Representation of classical solutions to a linear wave equation with pure delay

Universität Konstanz

Representation of classical solutions to a linear wave equation with pure delay

Denys Khusainov Michael Pokojovy Elvin Azizbayov

Konstanzer Schriften in Mathematik Nr. 322, November 2013

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Konstanzer Online-Publikations-System (KOPS)

URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-247552

with Pure Delay

Denys Ya. Khusainov

∗

, Mihael Pokojovy

†

, Elvin I. Azizbayov

‡

Otober3, 2013

Abstrat

Forawaveequationwithpuredelay,westudyaninhomogeneousinitial-boundaryvalueproblem

in abounded 1D domain. Under smoothnessassumptions, weproveuniqueexisteneof lassial

solutions foranygivennite time horizon and givetheir expliit representation. Continuousde-

pendeneonthedatainaweakextrapolatednormisalsoshown.

1 Introdution

The wave equation is a typial linear hyperboli seond-order partial dierential equationwhih nat-

urally arises when modeling phenomena ofontinuum mehanis suh assound, light, water or other

kind ofwavesinaoustis,(eletro)magnetis, elastiityanduiddynamis, et. (f. [6, 13℄). Provid-

ingaratheradequatedesriptionofphysialproesses,partialdierential equations,or equationswith

distributed parameters ingeneral,have foundnumerousappliations inmehanis, mediine,eology,

et. Introduing after-eets suh as delay into these equations has gained a lot of attention over

severalpastdeades. See e.g.,[2 ,3,7,8℄. Mathematialtreatment ofsuhsystemsrequiresadditional

arefulnesssine distributed systemswithdelayoften turnout to be even ill-posed(f. [4 ,5, 12℄).

Inthepresent paper,weonsideraninitial-boundaryvalueproblemfor agenerallinearwaveequation

withpuredelayand onstant oeientsina bounded intervalsubjetto non-homogeneous Dirihlet

boundary onditions. To solve the equation, we employ Fourier's separation method as well as the

speial funtions referred to as delay sine and osine funtions whih were introdued in [9 , 10℄. We

prove the existene of a unique lassial solution on any nite time interval, show its ontinuous

dependene on the dataina very weakextrapolated norm, give its representation asa Fourier series

and prove itsabsolute anduniform onvergene under ertainonditions onthedata.

2 Equation with pure delay

For

T > 0

^,

l > 0

^{, we onsider the following linear wave equation in a bounded interval}

(0, l)

^{with a}

single delaybeinga seondorderpartial dierene-dierential equation foran unknown funtion

η

∂ _tt η(t, x) = a ² ∂ _xx η(t − τ, x) + b∂ _x η(t − τ, x) + dη(t − τ, x) + g(t, x)

^for

(t, x) ∈ (0, T ) × (0, l)

^(2.1)

∗

DepartmentofCybernetis,KyivNationalTarasShevhenkoUniversity,Ukraine

†

DepartmentofMathematisandStatistis,UniversityofKonstanz,Germany

‡

DepartmentofMehanisandMathematis,BakuStateUniversity,Azerbaijan

η(t, 0) = θ ₁ (t), η(t, l) = θ ₂ (t)

^for

t > −τ,

η(t, x) = ψ(t, x)

^for

(t, x) ∈ (−τ, 0) × (0, l).

^(2.2)

Sine we are interested in studying lassial solutions, the following ompatibility onditions are re-

quired toassurefor smoothnessofthesolution on theboundary oftime-spae ylinder

ψ(t, 0) = θ ₁ (t), ψ(t, l) = θ ₂ (t)

^for

t > −τ.

Denition 2.1. Under a lassial solution to the problem (2.1 ), (2.2) we understand a funtion

η ∈ C ⁰ [−τ, T ] × [0, l]

whih satises

∂ _tt η, ∂ _tx η, ∂ _xx η ∈ C ⁰ [−τ, 0] × [0, l]

aswell as

∂ _tt η, ∂ _tx η, ∂ _xx η ∈ C ⁰ [0, T ] × [0, l]

and,being pluggedinto Equations (2.1 ),(2.2 ), turnsthem into identity.

Remark 2.2. The previous denition does not impose any ontinuity of time derivatives in

t = 0

^.

If the ontinuity or even smoothness are desired, additional ompatibility onditions on the data,

inluding

g

^{,arerequired.}

Let

k · k _k,2 := k · k _H k,2 ((0,l))

^,

k ∈ N ₀

, denote the standard Sobolev norm (f. [1℄) and

k · k _−k,2 :=

k · k _H −k,2 ((0,l))

^{denote the norm of} orresponding negative Sobolev spae. We introdue the norm

k · k X :=

s ∞

P

k=0

k · k ² _−k,2

^{and dene the Hilbert spae}

X

^{as a ompletion of}

L ² (0, l)

with respet

to

k · k X

^{. Obviously,}

X ֒ → D (0, l) ′

, i.e.,

X

^{an be} ontinuously embedded into the spae of distributions.

With this notation, we easily see that

A := a ² ∂ _x ² + b∂ _x + d

^(with

∂ _x

^{denoting the} distributional derivative)is abounded linearoperatoron

X

^sine

kAk _L(X) = sup

kuk X =1

kAuk _X = sup

kuk X =1

v u u t

∞

X

k=0

ka ² ∂ _x ² u + b∂ _x u + duk ² _−k,2

≤ sup

kuk X =1

∞

X

k=0

a ² kuk _−k−2,2 + bkuk _−k−1,2 + dkuk _−k,2

≤ sup

kuk X =1

(a ² + b + c)kuk X = a ² + b + d.

Theorem 2.3. Thereexistsaonstant

C > 0

^{,dependentonly on}

a, b, d, l, τ, T

^{,suhthattheestimate}

t∈[0,T] max kη(·, t)k ² _X + kη t (·, t)k ² _X

≤ C kψ(0, ·)k ² _X + kψ t (0, ·)k ² _X + C

Z ₀

−τ

kψ(s, ·)k ² _X + kψ t (s, ·)k ² _X dt+

C Z T

0

kg(s, ·)k ² _X + |θ ₁ (s)| ² + |θ ₂ (s)| ² ds

holds truefor anylassial solutionof Equations(2.1 ), (2.2 ).

Proof. Let

η

^{be the lassialsolution to Equations(2.1 ), (2.2 ). We dene}

w(t, x) :=

( η(t, x)

^for

(t, x) ∈ [−τ, 0] × [0, l],

η(t, x) − θ ₁ (t) − ^x _l (θ ₂ (t) − θ ₁ (t))

^for

(t, x) ∈ (0, T ] × [0, l].

Then

w

^satiseshomogeneous Dirihlet boundary onditions and solvestheequation

∂ _tt w(t, ·) = Aw(t − τ, ·) + f (t, ·)

^(2.3)

inthe extrapolated spae

X

^with

f(t, ·) = g(t, ·) + b (θ 2 (t) − θ ₁ (t)) + θ ₁ (t) + ^dx _l (θ 2 (t) − θ ₁ (t)) .

We multiply the equation with

w _t (t, ·)

^{in thesalar produt of}

X

^{and use Young's inequality to get}

theestimate

∂ _t kw t (t, ·)k ² _X = hAw(t − τ, ·), w t (t, ·)i X + hf (t, ·), w t (t, ·)i X

≤ kw(t − τ, ·)k ² _X +

1 + kAk ² _{L(X )}

kw t (t, ·)k ² _X + kf (t, ·)k ² _X .

(2.4)

As in[11℄, we introdue thehistory variable

z(s, t, x) := w(t − s, x)

^for

(s, t, x) ∈ [0, τ ] × [0, T ] × [0, l]

and obtain

z _t (s, t, x) + z _s (s, t, x) = 0

^for

(s, t, x) ∈ (0, τ ) × (0, T ) × (0, l).

Multiplying these identities with

w(t, ·)

ⁱⁿ

X

^{and performing apartial} integration, we nd

∂ _t Z τ

0

kz(s, t, ·)k ² _X ds = − Z τ

0

∂ _s kz(s, t, ·)k ² _X ds = kw(t, ·)k ² _X − kw(t − τ, ·)k ² _X .

^(2.5)

Addingequations (2.4 ) and (2.5 ) tothetrivial identity

∂ _t kw(t, ·)k ² _X ≤ kw(t, ·)k ² _X + kw _t (t, ·)k ² _X ,

weobtain

∂ _t

kw(t, ·)k ² _X + kw t (t, ·)k ² _X + Z ₀

−τ

kw(s, t, ·)k ² _X ds

≤ 2 + kAk _L(X)

kw(t, ·)k ² _X +kw t (t, ·)k ² _X +kf (t, ·)k ² _X .

Thus,we have shown

∂ _t E(t) ≤ 2 + kAk _L(X)

E(t) + kf (t, ·)k ² _X ,

^(2.6)

where

E(t) := kw(t, ·)k ² _X + kw t (t, ·)k ² _X + Z 0

−τ

kz(s, t, ·)k X ds.

FromEquation(2.6) we onlude

E(t) ≤ E(0) + 2 + kAk _{L(X )} Z _t

0

E(s)ds + Z _t

0

kf (t, ·)k ² _X ds.

Using nowtheintegralform ofGronwall's inequality, we obtain

E(t) ≤ E(0) + Z t

0

kf (t, ·)k ² _X ds + Z t

0

e ( ^2+kAk L(X) ) ^(t−s) E(0) +

Z s 0

kf (ξ, ·)k ² _X dξ

ds

≤ C ˜

E(0) + Z _T

0

kf (s, ·)k ² _X ds

(2.7)

for ertain

C > ˜ 0

^{. Taking into aount}

c ₁ kw(t, ·)k ² _X + |θ ₁ (t)| ² + |θ ₂ (t)| ²

≤ kη(t, ·)k ² _X ≤ C ₁ kw(t, ·)k ² _X + |θ ₁ (t)| ² + |θ ₂ (t)| ² , c ₂ kf (t, ·)k ² _X + |θ ₁ (t)| ² + |θ ₂ (t)| ²

≤ kg(t, ·)k ² _X ≤ C ₂ kf (t, ·)k ² _X + |θ ₁ (t)| ² + |θ ₂ (t)| ²

for some onstants

c ₁ , c ₂ , C ₁ , C ₂ > 0

^{and exploitingthe denitionof}

E(t)

^{,theproof is adiret onse-}

quene of Equation(2.7 ).

(ψ, g, θ ₁ , θ ₂ ) 7→ η

is well-dened,linear andontinuous inthenorms fromTheorem 2.3 .

Remark 2.5. It was essential to onsider the weak spae

X

^{. If the spae} orresponding to the usual wave equationisused, i.e.,

(η, η _t ) ∈ H ₀ ¹ (0, l)

× L ² (0, l)

,there follows from[5℄ thatEquation

(2.1 ), (2.2 ) isan ill-posed problemdue to the lakof ontinuous dependene on thedata even in the

homogeneous ase.

Next, we want to establish onditions on the data allowing for the existene of a lassial solution.

Performing thesubstitution

ξ(t, x) := e ^{− 2a b 2 x} η(t, x)

^for

(t, x) ∈ [−τ, T ] × [0, l]

^(2.8)

with a new unknown funtion

ξ

^{(p. [11℄), the initial boundary value problem (2.1 ), (2.2) an be}

written inthefollowing simpliedform withaself-adjoint operator ontheright-handside

∂ _tt ξ(t, x) = a ² ∂ _xx ξ(t − τ, x) + cξ(t − τ, x) + f (t, x)

^for

(t, x) ∈ (0, T ) × (0, l)

^(2.9)

with

c := d − _4a ^{b 2} 2

omplemented bythefollowing boundaryand initial onditions

ξ(t, 0) = µ ₁ (t), ξ(t, l) = µ ₂ (t)

^for

t > −τ

^with

µ ₁ (t) := θ ₁ (t), µ ₂ (t) := e ^{2a b 2 l} θ ₂ ,

^(2.10)

ξ(t, x) = ϕ(t, x)

^for

(t, x) ∈ (−τ, 0) × (0, l)

^with

ϕ(t, x) := e ^{2a b 2 x} ψ(t, x)

^(2.11)

and

f (t, x) := e ^{2a b 2 x} g(t, x)

^for

(t, x) ∈ [0, T ] × [0, l].

The solutionwill bedetermined intheform

ξ(t, x) = ξ ₀ (t, x) + ξ ₁ (t, x) + G(t, x).

Here,

G

^{is an arbitrary funtion with}

∂ _tt G, ∂ _tx G, ∂ _xx G ∈ C ⁰ [−τ, T ] × [0, l]

satisfying theboundary

onditions

G(t, 0) = µ ₁ (t), G(t, l) = µ ₂ (t).

Assuming

µ ₁ , µ ₂ ∈ C ² [−τ, T ]

,we let

G(t, x) := µ ₁ (t) + ^x _l (µ ₂ (t) − µ ₁ (t))

^for

(t, x) ∈ [−τ, T ] × [0, l].

^(2.12)

• ξ ₀

^solvesthe homogeneous equation

∂ _tt ξ ₀ (t, x) = a ² ∂ _xx ξ ₀ (t − τ, x) + cξ ₀ (t − τ, x)

^(2.13)

subjetto homogeneousboundary andnon-homogeneous initial onditions

ξ ₀ (t, 0) ≡ 0, ξ 0 (t, l) = 0

ⁱⁿ

(−τ, T ),

ξ ₀ (t, x) = Φ(t, x)

^for

(t, x) ∈ (−τ, 0) × (0, l)

^with

Φ(t, x) := ϕ(t, x) − G(t, x).

^(2.14)

In partiular,withthe funtion

G

^{seleted asinEquation(2.12 ), we obtain}

Φ(t, x) = ϕ(t, x) − µ ₁ (t) − ^x _l (µ 2 (t) − µ ₁ (t))

^for

(t, x) ∈ [−τ, 0] × [0, l].

^(2.15)

• ξ ₁

^solvesthe non-homogeneous equation

∂ _tt ξ ₁ (t, x) = a ² ∂ _xx ξ ₁ (t − τ, x) + cξ ₁ (t − τ, x) + F (t, x)

^for

(t, x) ∈ (0, T ) × (0, l)

^(2.16)

with

F(t, x) := a ² ∂ _xx G(t − τ, x) + cG(t − τ, x) − ∂ _tt G(t, x)

^for

(t, x) ∈ [0, T ] × [0, l]

^(2.17)

subjetto homogeneousboundary andinitial onditions. For

G

^{fromEquation (2.12 ),we have}

F(t, x) = f (t, x)+c µ ₁ (t − τ ) + ^x _l (µ ₂ (t − τ ) − µ ₁ (t − τ ))

− µ ¨ ₁ (t) + ^x _l (¨ µ ₂ (t) − µ ¨ ₁ (t))

.

^(2.18)

In this setion, we obtain a formal solution to the initial-boundary value problem (2.13 ) with initial

and boundary onditionsgiveninEquations(2.10),(2.11 ). Weexploit Fourier's separationmethod to

determine

ξ ₀

^{intheprodutform}

ξ ₀ (t, x) = T (t)X(x)

^{. Afterplugging thisansatzinto Equation(2.13),}

wend

X(x) ¨ T (t) = a ² X ^′′ (x)T (t − τ ) + cX(x)T (t − τ ).

Hene,

X(x)

T ¨ (t) − cT (t − τ )

= a ² X ^′′ (x)T (t − τ ).

By formallyseparating the variables,we dedue

X ^′′ (x)

X(x) = T ¨ (t) − cT (t − τ )

a ² T (t − τ ) = −λ ² .

Thus,theequation anbe deoupled asfollows

T ¨ (t) + a ² λ ² − c

T (t − τ ) = 0, X ^′′ (x) + λ ² X(x) = 0.

^(3.1)

These arelinearseondorder ordinary(delay) dierential equations withonstant oeients.

Due to thezero boundary onditions for

ξ ₀

^{,the boundary onditions for theseond equation in(3.1 )}

will also be homogeneous, i.e.,

X(0) = 0, X(l) = 0.

Therefore, we obtain a Sturm &Liouville problem admitting non-trivial solutions only for the eigen-

numbers

λ ² = λ ² _n = ^πn _l n

for

n ∈ N

and theorresponding eigenfuntions

X _n (x) = sin ^πn _l x

for

n ∈ N .

Assuming

πa l

2

− c > 0,

wedenote

ω _n = q

πn a

2

− c

^for

n ∈ N

and onsiderthe rst equationin(3.1 ), i.e.,

T ¨ (t) + ω ² _n T (t − τ ) = 0

^for

n ∈ N .

^(3.2)

The initial onditions foreah of theequations in(3.2 ) an beobtained byexpanding theinitial data

into a Fourier series withrespet totheeigenfuntion basisof theseondequation in(3.1 )

Φ(t, ·) =

∞

X

n=1

Φ _n (t) sin ^πn _l x

with

Φ _n (t) = 2 l

Z l 0

(ϕ(t, s) − G(t, s)) sin ^πn _l s ds,

∂ _t Φ(t, ·) =

∞

X

n=1

˙Φ n (t) sin ^πn _l x

with

Φ ˙ n (t) = 2 l

Z l 0

(∂ t ϕ(t, s) − ∂ _t G(t, s)) sin ^πn _l s ds

(3.3)

for

t ∈ [−τ, T ]

^{. Let us further determine the solution of theCauhy problem assoiated witheah of}

theequations in(3.2 ) subjetto theinitial onditionsfrom Equation(3.3 ).

with pure delay obtained in [9 ℄. The authors onsidered a linear homogeneous seond order delay

dierential equation

¨

x(t) + ω ² x(t − τ ) = 0

^for

t ∈ (0, ∞), x(t) = β(t)

^for

t ∈ [−τ, 0].

^(3.4)

Theyintrodued two speialfuntionsreferredto asdelayosine and sinefuntions. Exploitingthese

funtions, a uniquesolution to the initial valueproblem(3.4 ) wasobtained.

Denition 3.1. Delayosine isthe funtion given as

cos τ (ω, t) =



 

 

 



 

 

 



0, −∞ < t < −τ,

1, −τ ≤ t < 0,

1 − ω ^{2 t} _2! ² , 0 ≤ t < τ,

.

.

.

.

.

.

1 − ω ^{2 t} _2! ² + ω ^{4 ( t−τ)} _4! ⁴ − · · · + (−1) ^k ω ^{2k (t−(k−1)τ)} _(2k)! ^2k , (k − 1)τ ≤ t < kτ

(3.5)

with

2k

^-order polynomials on eah of the intervals

(k − 1)τ ≤ t < kτ

ontinuously adjusted at the nodes

t = kτ

^,

k ∈ N ₀

.

0 5 10 15

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

t co s τ (0 . 5 , t )

τ = 0.1 τ = 0.05 τ = 0 . 5

0 5 10 15

−10

−5 0 5 10 15

t co s τ (1 , t )

τ = 0.1 τ = 0.05 τ = 0 . 5

0 5 10 15

−1000

−500 0 500 1000 1500 2000 2500

t co s τ (2 , t )

τ = 0.1 τ = 0.05 τ = 0 . 5

Figure1: Delayosine funtion

Denition 3.2. Delaysine isthe funtion givenas

cos _τ (ω, t) =



 

 

 



 

 

 



0, −∞ < t < −τ,

ω(1 + τ ), −τ ≤ t < 0,

ω(1 + τ ) − ω ^{3 t} _3! ³ , 0 ≤ t < τ,

.

.

.

.

.

.

ω(1 + τ ) − ω ^{3 t} _3! ³ + · · · + (−1) ^k ω ^{2k+1 ( t−(k−1)τ)} _(2k+1)! ^2k+1 , (k − 1)τ ≤ t < kτ

(3.6)

with

(2k + 1)

^-order polynomials on eah of the intervals

(k − 1)τ ≤ t < kτ

ontinuously adjusted at thenodes

t = kτ

^,

k ∈ N ₀

.

0 5 10 15

−2

−1.5

−1

−0.5 0 0.5 1 1.5

t si n τ (0 . 5 , t )

τ = 0.1 τ = 0.05 τ = 0 . 5

0 5 10 15

−15

−10

−5 0 5 10

t si n τ (1 , t )

τ = 0.1 τ = 0.05 τ = 0.5

0 5 10 15

−2000

−1000 0 1000 2000 3000 4000 5000

t si n τ (2 , t )

τ = 0.1 τ = 0.05 τ = 0 . 5

Figure2: Delaysine funtion

There has furtherbeen provedthat delayosine uniquely solvesthelinearhomogeneous seondorder

ordinarydelaydierentialequationwithpuredelaysubjettotheunitinitialonditions

x ≡ 1

ⁱⁿ

[−τ, 0]

and thedelaysine inits turnsolvesEquation(3.4) subjettotheinitial onditions

x(t) = ω(t + τ )

^for

t ∈ [−τ, 0]

^.

Using the fat above, the solution of the Cauhy problem was represented in the integral form. In

partiular,thesolution

x

^tothehomogeneousdelaydierentialequation(3.4 )withtheinitialonditions

x ≡ β

ⁱⁿ

[−τ, 0]

^{for anarbitrary}

β ∈ C ² ([−τ, 0])

^{wasshownto be given as}

x(t) = β(−τ ) cos _τ (ω, t) + _ω ¹ β(−τ ˙ ) sin _τ (ω, t) + _ω ¹

Z ₀

−τ

sin _τ (ω, t − τ − s) ¨ β(s)ds.

^(3.7)

Turning bak to the delaydierential equation (3.2) withtheinitial onditions (2.4 ), we obtain their

unique solution inthe form

T _n (t) = Φ _n (−τ ) cos _τ (ω _n , t) + _ω ¹

n ˙Φ n (−τ ) sin _τ (ω _n , t) + _ω ¹

n

Z ₀

−τ

sin _τ (ω _n , t − τ − s) ¨ Φ(s)ds.

^(3.8)

Thus,assuming suient smoothnessof the datato be speied later,thesolution

ξ ₀

^tothehomoge-

neousequation(2.13 )satisfyinghomogeneousboundaryandnon-homogeneousinitialonditions

ξ ≡ Φ

in

[−τ, 0] × [0, l]

^{reads as}

ξ ₀ (t, x) =

∞

X

n=1

Φ _n (−τ ) cos _τ (ω _n , t) + _ω ¹

n

Φ ˙ _n (−τ ) sin _τ (ω _n , t)+

1 ω n

Z 0

−τ

sin τ (ω n , t − τ − s) ¨ Φ n (s)ds

sin ^πn _l x , Φ n (t) = 2

l Z l

0

(ϕ(t, s) − G(t, s)) sin ^πn _l s

ds

^for

n ∈ N .

(3.9)

Next,weonsiderthenon-homogeneous equation(2.16 )withtheright-handsidefromEquation(2.18 )

subjetto homogeneousinitial and boundaryonditions

∂ _tt ξ ₁ (t, x) = a ² ∂ _xx ξ ₁ (t − τ, x) + cξ ₁ (t − τ, x) + F (t, x)

^for

(t, x) ∈ (0, T ) × (0, l),

where

F (t, x) = f (t, x) + c µ ₁ (t − τ ) + ^x _l (µ ₂ (t − τ ) − µ ₁ (t − τ ))

− µ ¨ ₁ (t) − ^x _l (¨ µ ₂ (t) − µ ¨ ₁ (t)) .

The solution will be onstruted as asFourier series with respet to theeigenfuntions of the Sturm

&Liouville problemfrom theprevious setion,i.e.,

ξ ₁ (t, x) =

∞

X

n=1

T _n (t) sin ^πn _l x

.

^(4.1)

Plugging theansatz from (4.1) into Equation (2.6) and omparingthe time-dependent Fourier oe-

ients, we obtaina systemofountablymany seondorder delaydierential equations

T ¨ _n (t) + ω _n T _n (t − τ ) = F _n (t)

^for

t ∈ (0, T )

^with

F _n (t) = 2 l

Z l 0

F (t, s) sin ^πn _l x

ds.

^(4.2)

In [9℄,theinitial valueproblemfor thenon-homogeneous delay dierential equation

¨

x(t) + ω ² x(t − τ ) = f (t)

^for

t ≥ 0

withhomogeneous initial onditions

x ≡ 0

ⁱⁿ

[−τ, 0]

^{wasshownto be uniquely solved by}

x(t) =

Z t 0

sin τ (ω, t − τ − s)f (s)ds.

^(4.3)

Exploiting Equation(4.3 ),the equations in(4.2 ) subjetto zeroinitial onditions areuniquely solved

by

T _n (t) = Z t

0

sin _τ (ω _n , t − τ − s)F _n (s)ds.

^(4.4)

Therefore, the non-homogeneous partial delay dierential equation with homogeneous initial and

boundary onditionformallyreads as

ξ ₁ (t, x) =

∞

X

n=1

Z t 0

sin τ (ω n , t − τ − s)F n (s)ds

sin ^πn _l x

for

(t, x) ∈ [0, T ] × [0, l].

^(4.5)

5 General ase solution

The solutioninthe general ase an thus formally be representedasthefollowing series

ξ(t, x) =

∞

X

n=1

Φ _n (−τ ) cos _τ (ω _n , t) + _{ω n} ¹ ˙Φ n (−τ ) sin _τ (ω _n , t)+

1 ω n

Z 0

−τ

sin τ (ω n , t − τ − s) ¨ Φ n (s)ds

sin ^πn _l x +

∞

X

n=1

Z t 0

sin τ (ω n , t − τ − s)F n (s)ds

sin ^πn _l x

+ G(t, x).

(5.1)

Theorem 5.1. Let

T > 0

^,

τ > 0

^and

m := ⌈ ^T _τ ⌉

^{. Further, let the datafuntions}

ϕ

^,

µ ₁

^,

µ ₂

^and

f

^be

suhthattheir Fourier oeients

Φ _n

^and

F _n

^{giveninEquations(3.3 )and(4.5 ) satisfytheonditions}

n→∞ lim

|Φ _n (−τ )| + | Φ ˙ _n (−τ )|

n ^2m+3+α = 0, lim

n→∞ max

s∈[−τ,0] | Φ ¨ _n |n ^2m+3+α = 0,

n→∞ lim max

k=1,...,m max

t∈[(k−1)τ,max{kτ,T }] |F n (t)|n ^2k+3+α = 0

(5.2)

for an arbitrary,but xed

α > 0

^{. Let}

π l a 2

> c.

Then thelassialsolutionto problem(2.9 )(2.11 ) anberepresentedasan absolutelyand uniformly

onvergent FourierseriesgiveninEquation(5.1 ). Thelatterseriesisatwieontinuouslydierentiable

funtion withrespetto both variables. Itsderivativesoforderlessorequal twowithrespetto

t

^and

x

^{anbeobtainedbyaterm-wise}dierentiationoftheseriesandtheresultingseriesarealsoabsolutely and uniformlyonvergent in

[0, T ] × [0, l]

^.

Proof. Weregroup the series fromEquation(5.1) into the following sum

ξ(t, x) = S ₁ (t, x) + S ₂ (t, x) + S ₃ (t, x) + G(t, x),

where

S ₁ (t, x) =

∞

X

n=1

A n (t) sin ^πn _l x

, S ₂ (t, x) =

∞

X

n=1

B n (t) sin ^πn _l x

, S ₃ (t, x) =

∞

X

n=1

C n (t) sin ^πn _l x , A _n (t) = Φ _n (−τ ) cos _τ (ω _n , t) + _ω ¹

n ˙Φ n (−τ ) sin _τ (ω _n , t), B _n (t) = _ω ¹

n

Z 0

−τ

sin _τ (ω _n , t − τ − s) ¨ Φ _n ds, C _n (t) = _ω ¹

n

Z 0

−τ

sin τ (ω n , t − τ − s)F n (s)ds

and

ω _n = q

πn l a 2

− c

^for

n ∈ N .

1. First,weonsidertheoeient funtions

A _n

^{. Foranarbitrary}

t ∈ [0, T ]

^with

(k − 1)τ ≤ t < kτ

^,

we nd

A _n (t) = Φ _n (−τ ) cos _τ (ω _n , t) + _ω ¹

n ˙Φ n (−τ ) sin _τ (ω _n , t)

=

1 − ^πn _l a 2 t ³

2! + · · · + (−1) ^{k πn} _l a 2k (t−(k−1)τ) ^2k (2k)!

Φ _n (−τ )+

(1 + τ ) − ^πn _l a 2 t ²

3! + · · · + (−1) ^{k πn} _l a 2k (t−(k−1)τ ) ^2k+1 (2k+1)!

Φ n (−τ ).

If

Φ n (−τ )

^and

˙Φ n (−τ )

^,

n ∈ N

,aresuhthat the ondition

n→∞ lim

|Φ _n (−τ )| + | ˙Φ n (−τ )|

n ^2k+3+α = 0

holdstrue, the series

S ₁

^{aswellasits derivativesoforderlessorequal 2onvergeabsolutelyand}

uniformly. Note thatasingledierentiation withrespetto

x

orresponds,roughlyspeaking, to a multipliationwith

n

^.

2. Next, we onsider the oeients

B _n

^{. For an arbitrary}

t ∈ [0, T ]

^with

(k − 1)τ ≤ t < kτ

^{, we}

perform the substitution

t − τ − s = ξ

^{and exploitthe meanvaluetheoremto estimate}

|B _n (t)| =

1 ω n

Z t t−τ

sin _τ (ω _n , ξ) ¨ Φ _n (t − τ − ξ)dξ

≤ τ max

−τ≤s≤0 | Φ ¨ _n (s)| max

j=k−1,k max

t−τ≤s≤t

(s − τ ) − ^πn _l a 2 s ³ 3! + . . . +(−1) ^{j πn} _l a 2j (s−(j−1)τ) ^2j+1

(2j+1)!

.

Applying the theorem on dierentiation under the integral sign to

B _n

^{and taking into aount}

that

sin τ πn l a, ·

is twieweakly dierentiable in

[0, ∞)

^,namely:

sin τ πn l a, ·

∈ W _loc ^2,∞ (0, ∞)

,

its derivatives are polynomials of order lower than those of

sin _τ ^πn _l a, ·

and their onvolution

with

Φ ¨ n

^{isontinuous, analogous estimates an be obtainedfor}

˙Φ n

^and

Φ ¨ n

^{whih,intheir turn,}

also followto be ontinuousfuntions.

Now, ifthe ondition

n→∞ lim max

s∈[−τ,0] | Φ ¨ _n |n ^2m+3+α = 0

issatised,theseries

S ₂

^{aswellasits derivativesoforderlessorequal2onvergeabsolutelyand}

uniformly.

3. Finally,we lookat theFourier oeients

C _n

^{. Again, for anarbitrary period of time}

t ∈ [0, T ]

with

(k − 1)t ≤ t < kτ

^,

0 ≤ k ≤ m

^{, we substitute}

t − τ − ξ = s

^{. One again, using the mean}

valuetheorem, we estimate

|C n (t)| =

1 ω n

Z t t−τ

sin τ πn l a, ξ

F _n (t − τ − ξ)dξ

≤ τ max

t−τ≤s≤t | Φ ¨ n (s)| max

j=k−1,k max

t−τ≤s≤t

(s − τ ) − ^πn _l a 2 s ³ 3! + . . . +(−1) ^{j πn} _l a 2j (s−(j−1)τ) ^2j+1

(2j+1)!

.

Asbefore,

C _n

^{an beshown to be twie} ontinuouslydierentiable. If now

n→∞ lim max

k=1,...,m max

t∈[(k−1)τ,max{kτ,T }] |F _n (t)|n ^2k+3+α = 0

^(5.3)

is satised, then both

S ₃

^{and its} derivatives of order less or equal 2 onverge absolutely and uniformly.

Sine all three onditions areguaranteed bythe assumptions of theTheorem due to thefat

k ≤ m

^,

theproof isnished.

Remark 5.2. From the pratial point of view,the rapid deayondition onthe Fourier oeients

of thedata given in Equation(5.2 ) mean asuiently highSobolev regularityof thedata and orre-

sponding higherorder ompatibilityonditions at theboundaryof

(0, l)

^{(f. [11℄).}

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