An Operator Method for the Stability of Inhomogeneous Wave Equations

Abstract

In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, $u_{tt}(x,t)-c^2▵u(x,t)=f(x,t)$, for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples.

1. Introduction

Ulam [1] asked the following question: Under what conditions does there exist an additive function near an approximately additive function? in 1941, Hyers [2] provided an answer for this question, that for all $\epsilon \ge 0$ there exists an exact additive function F, such that $\parallel f(x)-F(x)\parallel \le \epsilon$ for all x, if a function f satisfies the inequality $\parallel f(x+y)-f(x)-f(y)\parallel \le \epsilon$ for all x. This theorem of Hyers was the origination for the terminology of the Hyers-Ulam stability.

To the best of our knowledge, Obłoza [3,4] also first investigated the Hyers-Ulam stability of the differential equations, and after that it was generalized by many mathematicians. We refer the reader to [1,2,5,6,7,8,9,10,11,12,13,14,15]. Notably, Prástaro and Rassias [16] first studied the Hyers-Ulam stability of the partial differential equations. Thereafter, the Hyers-Ulam stability of the partial differential equations was investigated in [8,10,17,18,19,20,21,22,23].

In this paper, we will study the wave equation in ${\mathbb{R}}^n$:

$$\begin{array}{c}\hfill u_{tt}(x,t)-c^2▵u(x,t)=f(x,t),\end{array}$$

(1)

where $c>0$ is a constant, $▵=\sum _{i=1}^n\frac{{\partial }^2}{\partial x_i^2}$ is the Laplace operator, and $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$. Actually, Choi and Jung [17] investigated the Hyers-Ulam stability of (1) by using the method of dilation invariance. For the fractional calculus, wavelet analysis, and fractal geometry, we refer the reader to [24,25,26,27].

In this paper, we will apply the operator method instead of the method of dilation invariance for investigating the generalized Hyers-Ulam stability of (1). One of the advantages of this present paper over [17] is that there are no limiting conditions in the definitions of $U_1^n$ and $U_2^n$ (see Section 3). We will also consider a more general form of the source term $f(x,t)$ than those in [17] to see the Hyers-Ulam stability of the Equation (1). In addition, concerning the domains of relevant functions, as [17] give a partial answer to the open problem raised in ([20] Remark 3), this paper will attempt to give a partial answer as well.

2. Preliminaries

In this section, we will introduce a modified version of ([18] Theorem 1) which is more suitable for practical applications. This modified version will be applied many times to the proof of our main theorems in the next section (cf. [12] Theorem 2.2). Indeed, the hypotheses of the original theorem ([18] Theorem 1) were formulated with a instead of $a_0$, which imposes a constraint on its usability. The proof of Theorem 1 precisely follows the lines of the proof of ([18] Theorem 1)—hence, we omit the proof.

Theorem 1.

([18] Theorem 1, Remark 3) Assume that X is a real Banach space, and $I=(a,b)$ is an open interval for arbitrary constants $a,b\in \mathbb{R}\cup \{\pm \infty \}$ with $a<b$. Let $p:I\to \mathbb{R}$ and $q:I\to X$ be continuous functions, such that there exists a constant $a_0\in [a,b)$ with the properties:

(i) 

${\int }_{a_0}^{t}p(s)ds$ exists for each $t\in I$;

(ii) 

${\int }_{a_0}^{t}q(y)exp\left\{{\int }_{a_0}^{y}p(s)ds\right\}dy$ exists for any $t\in I$.

Moreover, assume that $\phi :I\to [0,\infty )$ is a function such that:

(iii) 

${\int }_{a_0}^b\phi (y)exp\left\{{\int }_{a_0}^{y}p(s)ds\right\}dy$ exists.

If a continuously differentiable function $v:I\to X$ satisfies the differential inequality:

$$\begin{array}{c}\hfill \parallel v^{\prime }(t)+p(t)v(t)+q(t)\parallel \le \phi (t)\end{array}$$

for all $t\in I$, then there exists a unique, continuously differentiable function $v_0:I\to X$, such that: $v_0^{\prime }(t)+p(t)v_0(t)+q(t)=0$ for all $t\in I$ and

$$\begin{array}{c}\hfill \parallel v(t)-v_0(t)\parallel \le exp\left\{-{\int }_{a_0}^{t}p(s)ds\right\}{\int }_t^b\phi (y)exp\left\{{\int }_{a_0}^{y}p(s)ds\right\}dy\end{array}$$

for all $t\in I$.

3. Main Results

In this section, n is a fixed positive integer, and each point x in ${\mathbb{R}}^n$ is expressed as $x=(x_1,\dots ,x_i,\dots ,x_n)$, where $x_i$ denotes the ith coordinate of x. Moreover, $|x|$ denotes the Euclidean distance of x from the origin:

$$\begin{array}{c}\hfill |x|=\sqrt{x_1^2+\cdots +x_i^2+\cdots +x_n^2}\end{array}$$

Since the propagation speed of each solution $u(x,t)$ to the wave Equation (1) is c, the "shape" of the wave travels at the speed of c. Roughly speaking, $u(x,t)$ seems to have a similar shape at each $(x,t)$, provided $|x|/t=c$. That is, $u(x,t)$ depends on x and t, mainly through the term $|x|/t$. For this reason, we will search for approximate solutions to (1), which belong to a special class of scalar functions of the form $u(x,t)=tv(|x|/t)$, where v is a twice continuously differentiable function. Such a method will be called the “method of dilation invariance”—see [17,19,20].

Based on this argument, we define:

$$\begin{array}{c}\begin{array}{cc}\hfill D_1^n& :=\left\{(x,t)\in {\mathbb{R}}^n\setminus \{(0,\dots ,0)\}\times (0,\infty )||x|<ct\right\},\hfill \\ \hfill U_1^n& :=\{u:D_1^n\to \mathbb{R}|\text{ there exists a twice continuously differentiable}\hfill \\ & \text{function }v:(0,c)\to \mathbb{R}\text{ with }u(x,t)=tv\left(\frac{|x|}{t}\right)\text{ for all }(x,t)\in D_1^n\}.\hfill \end{array}\end{array}$$

We may compare these definitions above with the definitions of [17]. It is obvious that $(x,t)\in D_1^n$ if, and only if $0<|x|/t<c$. The conditions (2) and (5) may seem to be too strict at first glance. However, we shall see in Corollary 1 that they are not as strict as they look.

Theorem 2.

Let functions $\phi :(0,c)\to [0,\infty )$ and $\psi :(0,\infty )\to [0,\infty )$ be given such that:

$$\begin{array}{c}\hfill {\int }_{r_0}^c{\left(\frac{c^2}{y^2}-1\right)}^{(n-1)/2}{\int }_y^c\frac{\phi (s)}{s^2}{\left(\frac{c^2}{s^2}-1\right)}^{-(n+1)/2}dsdy\end{array}$$

(2)

exists for a fixed constant $r_0\in (0,c)$ and there exists a positive real number k with:

$$\begin{array}{c}\hfill k:=\underset{t>0}{inf}t\psi (t)>0.\end{array}$$

(3)

Assume that $f:D_1^n\to \mathbb{R}$ is a function for which there exists a continuous function $g:(0,c)\to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill f(x,t)=\frac{1}{t}g\left(\frac{|x|}{t}\right)\end{array}$$

(4)

and

$$\begin{array}{c}\hfill {\int }_{r_0}^{|x|/t}{\left(\frac{c^2}{y^2}-1\right)}^{(n-1)/2}{\int }_{r_0}^y\frac{|g(s)|}{s^2}{\left(\frac{c^2}{s^2}-1\right)}^{-(n+1)/2}dsdy\end{array}$$

(5)

exists for all $(x,t)\in D_1^n$. If a $u\in U_1^n$ satisfies the inequality:

$$\begin{array}{c}\hfill |u_{tt}(x,t)-c^2▵u(x,t)-f(x,t)|\le \phi \left(\frac{|x|}{t}\right)\psi (t)\end{array}$$

(6)

for all $(x,t)\in D_1^n$, then there exists a solution $u_0\in U_1^n$ of the wave Equation (1), such that:

$$\begin{array}{c}\hfill |u(x,t)-u_0(x,t)|\le kt{\int }_{|x|/t}^c{\left(\frac{c^2}{y^2}-1\right)}^{(n-1)/2}{\int }_y^c\frac{\phi (s)}{s^2}{\left(\frac{c^2}{s^2}-1\right)}^{-(n+1)/2}dsdy\end{array}$$

(7)

for all $(x,t)\in D_1^n$.

Proof. 

Our assumption $u\in U_1^n$ implies that there exists a twice continuously differentiable function $v:(0,c)\to \mathbb{R}$, such that:

$$\begin{array}{c}\hfill u(x,t)=tv(r)\end{array}$$

(8)

for all $(x,t)\in D_1^n$, where we set $r=|x|/t$. Then we have

$$\begin{array}{c}\hfill \frac{\partial r}{\partial t}=-\frac{r}{t},\frac{\partial |x|}{\partial x_i}=\frac{x_i}{|x|},\frac{\partial r}{\partial x_i}=\frac{1}{t}\frac{x_i}{|x|},\frac{\partial }{\partial x_i}\frac{x_i}{|x|}=\frac{{|x|}^2-x_i^2}{{|x|}^3}\end{array}$$

for $i\in \{1,\dots ,n\}$. Using these partial derivatives, we obtain:

$$\begin{array}{c}\hfill \begin{array}{cc}{u}_t(x,t)=v(r)-r{v}^{\prime }(r),\hfill & {\displaystyle u_{tt}(x,t)=\frac{r^2}{t}{v}^{″}(r)},\hfill \\ {\displaystyle u_{x_i}(x,t)=\frac{x_i}{|x|}{v}^{\prime }(r)},\hfill & {\displaystyle u_{x_i{x}_i}(x,t)=\frac{{|x|}^2-x_i^2}{{|x|}^3}{v}^{\prime }(r)+\frac{1}{t}\frac{x_i^2}{{|x|}^2}{v}^{″}(r)}\hfill \end{array}\end{array}$$

(9)

for $i\in \{1,\dots ,n\}$.

Therefore, in view of (4) and (9), we have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{tt}& (x,t)-c^2▵u(x,t)-f(x,t)\hfill \\ \hfill =& \frac{r^2}{t}{v}^{″}(r)-c^2\sum _{i=1}^n\left(\frac{{|x|}^2-x_i^2}{{|x|}^3}{v}^{\prime }(r)+\frac{1}{t}\frac{x_i^2}{{|x|}^2}{v}^{″}(r)\right)-\frac{1}{t}g(r)\hfill \\ \hfill =& -\frac{1}{t}\left(\left(c^2-r^2\right)v^{″}(r)+c^2\frac{n-1}{r}{v}^{\prime }(r)+g(r)\right)\hfill \end{array}\end{array}$$

and hence, by (6), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |u_{tt}& (x,t)-c^2▵u(x,t)-f(x,t)|\hfill \\ \hfill =& \frac{1}{t}|\left(c^2-r^2\right)v^{″}(r)+c^2\frac{n-1}{r}{v}^{\prime }(r)+g(r)|\hfill \\ \hfill \le & \phi (r)\psi (t)\hfill \end{array}\end{array}$$

and by considering (3), we further obtain:

$$\begin{array}{c}\hfill |\left(c^2-r^2\right)v^{″}(r)+\frac{(n-1)c^2}{r}{v}^{\prime }(r)+g(r)|\le k\phi (r)\end{array}$$

(10)

for all $r\in (0,c)$. (Indeed, for each fixed $r\in (0,c)$ and for any $t>0$, we can select an $x_t\in {\mathbb{R}}^n\setminus \{(0,\dots ,0)\}$ such that $(x_t,t)\in D_1^n$ and $r=|x_t|/t$).

We now apply a method for a decomposition of a second-order differential operator into two differential operators of first-order. Let us define the second-order differential operator ${\mathcal{L}}_n^2:{\mathrm{C}}^2(0,c)\to \mathrm{C}(0,c)$ by:

$$\begin{array}{c}\hfill {\mathcal{L}}_n^{2}v(r):=\left(c^2-r^2\right)v^{″}(r)+\frac{(n-1)c^2}{r}{v}^{\prime }(r),\end{array}$$

(11)

where $\mathrm{C}(0,c)$ and ${\mathrm{C}}^2(0,c)$ denote the set of all continuous real-valued functions and the set of all twice continuously differentiable real-valued functions defined on $(0,c)$, respectively.

We tried to decompose the differential operator ${\mathcal{L}}_n^2$ into the differential operators ${\mathcal{L}}_{A(r)}^{-}$ and ${\mathcal{L}}_{B(r)}^{+}$ such that:

$$\begin{array}{c}\hfill {\mathcal{L}}_n^{2}v(r)=\left({\mathcal{L}}_{A(r)}^{-}\circ {\mathcal{L}}_{B(r)}^{+}\right)v(r)\end{array}$$

(12)

for all $v\in {\mathrm{C}}^2(0,c)$, where we define

$$\begin{array}{c}\hfill {\mathcal{L}}_{A(r)}^{-}v(r):=(c-r)v^{\prime }(r)+A(r)v(r)\mathrm{and}{\mathcal{L}}_{B(r)}^{+}v(r):=(c+r)v^{\prime }(r)+B(r)v(r).\end{array}$$

(13)

Then we have:

$$\begin{array}{c}\begin{array}{cc}\hfill \left({\mathcal{L}}_{A(r)}^{-}\circ {\mathcal{L}}_{B(r)}^{+}\right)v(r)=& \left(c^2-r^2\right)v^{″}(r)+\left(c-r+(c+r)A(r)+(c-r)B(r)\right)v^{\prime }(r)\hfill \\ & +\left(A(r)B(r)+(c-r)B^{\prime }(r)\right)v(r).\hfill \end{array}\end{array}$$

(14)

Comparing both (12) and (14), we obtained:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(c+r)A(r)+(c-r)B(r)={\displaystyle \frac{(n-1)c^2}{r}-(c-r)},\hfill \\ A(r)B(r)+(c-r)B^{\prime }(r)=0.\hfill \end{array}\right.\end{array}$$

From the last system of equations, we got the Riccati equation:

$$\begin{array}{c}\hfill \left(c^2-r^2\right)B^{\prime }(r)+\left(\frac{(n-1)c^2}{r}-(c-r)\right)B(r)-(c-r)B{(r)}^2=0,\end{array}$$

(15)

one of whose solutions is $B(r)\equiv 0$ (see [28] §1.2.1). By making use of

$$\begin{array}{c}\hfill B(r)\equiv 0\mathrm{and}A(r)=\frac{(n-1)c^2}{r(c+r)}-\frac{c-r}{c+r},\end{array}$$

if we set:

$$\begin{array}{c}\hfill w(r):={\mathcal{L}}_{B(r)}^{+}v(r)=(c+r)v^{\prime }(r),\end{array}$$

(16)

it then follows from (10), (11) and (12) that:

$$\begin{array}{c}\hfill |{\mathcal{L}}_{A(r)}^{-}w(r)+g(r)|=|{\mathcal{L}}_n^{2}v(r)+g(r)|\le k\phi (r)\end{array}$$

for all $r\in (0,c)$. Thus by (13), we have:

$$\begin{array}{c}\hfill |w^{\prime }(r)+\left(\frac{(n-1)c^2}{r(c^2-r^2)}-\frac{1}{c+r}\right)w(r)+\frac{g(r)}{c-r}|\le \frac{k\phi (r)}{c-r}\end{array}$$

(17)

for all $r\in (0,c)$.

We can now apply Theorem 1 to our inequality (17) by considering the substitutions, as seen in the Table 1.

Table 1. Variable substitution table.

Theorem 1	X	t	a	b	${\mathit{a}}_0$	$\mathit{v}(\mathit{t})$	$\mathit{p}(\mathit{t})$	$\mathit{q}(\mathit{t})$	$\mathit{\phi }(\mathit{t})$
(17)	$\mathbb{R}$	r	0	c	$r_0$	$w(r)$	$\frac{(n-1)c^2}{r(c^2-r^2)}-\frac{1}{c+r}$	$\frac{g(r)}{c-r}$	$\frac{k\phi (r)}{c-r}$

Considering the above table, we easily verify that the following integral

$$\begin{array}{c}\hfill {\int }_{r_0}^{r}p(s)ds=\frac{n-1}{2}ln\left(\frac{r^2}{r_0^2}·\frac{c-r_0}{c-r}\right)-\frac{n+1}{2}ln\frac{c+r}{c+r_0}\end{array}$$

exists for any $r\in (0,c)$. Thus, the condition $(i)$ of Theorem 1 is fulfilled. Moreover, by considering

$$\begin{array}{c}\begin{array}{cc}\hfill exp\left\{{\int }_{r_0}^{r}p(s)ds\right\}& =exp\left\{{\int }_{r_0}^r\left(\frac{(n-1)c^2}{s(c^2-s^2)}-\frac{1}{c+s}\right)ds\right\}\hfill \\ & =\frac{r_0^2}{c-r_0}{\left(\frac{c^2}{r_0^2}-1\right)}^{(n+1)/2}·\frac{c-r}{r^2}{\left(\frac{c^2}{r^2}-1\right)}^{-(n+1)/2},\hfill \end{array}\end{array}$$

(18)

the Hypotheses (2) and (5) guarantee the validity of conditions $(ii)$ and $(iii)$ of Theorem 1, respectively.

According to Theorem 1 and (17), there exists a unique continuously differentiable function $w_0:(0,c)\to \mathbb{R}$, such that:

$$\begin{array}{c}\hfill w_0^{\prime }(r)+\left(\frac{(n-1)c^2}{r(c^2-r^2)}-\frac{1}{c+r}\right)w_0(r)+\frac{g(r)}{c-r}=0\end{array}$$

(19)

and

$$\begin{array}{c}\hfill |w(r)-w_0(r)|\le \frac{k{r}^2}{c-r}{\left(\frac{c^2}{r^2}-1\right)}^{(n+1)/2}{\int }_r^c\frac{\phi (y)}{y^2}{\left(\frac{c^2}{y^2}-1\right)}^{-(n+1)/2}dy\end{array}$$

for all $r\in (0,c)$.

It follows from (16) and the last inequality that:

$$\begin{array}{c}\hfill |v^{\prime }(r)-\frac{w_0(r)}{c+r}|\le k{\left(\frac{c^2}{r^2}-1\right)}^{(n-1)/2}{\int }_r^c\frac{\phi (y)}{y^2}{\left(\frac{c^2}{y^2}-1\right)}^{-(n+1)/2}dy\end{array}$$

(20)

for any $r\in (0,c)$.

We can again apply Theorem 1 to our inequality (20) by considering the substitutions seen in the following Table 2.

Table 2. Variable substitution table.

Theorem 1	X	t	a	b	${\mathit{a}}_0$	$\mathit{v}(\mathit{t})$	$\mathit{p}(\mathit{t})$	$\mathit{q}(\mathit{t})$	$\mathit{\phi }(\mathit{t})$
(20)	$\mathbb{R}$	r	0	c	$r_0$	$v(r)$	0	$-\frac{w_0(r)}{c+r}$	$\text{the right side of}$ (20)

By substituting $\omega =csin\theta$ $(0<\theta <\pi /2)$, we see that the integral

$$\begin{array}{c}\hfill {\int }_{r_0}^r{\left(\frac{c^2}{{\omega }^2}-1\right)}^{(n-1)/2}d\omega =c{\int }_{{sin}^{-1}(r_0/c)}^{{sin}^{-1}(r/c)}\frac{{cos}^n\theta }{{sin}^{n-1}\theta }d\theta \end{array}$$

(21)

exists for every $r\in (0,c)$. Indeed, according to the table of integrals (e.g., see [29] §2.518, §2.521), the last integral exists for any given $r,r_0\in (0,c)$. By considering the above table, (2), (5) and (18), we conclude that the conditions $(i)$, $(ii)$, and $(iii)$ of Theorem 1 are fulfilled.

Due to Theorem 1 and (20), there exists a unique continuously differentiable function $v_0:(0,c)\to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill v_0^{\prime }(r)-\frac{w_0(r)}{c+r}=0\end{array}$$

(22)

and

$$\begin{array}{c}\hfill |v(r)-v_0(r)|\le k{\int }_r^c{\left(\frac{c^2}{y^2}-1\right)}^{(n-1)/2}{\int }_y^c\frac{\phi (s)}{s^2}{\left(\frac{c^2}{s^2}-1\right)}^{-(n+1)/2}dsdy\end{array}$$

(23)

for any $r\in (0,c)$. Indeed, on account of (22) and the continuous differentiability of $w_0(r)$, $v_0(r)$ is a twice continuously differentiable function. If we define a function $u_0:D_1^n\to \mathbb{R}$ by $u_0(x,t):=t{v}_0(r)$, then $u_0\in U_1^n$, and inequality (7) follows immediately from (8) and (23). Furthermore, by using (19) and (22) and by following the first part of this proof, we can show that $u_0(x,t)$ is a solution to the wave Equation (1). □

In view of (21), if there exist positive real constants $k_1$ and $k_2$ such that

$$\begin{array}{c}\hfill \phi (r)\le k_1{r}^2{\left(\frac{c^2}{r^2}-1\right)}^{(n+1)/2}\end{array}$$

(24)

and

$$\begin{array}{c}\hfill |g(r)|\le k_2{r}^2{\left(\frac{c^2}{r^2}-1\right)}^{(n+1)/2}\end{array}$$

(25)

for all $r\in (0,c)$, then the conditions (2) and (5) of Theorem 2 are satisfied.

Corollary 1.

Assume that $f:D_1^n\to \mathbb{R}$ is a function for which there exists a continuous function $g:(0,c)\to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill f(x,t)=\frac{1}{t}g\left(\frac{|x|}{t}\right)\end{array}$$

for all $(x,t)\in D_1^n$ and

$$\begin{array}{c}\hfill |g(r)|\le {\epsilon }_3{r}^2{\left(\frac{c^2}{r^2}-1\right)}^{(n+1)/2}\end{array}$$

for all $r\in (0,c)$ and for some ${\epsilon }_3>0$. If a $u\in U_1^n$ satisfies the inequality

$$\begin{array}{c}\hfill |u_{tt}(x,t)-c^2▵u(x,t)-f(x,t)|\le \epsilon \frac{{|x|}^2}{t^3}{\left(\frac{c^2{t}^2}{{|x|}^2}-1\right)}^{(n+1)/2}\end{array}$$

for all $(x,t)\in D_1^n$ and for some $\epsilon >0$, then there exists a solution $u_0\in U_1^n$ of the wave equation (1), such that:

$$\begin{array}{c}\hfill |u(x,t)-u_0(x,t)|\le \epsilon t{\int }_{|x|/t}^c(c-y){\left(\frac{c^2}{y^2}-1\right)}^{(n-1)/2}dy\end{array}$$

for all $(x,t)\in D_1^n$.

Proof. 

Given positive real constants ${\epsilon }_1,{\epsilon }_2$ with $\epsilon ={\epsilon }_1{\epsilon }_2$, we define functions $\phi :(0,c)\to [0,\infty )$ and $\psi :(0,\infty )\to [0,\infty )$ by

$$\begin{array}{c}\hfill \phi (r):={\epsilon }_1{r}^2{\left(\frac{c^2}{r^2}-1\right)}^{(n+1)/2}\mathrm{and}\psi (t):=\frac{{\epsilon }_2}{t}.\end{array}$$

In view of (21), (24), (25) and with $r=|x|/t$, we can easily verify that the conditions (2) and (5) are satisfied. Thus, our assertion follows from Theorem 2 with $k={\epsilon }_2$. □

Let us define

$$\begin{array}{c}\begin{array}{cc}\hfill D_2^n& :=\left\{(x,t)\in {\mathbb{R}}^n\times (0,\infty )||x|>ct\right\},\hfill \\ \hfill U_2^n& :=\{u:D_2^n\to \mathbb{R}|\text{ there exists a twice continuously differentiable}\hfill \\ & \text{function }v:(c,\infty )\to \mathbb{R}\text{ with }u(x,t)=tv\left(\frac{|x|}{t}\right)\text{ for all }(x,t)\in D_2^n\}.\hfill \end{array}\end{array}$$

We may compare these definitions with those of $D_2^n$ and $U_2^n$ in [17]. Moreover, we see that $(x,t)\in D_2^n$ if, and only if $c<|x|/t<\infty$. Even if the conditions (26) and (29) below seem somewhat strict at first glance, they are indeed not so strict, as we shall see in Corollary 2.

Theorem 3.

Let functions $\phi :(c,\infty )\to [0,\infty )$ and $\psi :(0,\infty )\to [0,\infty )$ be given, such that:

$$\begin{array}{c}\hfill {\int }_{r_0}^{\infty }{\left(1-\frac{c^2}{y^2}\right)}^{(n-1)/2}{\int }_y^{\infty }\frac{\phi (s)}{s^2}{\left(1-\frac{c^2}{s^2}\right)}^{-(n+1)/2}dsdy\end{array}$$

(26)

exists for a fixed constant $r_0\in (c,\infty )$ and there exists a positive real number k with

$$\begin{array}{c}\hfill k:=\underset{t>0}{inf}t\psi (t)>0.\end{array}$$

(27)

Assume that $f:D_2^n\to \mathbb{R}$ is a function for which there exists a continuous function $g:(c,\infty )\to \mathbb{R}$, such that

$$\begin{array}{c}\hfill f(x,t)=\frac{1}{t}g\left(\frac{|x|}{t}\right)\end{array}$$

(28)

and

$$\begin{array}{c}\hfill {\int }_{r_0}^{|x|/t}{\left(1-\frac{c^2}{y^2}\right)}^{(n-1)/2}{\int }_{r_0}^y\frac{|g(s)|}{s^2}{\left(1-\frac{c^2}{s^2}\right)}^{-(n+1)/2}dsdy\end{array}$$

(29)

exists for all $(x,t)\in D_2^n$. If a $u\in U_2^n$ satisfies the inequality

$$\begin{array}{c}\hfill |u_{tt}(x,t)-c^2▵u(x,t)-f(x,t)|\le \phi \left(\frac{|x|}{t}\right)\psi (t)\end{array}$$

(30)

for all $(x,t)\in D_2^n$, then there exists a solution $u_0\in U_2^n$ of the wave Equation (1) such that

$$\begin{array}{c}\hfill |u(x,t)-u_0(x,t)|\le kt{\int }_{|x|/t}^{\infty }{\left(1-\frac{c^2}{y^2}\right)}^{(n-1)/2}{\int }_y^{\infty }\frac{\phi (s)}{s^2}{\left(1-\frac{c^2}{s^2}\right)}^{-(n+1)/2}dsdy\end{array}$$

(31)

for all $(x,t)\in D_2^n$.

Proof. 

Due to our assumption $u\in U_2^n$, there exists a twice continuously differentiable function $v:(c,\infty )\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill u(x,t)=tv(r)\end{array}$$

(32)

for all $(x,t)\in D_2^n$, where we set $r=|x|/t$.

We simply followed the first part of the proof of Theorem 2 and obtained the inequality

$$\begin{array}{c}\hfill |w^{\prime }(r)+\left(\frac{(n-1)c^2}{r(c^2-r^2)}-\frac{1}{c+r}\right)w(r)+\frac{g(r)}{c-r}|\le \frac{k\phi (r)}{r-c}\end{array}$$

(33)

for all $r\in (c,\infty )$ instead of (17), where we set $w(r)={\mathcal{L}}_{B(r)}^{+}v(r)=(c+r)v^{\prime }(r)$, as defined in (16).

We applied Theorem 1 to the inequality (33) by considering the substitutions seen in the Table 3.

Table 3. Variable substitution table.

Theorem 1	X	t	a	b	${\mathit{a}}_0$	$\mathit{v}(\mathit{t})$	$\mathit{p}(\mathit{t})$	$\mathit{q}(\mathit{t})$	$\mathit{\phi }(\mathit{t})$
(33)	$\mathbb{R}$	r	c	∞	$r_0$	$w(r)$	$\frac{(n-1)c^2}{r(c^2-r^2)}-\frac{1}{c+r}$	$\frac{g(r)}{c-r}$	$\frac{k\phi (r)}{r-c}$

Considering the above table and following the similar method as the proof of Theorem 2, we were able to easily verify that the following integral

$$\begin{array}{c}\hfill {\int }_{r_0}^{r}p(s)ds=\frac{n-1}{2}ln\left(\frac{r^2}{r_0^2}·\frac{r_0-c}{r-c}\right)-\frac{n+1}{2}ln\frac{r+c}{r_0+c}\end{array}$$

exists for any $r\in (c,\infty )$. Hence, the condition $(i)$ of Theorem 1 is satisfied. Moreover, by considering

$$\begin{array}{c}\hfill exp\left\{{\int }_{r_0}^{r}p(s)ds\right\}=\frac{r_0^2}{r_0-c}{\left(1-\frac{c^2}{r_0^2}\right)}^{(n+1)/2}·\frac{r-c}{r^2}{\left(1-\frac{c^2}{r^2}\right)}^{-(n+1)/2},\end{array}$$

the hypotheses (26) and (29) ensures the validity of conditions $(iii)$ and $(ii)$ of Theorem 1, respectively.

According to Theorem 1 and (33), there exists a unique continuously differentiable function $w_0:(c,\infty )\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill w_0^{\prime }(r)+\left(\frac{(n-1)c^2}{r(c^2-r^2)}-\frac{1}{c+r}\right)w_0(r)+\frac{g(r)}{c-r}=0\end{array}$$

(34)

and

$$\begin{array}{c}\hfill |w(r)-w_0(r)|\le \frac{k{r}^2}{r-c}{\left(1-\frac{c^2}{r^2}\right)}^{(n+1)/2}{\int }_r^{\infty }\frac{\phi (y)}{y^2}{\left(1-\frac{c^2}{y^2}\right)}^{-(n+1)/2}dy\end{array}$$

for all $r\in (c,\infty )$.

It follows from (16) and the last inequality that

$$\begin{array}{c}\hfill |v^{\prime }(r)-\frac{w_0(r)}{c+r}|\le k{\left(1-\frac{c^2}{r^2}\right)}^{(n-1)/2}{\int }_r^{\infty }\frac{\phi (y)}{y^2}{\left(1-\frac{c^2}{y^2}\right)}^{-(n+1)/2}dy\end{array}$$

(35)

for any $r\in (c,\infty )$.

We again applied Theorem 1 to inequality (35) by referring to the Table 4.

Table 4. Variable substitution table.

Theorem 1	X	t	a	b	${\mathit{a}}_0$	$\mathit{v}(\mathit{t})$	$\mathit{p}(\mathit{t})$	$\mathit{q}(\mathit{t})$	$\mathit{\phi }(\mathit{t})$
(35)	$\mathbb{R}$	r	c	∞	$r_0$	$v(r)$	0	$-\frac{w_0(r)}{c+r}$	$\text{the right side of}$ (35)

Since $p(t)\equiv 0$ in the above table, the condition $(i)$ of Theorem 1 is fulfilled. In connection with the condition $(ii)$, the integral ${\int }_{r_0}^r{w}_0(y)/(c+y)dy$ exists because $w_0$ is a continuously differentiable function. The validity of $(iii)$ is a simple consequence of (26).

Due to Theorem 1 and (35), there exists a unique continuously differentiable function $v_0:(c,\infty )\to \mathbb{R}$, such that

$$\begin{array}{c}\hfill v_0^{\prime }(r)-\frac{w_0(r)}{c+r}=0\end{array}$$

(36)

and

$$\begin{array}{c}\hfill |v(r)-v_0(r)|\le k{\int }_r^{\infty }{\left(1-\frac{c^2}{y^2}\right)}^{(n-1)/2}{\int }_y^{\infty }\frac{\phi (s)}{s^2}{\left(1-\frac{c^2}{s^2}\right)}^{-(n+1)/2}dsdy\end{array}$$

(37)

for any $r\in (c,\infty )$. Indeed, on account of (36), $v_0(r)$ is a twice continuously differentiable function. If we define a function $u_0:D_2^n\to \mathbb{R}$ by $u_0(x,t):=t{v}_0(r)$, then $u_0\in U_2^n$ and the inequality (31) follows immediately from (32) and (37). Furthermore, by using (34) and (36), and by following the first part of the proof of Theorem 2 (or by the last part of that proof), we can show that $u_0(x,t)$ is a solution to the wave Equation (1). □

If there exist positive real constants $k_1$, $k_2$, ${\theta }_1$, and ${\theta }_2$, such that

$$\begin{array}{c}\hfill \phi (r)\le \frac{k_1}{r^{{\theta }_1}}{\left(1-\frac{c^2}{r^2}\right)}^{(n+1)/2}\end{array}$$

(38)

and

$$\begin{array}{c}\hfill |g(r)|\le \frac{k_2}{r^{{\theta }_2}}{\left(1-\frac{c^2}{r^2}\right)}^{(n+1)/2}\end{array}$$

(39)

for any $r\in (c,\infty )$, then the conditions (26) and (29) of Theorem 3 are satisfied.

Corollary 2.

Assume that $f:D_2^n\to \mathbb{R}$ is a function for which there exists a continuous function $g:(c,\infty )\to \mathbb{R}$, such that

$$\begin{array}{c}\hfill f(x,t)=\frac{1}{t}g\left(\frac{|x|}{t}\right)\end{array}$$

for all $(x,t)\in D_2^n$ and

$$\begin{array}{c}\hfill |g(r)|\le \frac{{\epsilon }_3}{r^{{\theta }_2}}{\left(1-\frac{c^2}{r^2}\right)}^{(n+1)/2}\end{array}$$

for all $r\in (c,\infty )$ and for some ${\epsilon }_3>0$ and ${\theta }_2>0$. If a $u\in U_2^n$ satisfies the inequality

$$\begin{array}{c}\hfill |u_{tt}(x,t)-c^2▵u(x,t)-f(x,t)|\le \frac{\epsilon }{{|x|}^{{\theta }_1}{t}^{1-{\theta }_1}}{\left(1-\frac{c^2{t}^2}{{|x|}^2}\right)}^{(n+1)/2}\end{array}$$

for all $(x,t)\in D_2^n$ and for some $\epsilon >0$ and ${\theta }_1>0$, then there exists a solution $u_0\in U_2^n$ of the wave Equation (1), such that

$$\begin{array}{c}\hfill |u(x,t)-u_0(x,t)|\le \frac{\epsilon t}{1+{\theta }_1}{\int }_{|x|/t}^{\infty }\frac{1}{y^{1+{\theta }_1}}{\left(1-\frac{c^2}{y^2}\right)}^{(n-1)/2}dy\end{array}$$

for each $(x,t)\in D_2^n$.

Proof. 

We chose positive real numbers ${\epsilon }_1$ and ${\epsilon }_2$ with $\epsilon ={\epsilon }_1{\epsilon }_2$ and we defined functions $\phi :(c,\infty )\to [0,\infty )$ and $\psi :(0,\infty )\to [0,\infty )$ by

$$\begin{array}{c}\hfill \phi (r):=\frac{{\epsilon }_1}{r^{{\theta }_1}}{\left(1-\frac{c^2}{r^2}\right)}^{(n+1)/2}\mathrm{and}\psi (t):=\frac{{\epsilon }_2}{t}.\end{array}$$

On account of (38) and (39) with $r=|x|/t$, we can show that the conditions (26) and (29) are satisfied. Hence, our assertion follows from Theorem 3 with $k={\epsilon }_2$. □

Remark 1.

One may note that an approximate solution of (1) obtained by Corollary 1 converges to the solution of (1) as $r=|x|/t$ goes to c from the left side. The error estimate between two solutions is also less than $kt{c}^2{\left(\frac{c}{r}\right)}^{n-1}$ for any $(x,t)$ where $|x|<ct$. Also, one may note that an approximate solution of (1) obtained by Corollary 2 converges to the solution of (1), as $r=|x|/t$ goes to ∞. The error estimate between two solutions is also less than $kt{\left(\frac{1}{r}\right)}^{{\theta }_1}$ for any $(x,t)$ where $|x|>ct$.

4. Discussions

We were able to find strong symmetrical properties between the definitions of $D_1^n$ and $D_2^n$. In particular, the range of x, the spatial component of $D_1^n$, is clearly symmetrical with the range of x of $D_2^n$ with respect to the “light cone” $|x|=ct$. Furthermore, we could find distinct symmetry in the definitions of $U_1^n$ and $U_2^n$, which are the sets of twice continuously differentiable functions we considered in this paper. As can be clearly seen, the main results of this paper, Theorems 2 and 3, are also quite symmetrical to each other.