝑧, (23) for any 𝑟∈𝑅.
Due to (ii), it holds that lim_𝑧→𝑎+𝑣(𝑧)=0. Replacing 𝑟 with |𝑥|/𝑡 in the last inequalities, we get 𝑢(𝑥,𝑡)−𝛼𝑡_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝑑𝑧≤𝑡_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2∞}_𝑧𝑐²𝑠𝑞²−𝑐²𝑑𝑞𝑑𝑧, (24) for all 𝑥∈𝑈 and 𝑡∈𝑇.
If we define a function 𝑢₀∶𝑈×𝑇→ℝ by 𝑢₀(𝑥,𝑡)∶=𝛼𝑡_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝑑𝑧, (25) then we have 𝜕𝜕𝑥_𝑖𝑢₀(𝑥,𝑡)=𝛼𝑥_𝑖|𝑥|𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2},𝜕²𝜕𝑥²_𝑖𝑢₀()1𝑥,𝑡=𝛼|𝑥|−𝑥²_𝑖|𝑥|³𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}+(𝑛−1)𝛼𝑎²𝑐²𝑥²_𝑖𝑎²−𝑐²𝑡𝑟³|𝑥|²𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−3)/2},Δ𝑢₀()=()𝑥,𝑡𝑛−1𝛼𝑎²𝑎²−𝑐²|𝑥|𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−3)/2},𝜕𝑢𝜕𝑡⁰(𝑥,𝑡)=𝛼_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}−𝛼|𝑥|𝑑𝑧𝑡𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2},𝜕²𝜕𝑡²𝑢₀()=()𝑥,𝑡𝑛−1𝛼𝑎²𝑐²𝑎²−𝑐²|𝑥|𝑎²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−3)/2}, (26) 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 that 𝑢₀∈𝑊. Let 𝐹∶𝑅→ℝ be a function with the property 𝐹(𝑟)𝑎∶=²𝑟²⋅𝑟²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝑑𝑟. (27) Then we have 𝑢₀()𝐹|𝑥|𝑥,𝑡=𝛼𝑡𝑡−𝐹(𝑎), (28) which implies that 𝑢₀(𝑥,𝑡) can be expressed as 𝑡𝑣(|𝑥|/𝑡), where 𝑣(𝑟)=𝛼𝐹(𝑟)−𝛼𝐹(𝑎). Moreover, we get lim_{|𝑥|→𝑎𝑡2}lim_𝑡→𝑡2𝑢₀(𝑥,𝑡)=lim_{|𝑥|/𝑡→𝑎+}𝛼𝑡_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝑑𝑧=0, (29) which verifies that 𝑢₀∈𝑊. Finally, by (24), the inequality (10) holds true.

Assume now that 𝑏 and 𝑡₁ are given real numbers satisfying 0<𝑏<𝑐 and 0<𝑡₁<∞. We then set 𝑇^′∶=𝑡₁,∞,𝑈^′∶=𝑥∈ℝ^𝑛∶0<|𝑥|<𝑏𝑡₁,𝑅^′∶=(0,𝑏) (30) 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|→𝑏𝑡1}lim_𝑡→𝑡1𝑢(𝑥,𝑡)=0.

It might be remarked that (𝑥,𝑡)∈𝑈^′×𝑇^′ if and only if |𝑥|/𝑡∈𝑅^′. If we define 𝑢₁+𝑢₂(𝑥,𝑡)=𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡),𝜆𝑢₁(𝑥,𝑡)=𝜆𝑢₁(𝑥,𝑡), (31) for all 𝑢₁,𝑢₂∈𝑊^′and 𝜆∈ℝ, then 𝑊^′ is a vector space over real numbers.

Theorem 2. Let a function 𝜑∶𝑈^′×𝑇^′→[0,∞) be given such that there exists a positive real number 𝑠^′ with 𝑠^′∶=sup_{𝑥∈𝑈′,𝑡∈𝑇′}𝑡𝜑(𝑥,𝑡). (32) If a 𝑢∈𝑊^′ satisfies the inequality ()−1Δ𝑢𝑥,𝑡𝑐²𝑢_𝑡𝑡(𝑥,𝑡)≤𝜑(𝑥,𝑡), (33) 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 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤𝑡^𝑏_|𝑥|/𝑡𝑏²𝑧²⋅𝑐²−𝑧²𝑐²−𝑏^{2(𝑛−1)/2𝑧0}𝑐²𝑠^′𝑐²−𝑞²𝑑𝑞𝑑𝑧 (34) for all 𝑥∈𝑈^′ and 𝑡∈𝑇^′.

Proof. If 𝑣∶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 𝑤^′(𝑟)+𝑛−1𝑟𝑐²𝑐²−𝑟²𝑤(𝑟≤𝑐)²𝑠^′𝑐²−𝑟², (35) for all 𝑟∈𝑅^′, where 𝑤(𝑟)∶=𝑣^′(𝑟).
Set 𝑔(𝑟)∶=𝑛−1𝑟𝑐²𝑐²−𝑟²(𝑟)𝜙(𝑟)𝑐,ℎ∶=0,∶=²𝑠^′𝑐²−𝑟², (36) for any 𝑟∈𝑅^′. Then we get ^𝑏_𝑟𝑔𝑝𝑏𝑑𝑝=ln²𝑟²⋅𝑐²−𝑟²𝑐²−𝑏^{2(𝑛−1)/2}_𝑏0,𝜙(𝑟)expℜ^𝑟_𝑏𝑔𝑝=𝑑𝑝𝑑𝑟_𝑏0𝑟²𝑏²⋅𝑐²−𝑏²𝑐²−𝑟^{2(𝑛−1)/2}𝑐²𝑠^′𝑐²−𝑟²<𝑑𝑟_𝑏0𝑐²𝑠^′𝑐²−𝑟²𝑑𝑟<∞. (37) According to (35) and [13, Corollary 2], there exists a unique real number 𝛼 such that 𝑤(𝑟)𝑏−𝛼²𝑟²⋅𝑐²−𝑟²𝑐²−𝑏^{2(𝑛−1)/2}≤𝑏²𝑟²⋅𝑐²−𝑟²𝑐²−𝑏^{2(𝑛−1)/2𝑟0}𝑐²𝑠^′𝑐²−𝑞²𝑑𝑞 (38) or 𝑏²𝑟²⋅𝑐²−𝑟²𝑐²−𝑏^{2(𝑛−1)/2}𝛼−^𝑟₀𝑐²𝑠^′𝑐²−𝑞²𝑑𝑞≤𝑣^′(𝑟)≤𝑏²𝑟²⋅𝑐²−𝑟²𝑐²−𝑏^{2(𝑛−1)/2}𝛼+^𝑟₀𝑐²𝑠^′𝑐²−𝑞²𝑑𝑞, (39) for all 𝑟∈𝑅^′.
From the last inequalities, it follows that ^𝑏_𝑟𝑏²𝑧²⋅𝑐²−𝑧²𝑐²−𝑏^{2(𝑛−1)/2}𝛼−^𝑧₀𝑐²𝑠^′𝑐²−𝑞²𝑑𝑞𝑑𝑧≤lim_{𝑧→𝑏−}𝑣(𝑧)(𝑟)≤−𝑣_𝑏𝑟𝑏²𝑧²⋅𝑐²−𝑧²𝑐²−𝑏^{2(𝑛−1)/2}𝛼+^𝑧₀𝑐²𝑠^′𝑐²−𝑞²𝑑𝑞𝑑𝑧, (40) for each 𝑟∈𝑅^′.
On account of (iv), we have lim_𝑧→𝑏⁻𝑣(𝑧)=0. Replacing 𝑟 with |𝑥|/𝑡 in the last inequalities, we obtain 𝑢(𝑥,𝑡)−𝛼𝑡_𝑏^|𝑥|/𝑡𝑏²𝑧²⋅𝑐²−𝑧²𝑐²−𝑏^{2(𝑛−1)/2}𝑑𝑧≤𝑡^𝑏_|𝑥|/𝑡𝑏²𝑧²⋅𝑐²−𝑧²𝑐²−𝑏^{2(𝑛−1)/2𝑧0}𝑐²𝑠^′𝑐²−𝑞²𝑑𝑞𝑑𝑧, (41) for all𝑥∈𝑈^′and 𝑡∈𝑇^′.
Let us define a function 𝑢₀∶𝑈^′×𝑇^′→ℝ by 𝑢₀(𝑥,𝑡)∶=𝛼𝑡_𝑏^|𝑥|/𝑡𝑏²𝑧²⋅𝑐²−𝑧²𝑐²−𝑏^{2(𝑛−1)/2}𝑑𝑧. (42) 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 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤𝑡_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2∞}_𝑧𝑐²𝑠𝑞²−𝑐²𝑑𝑞𝑑𝑧≤𝑡_𝑎^|𝑥|/𝑡𝑎²𝑧²⋅𝑧²−𝑐²𝑎²−𝑐^{2(𝑛−1)/2}𝑐𝑠2ln𝑎+𝑐≤𝑎−𝑐𝑑𝑧𝑐𝑠𝑡2𝑎²𝑎²−𝑐^{2(𝑛−1)/2}ln𝑎+𝑐×𝑎−𝑐_{𝑎|𝑥|/𝑡}𝑐1−²𝑧^{2(𝑛−1)/2}𝑑𝑧, (43) for all 𝑥∈𝑈 and 𝑡∈𝑇. If we further substitute sin𝜃 for 𝑐/𝑧 in the previous inequality, then we obtain 𝑢(𝑥,𝑡)−𝑢₀(≤𝑥,𝑡)𝑐𝑠𝑡2𝑎²𝑎²−𝑐^{2(𝑛−1)/2}ln𝑎+𝑐×𝑎−𝑐_{sin−1(𝑐𝑡/|𝑥|)sin−1()𝑐/𝑎}cos^𝑛−1𝜃−𝑐cos𝜃sin²𝜃𝑐𝑑𝜃=−²𝑠𝑡2𝑎²𝑎²−𝑐^{2(𝑛−1)/2}ln𝑎+𝑐×−𝑎−𝑐cos^𝑛−1𝜃sin𝜃^{sin−1(𝑐𝑡/|𝑥|)}_{sin−1()𝑐/𝑎}+(𝑛−1)𝑐²𝑠𝑡2×𝑎²𝑎²−𝑐^{2(𝑛−1)/2}ln𝑎+𝑐×𝑎−𝑐_{sin−1(𝑐𝑡/|𝑥|)sin−1()𝑐/𝑎}cos^𝑛−2𝑐𝜃𝑑𝜃=−²𝑠𝑡2𝑎²𝑎²−𝑐^{2(𝑛−1)/2}ln𝑎+𝑐×𝑎𝑎−𝑐𝑐𝑐1−𝑎^{2(𝑛−1)/2}−|𝑥|𝑐𝑡1−𝑐𝑡|𝑥|^{2(𝑛−1)/2}+(𝑛−1)𝑐²𝑠𝑡2𝑎²𝑎²−𝑐^{2(𝑛−1)/2}×ln𝑎+𝑐𝑎−𝑐^{sin−1(𝑐𝑡/|𝑥|)}_sin⁻¹_(𝑐/𝑎)cos^𝑛−2𝜃𝑑𝜃, (44) for any 𝑥∈𝑈 and 𝑡∈𝑇.
For the case of 𝑛=3, the inequality (10) can be rewritten as 𝑢(𝑥,𝑡)−𝑢₀(≤𝑐𝑥,𝑡)₂𝑠𝑡2𝑎²𝑎²−𝑐²ln𝑎+𝑐×𝑎−𝑐𝑐𝑡|𝑥|+|𝑥|−𝑐𝑐𝑡𝑎−𝑎𝑐, (45) for all 𝑥∈𝑈 and 𝑡∈𝑇.

Remark 2. As in Remark 1, the inequality (34) in Theorem 2 can be rewritten as 𝑢(𝑥,𝑡)−𝑢₀(≤𝑥,𝑡)𝑐𝑠^′𝑡2𝑏²𝑐²−𝑏^{2(𝑛−1)/2}×ln𝑐+𝑏𝑐−𝑏^𝑏_|𝑥|/𝑡𝑐²𝑧²−1^(𝑛−1)/2𝑑𝑧, (46) for all 𝑥∈𝑈^′ and 𝑡∈𝑇^′. If we substitute 𝑐cos𝜃 for 𝑧 in the previous inequality, then we get 𝑢(𝑥,𝑡)−𝑢₀(𝑥,𝑡)≤−𝑐𝑠^′𝑡2𝑏²𝑐²−𝑏^{2(𝑛−1)/2}ln𝑐+𝑏×1𝑐−𝑏𝑡𝑐²𝑡²−|𝑥|^{2(𝑛−1)/2}|𝑥|^𝑛−2−𝑐²−𝑏^{2(𝑛−1)/2}𝑏^𝑛−2−(𝑛−1)𝑐²𝑠^′𝑡2𝑏²𝑐²−𝑏^{2(𝑛−1)/2}×ln𝑐+𝑏𝑐−𝑏^{cos−1(𝑏/𝑐)}_cos⁻¹_{(|𝑥|/𝑐𝑡)}sin^𝑛−2𝜃cos^𝑛−1𝜃𝑑𝜃, (47) for any 𝑥∈𝑈^′ and 𝑡∈𝑇^′.
For the case of 𝑛=3, the inequality (34) can be rewritten as 𝑢(𝑥,𝑡)−𝑢₀(≤𝑐𝑥,𝑡)₂𝑠^′𝑡2𝑏²𝑐²−𝑏²ln𝑐+𝑏𝑐−𝑏𝑐𝑡|𝑥|+|𝑥|−𝑏𝑐𝑡𝑐−𝑐𝑏, (48) 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∈ℝ^𝑛∶0<|𝑥|<𝑎𝑡₂} or 𝑇=(𝑡₁,∞) and 𝑈={𝑥∈ℝ^𝑛∶|𝑥|>𝑏𝑡₁} or 𝑇=(0,∞) and 𝑈={𝑥∈ℝ^𝑛∶|𝑥|>0}.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557). This work was also supported by the 2011 Hongik University Research Fund.