R E S E A R C H
Open Access
On the stability of the heat equation with an
initial condition
Soon-Mo Jung
**Correspondence:
[email protected] Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Korea
Abstract
In this paper, we prove the generalized Hyers-Ulam stability of the heat equation with an initial condition
u(x,t) =ut(x,t) (for allx∈Rnandt> 0),
u(x, 0) =g(x) (for allx∈Rn)
in a class of twice continuously differentiable functions under certain conditions.
1 Introduction
LetXbe a normed space, and letIbe an open interval. If for any functionf:I→X satis-fying the differential inequality
an(x)y(n)(x) +an–(x)y(n–)(x) +· · ·+a(x)y(x) +a(x)y(x) +h(x)≤ε
for allx∈Iand for someε≥ there exists a solutionf:I→Xof the differential equation
an(x)y(n)(x) +an–(x)y(n–)(x) +· · ·+a(x)y(x) +a(x)y(x) +h(x) =
such thatf(x) –f(x) ≤K(ε) for anyx∈I, whereK(ε) is an expression ofεonly, then we
say that the above differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replaceεandK(ε) byϕ(x) and(x), where
ϕ,:I→[,∞) are functions not depending onf andfexplicitly, then we say that the
corresponding differential equation has the generalized Hyers-Ulam stability. (This type of stability is sometimes called the Hyers-Ulam-Rassias stability.)
We may apply these terminologies to other differential equations and partial differential equations. For more detailed definitions of the Hyers-Ulam stability and the generalized Hyers-Ulam stability, refer to [–].
Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [, ]). Here, we introduce the result of Alsina and Ger (see []): If a differentiable functionf :I→Ris a solution of the differential inequality |y(x) –y(x)| ≤ε, whereIis an open subinterval ofR, then there exists a solutionf:I→R
of the differential equationy(x) =y(x) such that|f(x) –f(x)| ≤εfor anyx∈I. This result
was generalized by Miuraet al.(see [, ]).
In , Jung and Lee [] proved the Hyers-Ulam stability of the first-order linear par-tial differenpar-tial equation
aux(x,y) +buy(x,y) +cu(x,y) +d= ,
wherea,b∈Randc,d∈Care constants with(c)= . It seems that the first paper dealing with the Hyers-Ulam stability of partial differential equations was written by Prastaro and Rassias []. For a recent result on this subject, refer to [].
In this paper, using an idea from the papers [, ], we investigate the generalized Hyers-Ulam stability of the heat equation with an initial value condition
⎧ ⎨ ⎩
u(x,t) =ut(x,t) (for allx∈Rnandt> ),
u(x, ) =g(x) (for allx∈Rn) (.)
in the class of radially symmetric functions, wheredenotes the Laplace operator. The heat equation plays an important role in a number of fields of science. It is strongly related to the Brownian motion in probability theory. The heat equation is also connected with chemical diffusion, and it is sometimes called the diffusion equation.
2 Main result
For a given integern≥, xi denotes the ith coordinate of any point xin Rn, i.e., x= (x, . . . ,xi, . . . ,xn). We assume thattis a constant with <t≤ ∞, and we define
T:={t∈R| <t<t} and |x|:=
x
+· · ·+xn.
Due to an idea from [, Section ..], we may search for a solution of (.) of the form
u(x,t) = (/tn/)v(|x|/t/) for some functionv. Based on this argument, we define
U:=
u:Rn×T→R|u(x,t) =
tn/w(r) for allx∈R
n,t∈T
and for some functionw: [,∞)→Rwithr= |x|
t/
and lim
r→∞r
nw(r) = lim r→∞r
n–w(r) = .
Theorem . Letϕ,ϕ: [,∞)→[,∞)andψ:T→[,∞)be functions such that
D:=
∞
eu/ un–
∞
u
sn–ϕ(s)ds du<∞, (.)
c:=inf
t∈Tt
n/+ψ(t) > . (.)
Assume that g:Rn→Ris a function with g∈C(Rn)∩L∞(Rn).If a twice continuously
differentiable function u∈U satisfies
⎧ ⎨ ⎩
|u(x,t) –ut(x,t)| ≤ϕ(t|/x|)ψ(t) (for all x∈Rnand t∈T)
|u(x, ) –g(x)| ≤ϕ(|x|) (for all x∈Rn),
then there exist solutions u,u:Rn×T→Rof the heat equation and a real numberγ such that
u(x,t) –γu(x,t)≤ cD tn/e
–|x|/t, (.)
Rnu(x–y,t)g(y)dy–
γu(x,t)
≤(π)n/cD
Rnu(x–y,t)
g(y)dy (.)
for all x∈Rnand t∈T with|x|/t/> ,where
u(x,t) :=
(πt)n/e
–|x|/t and u
(x,t) :=
Rnu(x–y,t)g(y)dy. (.)
Proof Sinceu(x,t)∈U, there exists a functionw: [,∞)→Rsuch that
u(x,t) =
tn/w(r)
for anyx∈Rnandt∈T, where we setr=|x|/t/. Using this notation, we calculateu
tand u:
ut(x,t) = –
n
tn/+w(r) –
tn/+rw
(r),
uxi(x,t) =
tn/+/w
(r)xi |x|,
uxixi(x,t) =
tn/+/
t/w
(r) xi |x| +w
(r)
|x|– x
i |x|
.
So, we have
u(x,t) –ut(x,t)
=
tn/+
w(r) +n–
r w
(r) +r
w
(r) +n
w(r)
=
rn–
tn/+
rn–w(r) + (n– )rn–w(r)+
rnw(r) +nrn–w(r)
=
rn–
tn/+
rn–w(r) +r n w(r)
for allx∈Rn,t∈T withr> . Moreover, from the last equality and (.), it follows that
u(x,t) –ut(x,t)=
rn–
tn/+
rn–w(r) +r n w(r)
≤ϕ(r)ψ(t)
or
rn–w(r) +r n w(r)
for allr> andt∈T. In view of (.), we have
–crn–ϕ(r)≤
rn–w(r) +r n w(r)
≤crn–ϕ(r)
for anyr> .
We integrate each term of the last inequality fromrto∞and take account of the defi-nition ofUto get
–c
∞
r
sn–ϕ(s)ds≤–rn–w(r) – rn
w(r)≤c
∞
r
sn–ϕ(s)ds
or
w(r) + r w(r)
≤ c
rn–
∞
r
sn–ϕ(s)ds
for allr> .
According to [, Theorem ], together with (.), there exists a (unique) constantγ ∈R
such that
w(r) – γ (π)n/e
–r/
≤ce–r/
∞
r
eu/ un–
∞
u
sn–ϕ(s)ds du
for allr> , or equivalently
u(x,t) – γ (πt)n/e
–|x|/t
≤ c
tn/e
–|x|/t ∞
|x|/t/ eu/ un–
∞
u
sn–ϕ(s)ds du
for allx∈Rnandt∈T with|x|/t/> , which proves the validity of inequality (.), and
in view of (.), it is not difficult to prove thatu(x,t) is a solution of the heat equation, i.e.,u(x,t) –∂∂tu(x,t) = .
If we replacexwithx–yand multiply each term by|g(y)|, and if we integrate all terms in the last inequality overRn, then we obtain
–(π)n/cD
Rnu(x–y,t) g(y)dy
≤
Rn
u(x–y,t)g(y)dy–γ
Rn
u(x–y,t)g(y)dy
≤(π)n/cD
Rnu(x–y,t) g(y)dy
for allx,y∈Rnandt∈T. If we defineu
(x,t) as in (.) for anyx∈Rnandt∈T, following
the proof of [, Theorem in Section .], we can then easily prove that
u(x,t) – ∂
∂tu(x,t) =
for allx∈Rnandt∈Twith|x|/t/> , which shows the validity of inequality (.).
Remark .
(i) The linearity of solutions of heat equation (.) implies thatγu(x,t)andγu(x,t)
(ii) As in the proof of [, Theorem in Section .], we can show that
lim (x,t)→(x,)
x∈Rn,t>
u(x,t) =g(x)
for eachx∈Rn.
(iii) If a functionϕ: [,∞)→[,∞)satisfies
∞
u
sn–ϕ(s)ds=O
une–αu
for allu≥and for someα> /, then there exists a positive numberAsuch that
D=
∞
eu/ un–
∞
u
sn–ϕ(s)ds du
≤ ∞
Aue(/–α)udu
= A
α– <∞,
i.e.,ϕsatisfies condition (.).
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
The author declares that this paper is his original paper.
Acknowledgements
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).
Received: 4 August 2013 Accepted: 1 October 2013 Published:07 Nov 2013
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