# Stability of the Diffusion Equation with a Source

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

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## Abstract

We will prove the generalized Hyers-Ulam stability of the (inhomogeneous) diffusion equation with a source,ut(x,t)-k△u(x,t)=f(x,t), for a class of scalar functions with continuous second partial derivatives.


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## Body

## 1. Introduction

The stability problem for functional equations or differential equations started with the well-known question of Ulam [1]:Under what conditions does there exist an additive function near an approximately additive function? In 1941, Hyers [2] gave an affirmative answer to the question of Ulam 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 f(x+y)-f(x)-f(y)≤ε for all x, then there exists an exact additive function F such that f(x)-F(x)≤ε for all x. In that case, the Cauchy additive functional equation, f(x+y)=f(x)+f(y), 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 condition is satisfied 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 u(x)-u0(x)≤K(ε) for any x∈I and limε→0K(ε)=0.When the above statement is true even if we replaceε and K(ε) by φ(x) and Φ(x), 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 of these terminologies, refer to [1–9].To the best of our knowledge, Obloza 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 ∫ab‍|g(x)|dx<∞. Suppose ε is an arbitrary positive real number. Obloza proved that there exists a constant δ>0 such that |y(x)-y0(x)|≤δ for all x∈(a,b) whenever a differentiable function y:(a,b)→R satisfies |y′(x)+g(x)y(x)-r(x)|≤ε for all x∈(a,b) and a function y0:(a,b)→R satisfies y0′(x)+g(x)y0(x)=r(x) for all x∈(a,b) and y(τ)=y0(τ) for some τ∈(a,b). Since then, a number of mathematicians have dealt with this subject (see [3, 12]).Prástaro and Rassias seem to be the first authors who investigated the Hyers-Ulam stability of partial differential equations (see [13]). Thereafter, Jung and Lee [14] proved the Hyers-Ulam stability of the first-order linear partial differential equation of the form (3)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, Hegyi and Jung proved the generalized Hyers-Ulam stability of the diffusion equation on the restricted domain or with an initial condition (see [15, 16]).In this paper, applying ideas from [15, 17], we investigate the generalized Hyers-Ulam stability of the (inhomogeneous) diffusion equation with a source(4)utx,t-k△ux,t=fx,tfor x∈Rn∖0,…,0 and t>0, where k is a positive constant, △=∂2/∂x12+⋯+∂2/∂xn2, and n is a positive integer.The main advantages of this present paper over the previous works [15, 16] are that this paper deals with the inhomogeneous diffusion equation with a source and it describes the behavior of approximate solutions of diffusion equation in the vicinity of origin (roughly speaking, an approximate solution is a solution to a perturbed equation), while the previous works deal with domains not including the vicinity of origin or the homogeneous diffusion equation (without source term).


## 2. Preliminaries

Ifu(x,t) is a solution to the diffusion equation (4) with n=1 and a is a positive constant, then the dilated function w(x,t)≔u(ax,at) satisfies the equality, wt(x,t)-kwxx(x,t)=af(ax,at), for all x>0 and t>0. When the source term f(x,t) satisfies the additional condition(5)afax,at=fx,t,w(x,t) is also a solution to (4) with n=1. This property is called the invariance under dilation. Hence, it is worth searching for approximate solutions to (4), which are scalar functions of the form (6)ux,t=14ktαvx4kt,where α is a real parameter which will be determined later and v is a twice continuously differentiable function. That is, u(x,t) depends on x and t primarily through the term |x|/4kt. We note that the intention of inclusion of the factor 1/4k in the above formula is to simplify our formulations later.Throughout this paper, letn be a fixed positive integer if there is no specification. Each point x in Rn is expressed as x=(x1,…,xi,…,xn), where xi denotes the ith coordinate of x. Moreover, |x| denotes the Euclidean distance of x from the origin; i.e., (7)x=x12+⋯+xi2+⋯+xn2.Based on this argument, we define(8)Uαn≔u:Rn∖0,…,0×0,∞→R∣there  exists  a  twice  continuously  differentiable  functionv:0,∞→Rwithux,t=14ktαvx4ktfor  allx∈Rn∖0,…,0,t>0,where n is a positive integer and α is a parameter.The proof of the following lemma runs in the usual and routine way. Hence, we omit the proof.Lemma 1.
If a functionu belongs to Uαn and a twice continuously differentiable function v:(0,∞)→R is correspondingly given by (9)ux,t=14ktαvx4kt,then (10)utx,t-k△ux,t=-k4ktα+1v′′r+2r+n-1rv′r+4αvrfor all x∈Rn∖0,…,0, t>0, and for all r>0 obeying the relation r=|x|/4kt.Let us define the second-order differential operatorLα2:C2(0,∞)→C(0,∞) by(11)Lα2vr≔v′′r+2r+n-1rv′r+4αvr,where C(0,∞) and C2(0,∞) denote the set of all continuous real-valued functions and the set of all twice continuously differentiable real-valued functions defined on (0,∞), respectively.We now try to decompose the differential operatorLα2 into differential operators LA(r) and LB(r) of first order such that(12)Lα2vr=LAr∘LBrvrfor all v∈C2(0,∞), where we define(13)LArvr≔v′r+Arvr,LBrvr≔v′r+Brvr.Then we have(14)LAr∘LBrvr=v′′r+Ar+Brv′r+ArBr+B′rvr.Comparing both (12) and (14), we obtain (15)Ar+Br=2r+n-1r,ArBr+B′r=4α.From the last system of equations, we get a Riccati equation(16)B′r+2r+n-1rBr-Br2=4α,one of whose solutions has the form (n-1)β/r, where β≠0 is a real constant: If we put Bp(r)=(n-1)β/r in the Riccati equation (16), then we have(17)Bp′r+2r+n-1rBpr-Bpr2=2n-1β+n-1βn-2-n-1βr2.Comparing (16) with (17) and considering that 4α is a constant, we conclude that (18)α=n-22,β=n-2n-1,Bpr=n-2rfor all integers n≥2. (Even if β is not defined for n=1, we can also verify the truth of the formulas for α and Bp(r) for n=1 by a direct calculation.)Using this particular solutionBp(r) and in view of [18, § 1.2.1], the general solution of the Riccati equation (16) with 4α=2n-4 is given by (19)Br=n-2r+rn-3e-r2Cn-∫r0rωn-3e-ω2dω,where r0 is a nonnegative fixed real number, Cn is a constant, and we set Cn=∞ for the particular solution Bp(r)=(n-2)/r.Lemma 2.
Letn be a positive integer. Then (20)Ln-2/22vr=L2r+n-1/r-Br∘LBrvrfor all v∈C2(0,∞), where B(r) is defined in (19).


## 3. Main Results

Before starting with our main theorem, we modify the theorem ([19, Theorem 1]) into a more suitable form for practical applications and we will apply this modified version to the proof of our main theorem (cf. [20, Theorem 2.2]). Indeed, the hypotheses of the original theorem [19, Theorem 1] were formulated with a instead of a0 which impose a constraint on its usability. The proof of Theorem 3 exactly follows the lines of the proof of [19, Theorem 1]. Hence, we omit the proof.Theorem 3 ([19, Theorem 1, Remark 3]).
Assume thatX is a real Banach space and I=(a,b) is an open interval for arbitrary constants a,b∈R∪±∞ with a<b. Let p:I→R and q:I→X be continuous functions such that there exists a constant a0∈[a,b) with the following properties: (i)∫a0t‍p(s)ds exists for each t∈I.(ii)∫a0t‍qyexp⁡∫a0y‍psdsdy exists for any t∈I.
Moreover, assume thatφ:I→[0,∞) is a function such that (iii)∫a0b‍φyexp⁡∫a0y‍psdsdy exists.
If a continuously differentiable functionv:I→X satisfies the differential inequality (21)v′t+ptvt+qt≤φtfor all t∈I, then there exists a unique continuously differentiable function v0:I→X such that v0′(t)+p(t)v0(t)+q(t)=0 for all t∈I and (22)vt-v0t≤exp⁡-∫a0tpsds∫tbφyexp⁡∫a0ypsdsdyfor all t∈I.The following theorem is the main result of this paper which deals with the generalized Hyers-Ulam stability of the diffusion equation (4) when n is an integer larger than 2.Theorem 4.
Letn≥3 be an integer and assume that φ,ψ:(0,∞)→[0,∞) are functions satisfying the conditions(23)∫0∞zez2φzdz<∞and(24)c≔inft>04ktn/2kψt>0.Suppose f:Rn∖0,…,0×(0,∞)→R is a function for which there exists a continuous function g:(0,∞)→R such that(25)fx,t=k4ktn/2gx4ktfor all x∈Rn∖0,…,0 and t>0 and(26)∫0∞zez2gzdz<∞.Assume moreover that the constant Cn in (19) is chosen such that(27)m∫0∞ωn-3e-ω2dω≤Cn<m+1∫0∞ωn-3e-ω2dωfor some integer m≥2. For any function u∈U(n-2)/2n satisfying the inequality(28)utx,t-k△ux,t-fx,t≤φx4ktψtfor all x∈Rn∖0,…,0 and t>0, there exists a solution u0:Rn∖0,…,0×(0,∞)→R of the diffusion equation (4) such that u0∈U(n-2)/2n and(29)ux,t-u0x,t≤ce2/m-1xn-2∫0∞zez2φzdz∫0∞yn-3e-y2dyfor all x∈Rn∖0,…,0 and t>0.Proof.
Sinceu∈U(n-2)/2n, it follows from Lemma 1 with α=(n-2)/2, (25), and (28) that (30)utx,t-k△ux,t-fx,t=k4ktn/2v′′r+2r+n-1rv′r+2n-4vr+gr≤φrψtfor all x∈Rn∖0,…,0 and t>0, where v:(0,∞)→R is the twice continuously differentiable function given in the definition of U(n-2)/2n and we set r=|x|/4kt. In view of (11), (24), and the last inequality, we have (31)Ln-2/22vr+gr=v′′r+2r+n-1rv′r+2n-4vr+gr≤cφrfor all r>0. We here note that {r:x∈Rn∖0,…,0,t>0}=(0,∞).
On account of Lemma2, we further have (32)L2r+n-1/r-Br∘LBrvr+gr≤cφrfor all r>0, where B(r) is defined in (19). If we define a continuously differentiable function w:(0,∞)→R by w(r)≔LBrv(r)=v′(r)+B(r)v(r), then it follows from the last inequality that(33)w′r+2r+n-1r-Brwr+gr≤cφrfor all r>0, where B(r) is given in (19) with a positive real constant r0.
We can now apply Theorem3 to our inequality (33) by considering the substitutions as we see in Table 1.
Our hypothesis thatn is an integer not less than 3 implies that (34)∫0∞ωn-3e-ω2dω<∞.It then follows from (19) and (27) that(35)n-2s+sn-3e-s2m+2∫0∞ωn-3e-ω2dω<Bs<n-2s+sn-3e-s2m-1∫0∞ωn-3e-ω2dω.Hence, we have (36)∫r0rpsds=∫r0r2s+n-1s-Bsds<r2-r02+ln⁡r-ln⁡r0-∫r0rsn-3e-s2dsm+2∫0∞ωn-3e-ω2dω<r2-r02+ln⁡r-ln⁡r0+1m+2,which implies that the condition (i) of Theorem 3 is satisfied.
Moreover, it follows from the last inequality that(37)exp⁡∫r0ypsds<e1/m+2r0er02yey2and, by (26), we get (38)∫r0rexp⁡∫r0ypsdsgydy<e1/m+2r0er02∫r0ryey2gydy<∞for all r>0, which means that the condition (ii) of Theorem 3 is satisfied.
Similarly, it also follows from (23) that (39)∫r0∞cφyexp⁡∫r0ypsdsdy<ce1/m+2r0er02∫r0∞yey2φydy<∞,by which we conclude that the condition (iii) of Theorem 3 is satisfied.
According to Theorem3 (or [19, Theorem 1]) and (33), there exists a unique continuously differentiable function w0:(0,∞)→R such that(40)w0′r+2r+n-1r-Brw0r+gr=0for all r>0 and(41)wr-w0r≤exp⁡-∫r0rpsds∫r∞cφyexp⁡∫r0ypsdsdy=c∫r∞yn-1ey2rn-1er2φyexp⁡-∫ryBsdsdyfor all r>0. In particular, by [19, Theorem 1], w0(r) is explicitly given by (42)w0r=σr0n-1er02rn-1er2exp⁡∫r0rBsds-∫r0rzn-1ez2rn-1er2exp⁡-∫rzBsdsgzdzwith some σ∈R.
By (35), we have (43)-Bs<-n-2s,∫yrBsds<ln⁡ryn-2for any r,y>0 with r≤y. Therefore, since r≤y<∞, it follows from (41) that (44)wr-w0r≤c∫r∞yey2rer2φydyor(45)v′r+Brvr-w0r≤c∫r∞yey2rer2φydyfor all r>0.
We apply Theorem3 to our inequality (45) by considering the substitutions as we see in Table 2.
First, in view of (19) and (35), we get (46)-1m-1<∫r0rsn-3e-s2Cn-∫r0sωn-3e-ω2dωds<1m-1and hence (47)ln⁡rr0n-2-1m-1<∫r0rBsds<ln⁡rr0n-2+1m-1for all r>0, by which we see that the condition (i) of Theorem 3 is satisfied.
Further, it follows from the last inequality that(48)exp∫r0rBsds<rr0n-2e1/m-1for any r,r0>0. By (42) and (48), we easily get(49)w0r≤σr0n-1er02rn-1er2exp⁡∫r0rBsds+∫0∞zn-1ez2rn-1er2exp⁡-∫rzBsdsgzdz≤σr0n-1er02rn-1er2rr0n-2e1/m-1+∫0∞zn-1ez2rn-1er2rzn-2e1/m-1gzdz=e1/m-1rer2σr0er02+∫0∞zez2gzdz.It now follows from (26), (48), and (49) that (50)∫r0rexp⁡∫r0yBsds-w0ydy≤∫0∞exp⁡∫r0yBsdsw0ydy≤e2/m-1r0n-2σr0er02+∫0∞zez2gzdz∫0∞yn-3e-y2dy<∞,which means that the condition (ii) of Theorem 3 is satisfied.
Analogously, it follows from (23) and (48) that (51)∫r0∞cyey2∫y∞zez2φzdzexp⁡∫r0yBsdsdy≤∫r0∞cyey2∫0∞zez2φzdzyn-2r0n-2e1/m-1dy≤ce1/m-1r0n-2∫0∞zez2φzdz∫0∞yn-3e-y2dy<∞,by which we conclude that the condition (iii) of Theorem 3 is satisfied.
Due to Theorem3, (45), and (48), there exists a unique continuously differentiable function v0:(0,∞)→R such that(52)v0′r+Brv0r-w0r=0for all r>0 and(53)vr-v0r≤exp⁡-∫r0rBsds∫r∞cyey2∫y∞zez2φzdzexp⁡∫r0yBsdsdy≤r0n-2e1/m-1rn-2·ce1/m-1r0n-2∫0∞zez2φzdz∫0∞yn-3e-y2dy≤ce2/m-1rn-2∫0∞zez2φzdz∫0∞yn-3e-y2dyfor all r>0. In particular, by [19, Theorem 1], v0(r) is explicitly given by (54)v0r=τexp⁡-∫r0rBsds+exp⁡-∫r0rBsds∫r0rexp⁡∫r0yBsdsw0ydywith some τ∈R. We remark that v0(r) is indeed a twice continuously differentiable function.
Now, let us define the twice continuously differentiable functionu0:Rn∖0,…,0×(0,∞)→R by (55)u0x,t≔14ktn-2/2v0r,where we set r=|x|/4kt. Then, u0∈U(n-2)/2n and inequality (29) follows directly from (53). By Lemma 1 with (n-2)/2 and u0 and v0 in place of α, u, and v, respectively, we further have (56)∂∂tu0x,t-k∑i=1n∂2∂xi2u0x,t=-k4ktn/2v0′′r+2r+n-1rv0′r+2n-2v0rfor all x∈Rn∖0,…,0, t>0, and for all r>0 obeying the relation r=|x|/4kt.
Finally, it follows from Lemma2, (11), (13), (19), (40), and (52) that (57)v0′′r+2r+n-1rv0′r+2n-2v0r=Ln-2/22v0r=L2r+n-1/r-Br∘LBrv0r=L2r+n-1/r-Brw0r=-grfor all r>0. Hence, by (25), we have (58)∂∂tu0x,t-k∑i=1n∂2∂xi2u0x,t=fx,tfor all x∈Rn∖0,…,0 and t>0.Table 1 Theorem3 X t a b a 0 v ( t ) p ( t ) q ( t ) φ ( t ) (33) R r 0 ∞ r 0 w ( r ) 2 r + n - 1 r - B ( r ) g ( r ) c φ ( r )Table 2 Theorem3 X t a b a 0 v ( t ) p ( t ) q ( t ) φ ( t ) (45) R r 0 ∞ r 0 v ( r ) B ( r ) - w 0 ( r ) c  ∫ r ∞ y e y 2 r e r 2 φ ( y ) d yIf we setB(r)=Bp(r)=(n-2)/r in the proof of Theorem 4, i.e., if we set Cn=∞ in (19), then we obtain the following corollary by letting m→∞ in (27) and (29) of Theorem 4.Corollary 5.
Under the same hypotheses and conditions of Theorem4 except condition (27), there exists a solution u0:Rn∖0,…,0×(0,∞)→R of the diffusion equation (4) such that u0∈U(n-2)/2n and (59)ux,t-u0x,t≤cxn-2∫0∞zez2φzdz∫0∞yn-3e-y2dyfor all x∈Rn∖0,…,0 and t>0.We now introduce a concrete example for Theorem4 and Corollary 5 in the following corollary.Corollary 6.
Letn≥3 be an integer and let c,μ1,μ2,ν1,ν2 be positive real constants and let a function u:Rn∖0,…,0×(0,∞)→R belong to U(n-2)/2n. If the function u satisfies the inequality (60)utx,t-k△ux,t-μ2kx4ktn-1/2exp⁡-x24kt-ν2x4kt≤cμ1kx4ktn-1/2exp⁡-x24kt-ν1x4ktfor all x∈Rn∖0,…,0 and t>0, then there exists a solution u0:Rn∖0,…,0×(0,∞)→R of the diffusion equation (4) such that u0∈U(n-2)/2n and (61)ux,t-u0x,t≤cμ1ν1xn-2∫0∞yn-3e-y2dyfor all x∈Rn∖0,…,0 and t>0.Proof.
First, we set(62)φr≔μ1e-ν1rrer2,ψt≔ck4ktn/2,fx,t≔μ2kx4ktn-1/2exp⁡-x24kt-ν2x4kt,gr≔μ2e-ν2rrer2.Then, all the conditions and hypotheses (23), (24), (25), (26), and (28) are fulfilled.
Due to Corollary5, there exists a solution u0:Rn∖0,…,0×(0,∞)→R of the diffusion equation (4) such that u0∈U(n-2)/2n and (63)ux,t-u0x,t≤cxn-2∫0∞zez2φzdz∫0∞yn-3e-y2dy≤cμ1ν1xn-2∫0∞yn-3e-y2dyfor all x∈Rn∖0,…,0 and t>0.


## 4. Discussions and Conclusions

The diffusion equation is sometimes called a heat equation or a continuity equation and it plays an important role in a number of fields of science. For example, the diffusion equation describes the conduction of heat, the signal transmission in communication systems, and diffusion models of chemical diffusion phenomena and it is also connected with Brownian motion in probability theory.This paper was partially motivated by a previous work [21] in which the generalized Hyers-Ulam stability of the one-dimensional wave equation with a source, utt(x,t)-c2uxx(x,t)=f(x,t) was investigated by using the method of characteristic coordinates. On the other hand, we prove in this paper the generalized Hyers-Ulam stability of the n-dimensional diffusion equation with a source, ut(x,t)-k△u(x,t)=f(x,t), by applying a kind of method for decomposition of differential operators.The main advantages of this present paper over the existing results [15, 16] are that this paper deals with the (inhomogeneous) diffusion equation with a source and it can describe the behavior of approximate solutions of (inhomogeneous) diffusion equation in the vicinity of origin, while the previous work [15] deals with the case of domain D=x∈Rn∣a<|x|<b with 0<a<b≤∞ and n≥2, as we see that the domain D does not include the vicinity of origin and while the other existing result [16] deals with the generalized Hyers-Ulam stability of the homogeneous diffusion equation with an initial condition (but without source term).



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