# Stability of the Wave Equation with a Source

Authors: Soon-Mo Jung; Seungwook Min
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We prove the generalized Hyers-Ulam stability of the wave equation with a source,uttx,t-c2uxxx,t=fx,t, for a class of real-valued functions with continuous second partial derivatives in x and t.


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## Body

## 1. Introduction

The stability problem for functional equations or (partial) differential equations started with the question of Ulam [1]:Under what conditions does there exist an additive function near an approximately additive function? In 1941, Hyers [2] answered the question of Ulam in the affirmative for the Banach space cases. Indeed, Hyers’ theorem states that the following statement is true for all ε≥0: if a function f satisfies the inequality fx+y-fx-fy≤ε for all x, then there exists an exact additive function F such that fx-Fx≤ε for all x. In that case, the Cauchy additive functional equation, fx+y=fx+fy, is said to have (satisfy) the Hyers-Ulam stability.Assume thatV is a normed space and I is an open interval of R. The nth-order linear differential equation(1)anxynx+an-1xyn-1x+⋯+a1xy′x+a0xyx+hx=0is said to have (satisfy) the Hyers-Ulam stability provided the following statement is true for all ε≥0: if a function u:I→V satisfies the differential inequality (2)anxunx+an-1xun-1x+⋯+a1xu′x+a0xux+hx≤εfor all x∈I, then there exists a solution u0:I→V to the differential equation (1) and a continuous function K such that ux-u0x≤Kε for any x∈I and limε→0Kε=0.When the above statement is true even if we replaceε and Kε by φx and Φx, where φ,Φ:I→0,∞ are functions not depending on u and u0 explicitly, the corresponding differential equation (1) is said to have (satisfy) the generalized Hyers-Ulam stability. (This type of stability is sometimes called the Hyers-Ulam-Rassias stability.)These terminologies will also be applied for other differential equations and partial differential equations. For more detailed definitions, we refer the reader to [1–9].To the best of our knowledge, Obłoza was the first author who investigated the Hyers-Ulam stability of differential equations (see [10, 11]): assume that g,r:a,b→R are continuous functions with ∫abgxdx<∞ and ε is an arbitrary positive real number. Obłoza’s theorem states that there exists a constant δ>0 such that yx-y0x≤δ for all x∈a,b whenever a differentiable function y:a,b→R satisfies the inequality y′x+gxyx-rx≤ε for all x∈a,b and a function y0:a,b→R satisfies y0′x+gxy0x=r(x) for all x∈a,b and yτ=y0τ for some τ∈a,b. Since then, a number of mathematicians have dealt with this subject (see [3, 12, 13]).Prástaro and Rassias are the first authors who investigated the Hyers-Ulam stability of partial differential equations (see [14]). Thereafter, the first author [15], together with Lee, proved the Hyers-Ulam stability of the first-order linear partial differential equation of the form, auxx,y+buyx,y+cux,y+d=0, where a,b∈R and c,d∈C are constants with R(c)≠0. As a further step, the first author proved the generalized Hyers-Ulam stability of the wave equation without source (see [16, 17]).One of typical examples of hyperbolic partial differential equations is the wave equation with a spatial variablex and a time variable t,(3)uttx,t-c2uxxx,t=fx,twhere c>0 is a constant, whose solution is a scalar function u=ux,t describing the propagation of a wave at a speed c in the spatial direction.In this paper, applying ideas from [16, 18], we investigate the generalized Hyers-Ulam stability of the wave equation (3) with a source, where x>a and t>b with a,b∈R∪-∞. The main advantages of this present paper over the previous papers [16, 17] are that this paper deals with the wave equation with a source and it describes the behavior of approximate solutions of wave equation in the vicinity of origin while the previous one [17] can only deal with domains excluding the vicinity of origin. (Roughly speaking, a solution to a perturbed equation is called an approximate solution.)


## 2. Main Results

We know that if we introduce the characteristic coordinates(4)ξ=x+ct,η=x-ct,then the wave equation, uttx,t=c2uxxx,t, is transformed into uξηξ,η=0, which seems to be handled easily.Given real constantsa and b with a,b∈R∪-∞, we define (5)U≔x,t∈R×R:x+ct>a,x-ct>b,W≔ξ,η∈R×R:ξ>a,η>b.We note that the map x,t↦ξ,η, where ξ=x+ct and η=x-ct, is a one-to-one correspondence from U onto W (see Figure 1).Figure 1Theorem 1.
Assume thatf:U→R and φ:U→0,∞ are continuous functions with the properties(6)∫∫Ufx,tdxdt<∞,∫∫Uφx,tdxdt<∞.If a function u:U→R has continuous second partial derivatives and satisfies the inequality(7)uttx,t-c2uxxx,t-fx,t≤φx,tfor all (x,t)∈U, then there exists a function v:U→R with continuous second partial derivatives such that v is a solution to the wave equation (3) and(8)ux,t-vx,t≤12c∫∫Ua,b,x,tφp,qdpdqfor all x,t∈U, where Ua,b,x,t is the interior of the parallelogram having the points (9)a+b2,a-b2c,x+a-ct2,-x+a+ct2c,x,t,x+b+ct2,x-b+ct2cas its vertices.Proof.
We introduce the characteristic coordinates (4) and we set(10)wξ,η≔uξ+η2,ξ-η2c=ux,t,gξ,η≔fξ+η2,ξ-η2c=fx,t,ψξ,η≔φξ+η2,ξ-η2c=φx,tfor all x,t∈U and corresponding ξ,η∈W with the relations in (4).
By the chain rule, we get(11)uxx,t=wξξ,η∂ξ∂x+wηξ,η∂η∂x=wξξ,η+wηξ,η,utx,t=wξξ,η∂ξ∂t+wηξ,η∂η∂t=cwξξ,η-cwηξ,ηand hence,(12)uttx,t-c2uxxx,t=∂∂t-c∂∂x∂∂t+c∂∂xux,t=∂∂t-c∂∂x2cwξξ,η=-4c2wξηξ,ηfor all (x,t)∈U and corresponding ξ,η∈W with the relations in (4).
It then follows from (7) and (10) that (13)uttx,t-c2uxxx,t-fx,t=-4c2wξηξ,η-gξ,η≤ψξ,ηor (14)wξηξ,η+14c2gξ,η≤14c2ψξ,ηor (15)-14c2ψξ,η-14c2gξ,η≤wξηξ,η≤14c2ψξ,η-14c2gξ,ηfor any ξ,η∈W.
Considering the conditions in (6) and Figure 2, we can integrate each term of the last inequality from a to ξ with respect to the first variable and then we integrate each term of the resulting inequality from b to η with respect to the second variable to obtain (16)-14c2∫bη∫aξψσ,τdσdτ-14c2∫bη∫aξgσ,τdσdτ≤∫bη∫aξwστσ,τdσdτ≤14c2∫bη∫aξψσ,τdσdτ-14c2∫bη∫aξgσ,τdσdτfor any ξ,η∈W.
If we define the functionz:W→R by (17)zξ,η≔-14c2∫bη∫aξgσ,τdσdτ+wξ,b+wa,η-wa,b,then we have(18)wξ,η-zξ,η≤14c2∫bη∫aξψσ,τdσdτfor all ξ,η∈W. Moreover, we get(19)zηξξ,η=-14c2gξ,η.
We now setvx,t≔zξ,η=zx+ct,x-ct and, analogously to (11), we compute the partial derivatives:(20)vxx,t=zξξ,η+zηξ,η,vxxx,t=zξξξ,η+2zξηξ,η+zηηξ,η,vtx,t=czξξ,η-czηξ,η,vttx,t=c2zξξξ,η-2c2zξηξ,η+c2zηηξ,η.In view of (10), (12), (19), and (20), we get (21)vttx,t-c2vxxx,t=-4c2zξηξ,η=gξ,η=fx,tfor all x,t∈U, that is, v is a solution to wave equation (3).
We compute the Jacobian determinant(22)Jx,t=det⁡∂ξ∂x∂ξ∂t∂η∂x∂η∂t=det⁡1c1-c=-2c.By (10) and (18), we obtain (23)ux,t-vx,t≤14c2∫bη∫aξψσ,τdσdτ=12c∫∫Ua,b,x,tφp,qdpdqfor all (x,t)∈U (see Figure 2).Figure 2Remark 2.
In general, it is somewhat tedious to estimate the upper bound of inequality (8). However, in view of (10) and (18), we can compute the upper bound less tediously: (24)ux,t-vx,t≤∫bη∫aξ14c2ψσ,τdσdτ=∫bx-ct∫ax+ct14c2φσ+τ2,σ-τ2cdσdτfor all (x,t)∈U.Whena=b=-∞ in Theorem 1, U=W=R×R. In that case, by Theorem 1 and Remark 2, we have the following corollary.Corollary 3.
Assume thatf:R×R→R and φ:R×R→0,∞ are continuous functions satisfying the conditions (25)∫-∞∞∫-∞∞fx,tdxdt<∞,∫-∞∞∫-∞∞φx,tdxdt<∞.If a function u:R×R→R has continuous second partial derivatives and satisfies the inequality (26)uttx,t-c2uxxx,t-fx,t≤φx,tfor all x,t∈R, then there exists a function v:R×R→R with continuous second partial derivatives such that v is a solution to the wave equation (3) and (27)ux,t-vx,t≤14c2∫-∞x-ct∫-∞x+ctφσ+τ2,σ-τ2cdσdτfor all x,t∈R.



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*Source: 8274159-2018-06-03.xml*