S udies in Applied Ma hema ics
ORIGINAL ARTICLE
On he Double Moo e–Gibson–Thompson Sys em o
The mo iscoelas ici y
Filippo Dell’O o1Lo enzo Li e ani2Vi o ino Pa a1Ramon Quin anilla3
1Poli ecnico di Milano, Dipa imen o di Ma ema ica, Milano, I aly 2Depa men o Da a Science, F ied ich-Alexande -Uni e si ä E langen-Nü nbe g,
E langen, Ge many 3Depa amen de Ma emà iques, Uni e si a Poli ècnica de Ca alunya, Te assa, Ba celona, Spain
Co espondence: Filippo Dell’O o ([email p o ec ed])
Recei ed: 13 May 2024 Re ised: 10 Oc obe 2024 Accep ed: 23 Oc obe 2024
Funding: L.L. has been suppo ed by he Alexande on Humbold Founda ion. F.D. and V.P. ha e been pa ially suppo ed by he I alian MIUR-PRIN G an
2020F3NCPX “Ma hema ics o indus y 4.0 (Ma h4I4)”.
Keywo ds: exponen ial s abili y | Moo e–Gibson–Thompson sys em | solu ion semig oup | he mo iscoelas ici y
ABSTRACT
In his pape , we add ess he sys em made by wo coupled one-dimensional Moo e–Gibson–Thompson equa ions
{
𝑢+𝛼
𝑢−𝛽
𝑢𝑥𝑥 −𝛾𝑢𝑥𝑥 =𝑝(𝛼
𝑤𝑥+
𝑤𝑥)
𝑤+
𝛼
𝑤−
𝛽
𝑤𝑥𝑥 −
𝛾𝑤𝑥𝑥 =
𝑝(
𝛼
𝑢𝑥+
𝑢𝑥)
a ising in he desc ip ion o he mo iscoelas ic ma e ials. He e, 𝛼,𝛽,𝛾,
𝛼,
𝛽,
𝛾>0while 𝑝
𝑝>0. When bo h he MGT equa ions lie
in he subc i ical egime, ha is,
𝛽−𝛾
𝛼>0and
𝛽−
𝛾
𝛼>0
we p o e ha he sys em gene a es an exponen ially s able solu ion semig oup. This imp o es some ecen esul s in he li e a u e,
whe e he exponen ial s abili y is a ained only wi hin ei he a s onge condi ion han subc i icali y o bo h equa ions, o when 𝛼
and
𝛼a e su icien ly close. The key idea is o deduce he exponen ial s abili y om ha o a ela ed sys em, made by wo coupled
equa ions o he iscoelas ici y ype. The la e esul has also an independen in e es .
2010 MATHEMATICS SUBJECT CLASSIFICATION: 35B40, 35Q74, 74D05, 74F05
1P eamble
The mo iscoelas ici y deals wi h ma e ials ha , besides ha ing
a endency o eco e hei o iginal s a e (elas ic e ec s), also
exhibi mechanical iscous e ec s, as well as he mal dissipa ion.
Di e en desc ip ions o iscoelas ici y can be ound in he
li e a u e, such as he Kel in–Voig o he Zene ones, he la e
o en e e ed o as iscoelas ici y o Moo e–Gibson–Thompson
ype, o models depending on he his o y o he de o ma ion
g adien . Likewise, se e al di e en ypes o he mal dissipa ion
ha e been p oposed h ough he yea s, mainly o o e come he
pa adox o he ins an aneous p opaga ion o he mal wa es,
ypical o he classical Fou ie law [1, 2]. Wi h no claim o be
exhaus i e, we jus ecall he hea laws o Maxwell–Ca aneo
[3, 4], Lo d-Shulman [5], Gu in–Pipkin [6], G een–Lindsay [7],
G een–Naghdi [8–10], Moo e–Gibson–Thompson [11–13].
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
© 2024 The Au ho (s). S udies in Applied Ma hema ics published by Wiley Pe iodicals LLC.
S udies in Applied Ma hema ics,2025;154:e12784
h ps://doi.o g/10.1111/sapm.12784
1o 10
In his pape , we ocus on he mo iscoelas ic ma e ials whe e
he dissipa i e e ec s, bo h iscous and he mal, a e desc ibed by
he Moo e–Gibson–Thompson (MGT he ea e ) equa ion which,
w i eninanabs ac o m, eads
𝑢+𝛼
𝑢+𝛽𝖠
𝑢+𝛾𝖠𝑢 =0,
in he unknown 𝑢=𝑢(𝑡). He e, 𝖠is a s ic ly posi i e sel -
adjoin ope a o on some Hilbe space, while 𝛼,𝛽,𝛾 >0a e
gi en pa ame e s, he do s anding o de i a i e wi h espec o
ime. Al hough o iginally in oduced by S okes in he mid-19 h
cen u y [14], he MGT equa ion is named a e he wo ks [15,
16], and plays a pa amoun ole in he desc ip ion o se e al
physical phenomena (see, e.g., [17–19], and e e ences he ein).
Fo ins ance, om he mechanical poin o iew, i se es as a
model o he Zene heo y o iscous ma e ials, whe eas om
he he mal poin o iew i can be ob ained ia he in oduc ion
o a elaxa ion ime in he G een-Naghdi ype III hea conduc ion
heo y. Ano he way o end up o wi h he MGT equa ion is
o conside ce ain second-o de equa ions wi h memo y (e.g.,
he equa ion o iscoelas ici y), o he pa icula choice o he
nega i e exponen ial ke nel [20]. This will be pa icula ly ele an
in connec ion wi h his pape .
I is well known ha he asymp o ic beha io o he solu ions
o he MGT equa ion is s ongly in luenced by he alues o he
pa ame e s 𝛼,𝛽,𝛾. Indeed, de ining he s abili y numbe
𝜘=𝛽−𝛾
𝛼,
he solu ions decay o ze o i and only i 𝜘>0, which de ines he
so-called subc i ical egime o he model (see, e.g., [21, 22]). In
ac , since he desc ibed phenomena ( iscoelas ici y and he mal
conduc ion) a e in insically dissipa i e, i will be o in e es o
ou scopes o conside he case 𝜘>0only.
2The Double MGT Sys em
The main objec o ou in es iga ion is he sys em made by wo
one-dimensional MGT equa ions coupled oge he , namely,
{
𝑢+𝛼
𝑢−𝛽
𝑢𝑥𝑥 −𝛾𝑢𝑥𝑥 =𝑝(𝛼
𝑤𝑥+
𝑤𝑥),
𝑤+
𝛼
𝑤−
𝛽
𝑤𝑥𝑥 −
𝛾𝑤𝑥𝑥 =
𝑝(
𝛼
𝑢𝑥+
𝑢𝑥), (2.1)
in he unknowns 𝑢=𝑢(𝑥, 𝑡) and 𝑤=𝑤(𝑥,𝑡),wi h(𝑥, 𝑡) ∈
(0, 𝜋) ×ℝ+,whe e𝛼,𝛽,𝛾 and hei co esponding ha s a e s ic ly
posi i e cons an s, while 𝑝,
𝑝∈ℝ ul ill 𝑝
𝑝>0. Clea ly, he
in e al (0, 𝜋) is aken jus o con enience, and could be eplaced
by any o he bounded open in e al. Sys em (2.1) is subjec o he
homogeneous Di ichle bounda y condi ions
𝑢(0,𝑡) =𝑢(𝜋,𝑡) =𝑤(0,𝑡) =𝑤(𝜋,𝑡) =0, (2.2)
and is supplemen ed wi h he ini ial condi ions a he ini ial ime
𝑡=0
⎧
⎪
⎨
⎪
⎩
𝑢(𝑥, 0) =𝑢0(𝑥),
𝑢(𝑥, 0) =𝑢1(𝑥),
𝑢(𝑥, 0) =𝑢2(𝑥), ⎧
⎪
⎨
⎪
⎩
𝑤(𝑥,0) =𝑤0(𝑥),
𝑤(𝑥,0) =𝑤1(𝑥),
𝑤(𝑥,0) =𝑤2(𝑥),
(2.3)
whe e 𝑢𝚤and 𝑤𝚤a e assigned unc ions on [0, 𝜋].Wealso equi e
ha he s abili y numbe s o he wo in ol ed MGT equa ions be
s ic ly posi i e, o wi ,
𝜘=𝛽−𝛾
𝛼>0and 𝜘=
𝛽−
𝛾
𝛼>0. (2.4)
This choice is consis en wi h he physical meaning o he wo
equa ions, which, as we will see sho ly, a e bo h dissipa i e.
Rema k 2.1. Sys em (2.1)isessen ially one-dimensional due o
he pa icula na u e o he coupling, which in ol es he i s
de i a i e in space. In ac , eplacing 𝜕𝑥wi h (−Δ)1∕2, al hough
in his case he physical meaning is less clea , he analysis ca ied
ou in his pape ex ends e ba im o he 𝑁-dimensional case.
3 De i a ion o he Physical Model
Gi en a he mo iscoelas ic ba o linea mass densi y 𝜌>0
occupying he in e al [0, 𝜋], he equa ions uling he e olu ion
o he ans e sal displacemen 𝑢=𝑢(𝑥,𝑡) and he en opy 𝜂=
𝜂(𝑥,𝑡),wi h(𝑥, 𝑡) ∈ (0, 𝜋) ×ℝ, ead
{𝜌
𝑢=𝗍𝑥,
𝜏
𝜂=𝗊𝑥,(3.1)
he subsc ip 𝑥s anding o space de i a i e, whe e 𝗍=𝗍(𝑥, 𝑡) is
he s ess, 𝗊=𝗊(𝑥, 𝑡) is he hea lux, and 𝜏>0is he e e ence
empe a u e which, wi hou loss o gene ali y, will be con en-
ionally se equal o 1 he ea e . We will conside dissipa i e
e ec s ( iscous and he mal) which a e due o he in luence
o he pas his o y o he ma e ial on he u u e dynamics.
F om he classical axioms o he momechanics, i is no ac ually
immedia e o p opose a he mo iscoelas ic heo y complying
wi h such an assump ion. In his espec , one o he bes and i s
appea ed app oaches is he linea heo y o Gu in [23], based
on he in a iance o en opy unde ime e e sal. The dis inc i e
cha ac e o his heo y lies in he choice o he cons i u i e
equa ions o 𝗍,𝜂,and𝗊, which, o he case o cen osymme ic
ma e ials, ake he o m (omi ing he dependence on he space
a iable)
𝗍=∫𝑡
−∞ [𝑓(𝑡 −𝑠)
𝑢𝑥(𝑠) +𝑎(𝑡 −𝑠)
𝜃(𝑠)]𝑑𝑠,
𝜂=∫𝑡
−∞ [−𝑎(𝑡 −𝑠)
𝑢𝑥(𝑠) +𝑏(𝑡 −𝑠)
𝜃(𝑠)]𝑑𝑠,
𝗊=∫𝑡
−∞
𝑔(𝑡 −𝑠)𝜃𝑥(𝑠)𝑑𝑠.
He e, 𝜃=𝜃(𝑥,𝑡) is he ela i e empe a u e, while he con olu-
ion ke nels 𝑓,𝑔,𝑎,𝑏, a e posi i e con ex dec easing unc ions,
ha ing s ic ly posi i e limi a in ini y, which do no depend on
he ma e ial poin 𝑥. In ac , being a coupling e m, he ole o
𝑎and −𝑎can be in e changed. Howe e , in his wo k we shall
assume ha he unc ions 𝑎and 𝑏a e cons an : namely, 𝑎≠0and
𝑏>0. Acco dingly, wi hin he physically easonable hypo heses
ha bo h 𝑢and 𝜃 anish as 𝑡→−∞, plugging he cons i u i e
2o 10 S udies in Applied Ma hema ics,2025
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equa ions in o Equa ion (3.1) we deduce he sys em
⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝜌
𝑢−∫𝑡
−∞
𝑓(𝑡 −𝑠)
𝑢𝑥𝑥(𝑠)𝑑𝑠 =𝑎𝜃𝑥,
𝑏
𝜃−∫𝑡
−∞
𝑔(𝑡 −𝑠)𝜃𝑥𝑥(𝑠)𝑑𝑠 =𝑎
𝑢𝑥.
(3.2)
A his poin , bo owing he concep om G een and Naghdi [8–
10], we in oduce he he mal displacemen
𝑤(𝑥,𝑡) =𝑤(𝑥,0) +∫𝑡
0
𝜃(𝑥,𝑠)ds,
hence sa is ying he ela ion
𝑤=𝜃. Wi h his posi ion, sys-
em (3.2) akes he mo e symme ic o m
⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝜌
𝑢−∫𝑡
−∞
𝑓(𝑡 −𝑠)
𝑢𝑥𝑥(𝑠)𝑑𝑠 =𝑎
𝑤𝑥,
𝑏
𝑤−∫𝑡
−∞
𝑔(𝑡 −𝑠)
𝑤𝑥𝑥(𝑠)𝑑𝑠 =𝑎
𝑢𝑥.
(3.3)
The nex s ep is o pe o m an in eg a ion by pa s, oge he wi h
a change o a iables, yielding he equali ies
∫𝑡
−∞
𝑓(𝑡 −𝑠)
𝑢𝑥𝑥(𝑠)𝑑𝑠 =𝑓(∞)𝑢𝑥𝑥 −∫∞
0
𝑓′(𝑠)[𝑢𝑥𝑥(𝑡) −𝑢𝑥𝑥(𝑡 −𝑠)]𝑑𝑠,
∫𝑡
−∞
𝑔(𝑡 −𝑠)
𝑤𝑥𝑥(𝑠)𝑑𝑠 =𝑔(∞)𝑤𝑥𝑥 −∫∞
0
𝑔′(𝑠)[𝑤𝑥𝑥(𝑡) −𝑤𝑥𝑥(𝑡 −𝑠)]𝑑𝑠,
he p ime deno ing he de i a i e wi h espec o 𝑠. Then, de ining
he dec easing (and summable o physical easons) nonnega i e
unc ions
𝜇(𝑠) =−
1
𝜌𝑓′(𝑠) and
𝜇(𝑠) =−
1
𝑏𝑔′(𝑠),
and se ing
𝜚=1
𝜌𝑓(∞),
𝜚=1
𝑏𝑔(∞), 𝑝 =𝑎
𝜌,
𝑝=𝑎
𝑏,
we ob ain om Equa ion (3.3)
⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝑢−𝜚𝑢𝑥𝑥 −∫∞
0
𝜇(𝑠)[𝑢𝑥𝑥(𝑡) −𝑢𝑥𝑥(𝑡 −𝑠)]𝑑𝑠 =𝑝
𝑤𝑥,
𝑤−
𝜚𝑤𝑥𝑥 −∫∞
0
𝜇(𝑠)[𝑤𝑥𝑥(𝑡) −𝑤𝑥𝑥(𝑡 −𝑠)]𝑑𝑠 =
𝑝
𝑢𝑥.
(3.4)
Sys em (3.4) is made by wo equa ions o he iscoelas ici y
ype (al hough he e he second equa ion has a di e en physical
meaning) coupled oge he , whe e all he physical pa ame e s in
play a e ee. Ac ually, he mos signi ican con olu ion ke nels 𝜇
and
𝜇 om he physical iewpoin a e he nega i e exponen ials.
Acco dingly, le us ake wo ke nels o he o m
𝜇(𝑠) =𝜅
𝜀e−𝑠
𝜀and
𝜇(𝑠) =
𝜅
𝜀e−𝑠
𝜀,(3.5)
o some 𝜅,
𝜅,𝜀,
𝜀>0. Then, aking he sum o he i s equa-
ion o Equa ion (3.4)wi h𝜀- imes i s ime de i a i e, and he sum
o he second equa ion wi h
𝜀- imes i s ime de i a i e, we end up
wi h he sys em
⎧
⎪
⎨
⎪
⎩
𝑢+1
𝜀
𝑢−(𝜚 +𝜅)
𝑢𝑥𝑥 −𝜚
𝜀𝑢𝑥𝑥 =𝑝
𝜀
𝑤𝑥+𝑝
𝑤𝑥,
𝑤+1
𝜀
𝑤−(
𝜚+
𝜅)
𝑤𝑥𝑥 −
𝜚
𝜀𝑤𝑥𝑥 =
𝑝
𝜀
𝑢𝑥+
𝑝
𝑢𝑥,
(3.6)
whe e now all he memo y e ms ha e disappea ed. Se ing
𝛼=1
𝜀,𝛽=𝜚+𝜅, 𝛾 =𝜚
𝜀,
𝛼=1
𝜀,
𝛽=
𝜚+
𝜅,
𝛾=
𝜚
𝜀,
sys em (3.6) is no hing bu Equa ion (2.1). No e ha condi-
ion (2.4) is au oma ically sa is ied, as he abo e choice yields
𝜘=𝛽−𝛾
𝛼=𝜅and 𝜘=
𝛽−
𝛾
𝛼=
𝜅.
In conclusion, his p ocedu e allows o eco e all possible
𝛼,𝛽,𝛾,
𝛼,
𝛽,
𝛾>0only i bo h he MGT equa ions in Equa ion (2.1)
lie in he subc i ical egime, ha is, only i Equa ion (2.4) holds.
4 Well-Posedness
4.1 No a ion
We deno e by 𝐻 he Hilbe space 𝐿2(0, 𝜋) wi h he inne p oduc
and no m ⟨⋅, ⋅⟩and ‖⋅‖, espec i ely. Calling by 𝖠=−𝜕𝑥𝑥 he
s ic ly posi i e sel -adjoin Laplace–Di ichle ope a o on 𝐻,
hence wi h dom(𝖠) =𝐻2(0,𝜋)∩𝐻
1
0(0, 𝜋), we will conside o
𝚤=−1, 1, 2 he u he Hilbe spaces
𝐻𝚤=dom(𝖠 𝗂
𝟤),
endowed wi h he inne p oduc and no ms
⟨𝑢,𝑣⟩𝚤=⟨𝖠𝗂
𝟤𝑢,𝖠 𝗂
𝟤𝑣⟩and ‖𝑢‖𝚤=‖𝖠𝗂
𝟤𝑢‖.
Thus,
𝐻−1=𝐻−1(0, 𝜋), 𝐻1=𝐻1
0(0, 𝜋), 𝐻2=𝐻2(0, 𝜋) ∩ 𝐻1
0(0, 𝜋),
he igh -hand sides being he usual Sobole spaces on (0, 𝜋).In
pa icula ,
‖𝑢𝑥‖−1=‖𝑢‖,‖𝑢‖1=‖𝑢𝑥‖,‖𝑢‖2=‖𝑢𝑥𝑥‖.
We also ecall he Poinca é inequali y
‖𝑢‖≤‖𝑢‖1,∀𝑢∈𝐻
1.
He e, he Poinca é cons an equals 1 due o he choice o he
in e al (0, 𝜋). The dynamics o ou sys em (2.1) will ake place
in he phase space
=𝐻1×𝐻1×𝐻×𝐻1×𝐻1×𝐻,
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endowed wi h he s anda d p oduc no m, deno ed by ‖⋅‖.We
will also encoun e he mo e egula space
1=𝐻2×𝐻2×𝐻1×𝐻2×𝐻2×𝐻1.
4.2 The Solu ion Semig oup
In oducing he six-componen ec o
𝒖=(𝑢, 𝑢∗,𝑢∗∗,𝑤,𝑤∗,𝑤∗∗),
p oblem (2.1)–(2.2) can be w i en as he O dina y Di e en ial
Equa ion (ODE) on
𝑑
𝑑𝑡𝒖(𝑡) =𝔸𝒖(𝑡), (4.1)
while he ini ial condi ion (2.3) ansla es in o
𝒖(0) =𝒖0=(𝑢0,𝑢
1,𝑢
2,𝑤
0,𝑤
1,𝑤
2)∈.
The linea ope a o 𝔸is gi en by
𝔸(𝑢, 𝑢∗,𝑢∗∗,𝑤,𝑤∗,𝑤∗∗)=(𝑢∗,𝑢∗∗,𝑢∗∗∗,𝑤∗,𝑤∗∗,𝑤∗∗∗),
whe e
𝑢∗∗∗ =−𝛼𝑢∗∗ +(𝛽𝑢∗+𝛾𝑢)𝑥𝑥 +𝑝(𝛼𝑤∗
𝑥+𝑤∗∗
𝑥),
𝑢∗∗∗ =−
𝛼𝑤∗∗ +(
𝛽𝑤∗+
𝛾𝑤)𝑥𝑥 +
𝑝(
𝛼𝑢∗
𝑥+𝑢∗∗
𝑥),
wi h domain
dom(𝔸) =⎧
⎪
⎨
⎪
⎩
𝒖∈||||||||
𝑢∗∗,𝑤∗∗ ∈𝐻
1
𝛽𝑢∗+𝛾𝑢 ∈ 𝐻2
𝛽𝑤∗+
𝛾𝑤 ∈ 𝐻2⎫
⎪
⎬
⎪
⎭
.
Obse e ha we ha e he inclusion 1⊂dom(𝔸).
Equa ion (2.1) u ns ou o gene a e a s ongly con inuous
semig oup o bounded linea ope a o s 𝑆(𝑡) ∶ →,whose
in ini esimal gene a o is 𝔸. Acco dingly, o all ini ial da a 𝒖0∈
he solu ion a ime 𝑡is gi en by
𝒖(𝑡) =𝑆(𝑡)𝒖0,
whose ela ed ene gy eads
𝖤(𝑡) =1
2‖𝑆(𝑡)𝒖0‖2
.
The p oo o his ac is de ailed in [24] ia semig oup echniques.
Howe e , i could also be ob ained by means o ene gy es ima es,
pe o ming he s anda d MGT mul iplica ion [20], ha is, mul i-
plying in 𝐻 he i s equa ion in Equa ion (2.1)by
𝑢+𝛼
𝑢,and he
second one by
𝑤+
𝛼
𝑤( imes 𝑝∕
𝑝, in o de o ha e a cancella ion
o he highe -o de e ms in he coupling). This, o egula ini ial
da a 𝒖0∈dom(𝔸),gi es
𝑑
𝑑𝑡𝖥+𝛼𝜘‖
𝑢‖2
1+𝑝
𝑝
𝛼𝜘‖
𝑤‖2
1=𝖦,
whe e
𝖥=1
2[𝛾
𝛼‖
𝑢+𝛼𝑢‖2
1+‖
𝑢+𝛼
𝑢‖2+𝜘‖
𝑢‖2
1]
+𝑝
2
𝑝[
𝛾
𝛼‖
𝑤+
𝛼𝑤‖2
1+‖
𝑤+
𝛼
𝑤‖2+𝜘‖
𝑤‖2
1]
is equi alen o he s anda d ene gy unc ional 𝖤(see [20] o
mo e de ails), while
𝖦=𝑝𝛼⟨
𝑤𝑥,
𝑢+𝛼
𝑢⟩−𝑝𝛼⟨
𝑢𝑥,
𝑤⟩+𝑝
𝛼⟨
𝑢𝑥,
𝑤+
𝛼
𝑤⟩−𝑝
𝛼⟨
𝑤𝑥,
𝑢⟩.
No e ha 𝖦is con olled by 𝖥. This, by a s aigh o wa d
applica ion o he G onwall lemma, yields
𝖤(𝑡) ≤𝐾1𝖤(0)e𝐾2𝑡,(4.2)
o some posi i e cons an s 𝐾1,𝐾
2. Howe e , a his s age, unless
we a e in he simple si ua ion 𝛼=
𝛼implying 𝖦=0(hence
𝐾2=0), we do no e en know whe he he ene gy (hence he
semig oup) is bounded.
Rema k 4.1. By he same mul iplica ion, his ime in 𝐻1,i iseasy
o see ha 𝑆(𝑡) is a s ongly con inuous semig oup on he space
1as well.
5The Main Resul
The main esul o he pape conce ns wi h he exponen ial
s abili y o 𝑆(𝑡).
Theo em 5.1. Wi hin assump ion (2.4), he e exis cons an s
𝑀≥1and 𝜔>0such ha he exponen ial es ima e
‖𝑆(𝑡)𝒖0‖≤𝑀‖𝒖0‖𝑒−𝜔𝑡
holds o e e y 𝑡≥0and e e y 𝒖0∈.
Theo em 5.1 has been p o ed in [25] unde he se e e es ic ion
ha
min{𝛼,
𝛼} >max {𝛾
𝛽,
𝛾
𝛽}.
In o he wo ds, no only bo h he MGT equa ions mus be sub-
c i ical, bu hey mus emain subc i ical e en i he coe icien s
𝛼and
𝛼a e swi ched. Indeed, he key idea o [25] is o employ he
same coe icien 𝛼(o
𝛼) o bo h equa ions, hence exploi ing he
same s uc u e o he highe -o de ime de i a i es, ea ing he
emaining e ms as a pe u ba ion. Ano he impo an aspec is
he ole o he coupling and i s connec ion wi h 𝛼and
𝛼.When
𝛼=
𝛼, he dissipa i e na u e o he sys em eme ges immedia ely
om Equa ion (4.2), since in ha case 𝐾2=0. On he con a y,
when 𝛼and
𝛼a e di e en , i is e y di icul o exhibi he
dissipa i e cha ac e o he model. In his si ua ion, i has been
shown in he e y ecen pape [24] ha he solu ion semig oup
𝑆(𝑡) decays exponen ially o ze o whene e 𝛼and
𝛼a e su i-
cien ly close. Acco dingly, he gene al p oblem o coe icien s
subjec only o Equa ion (2.4) wi h no u he es ic ions equi es
a comple ely di e en app oach, as he s anda d MGT es ima es
4o 10 S udies in Applied Ma hema ics,2025
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FIGURE 1 Scheme o he p oo o Theo em 5.1.
seem o una oidably ail. The main idea o his wo k is o bypass
comple ely his s ep, by exploi ing in a deepe way he connec ion
be ween ou sys em (2.1)and(3.4), con en ionally e e ed o
as he double iscoelas ic sys em he ea e . Indeed, in Sec ions 6
and 7, we will show ha sys em (3.4) gene a es a s ongly
con inuous semig oup 𝑇(𝑡) in a sui able unc ional amewo k,
and such a semig oup is exponen ially s able. Inciden ally, his
ac has an independen in e es . Then, in Sec ion 8, de o ed o
he p oo o Theo em 5.1, we will show ha pa o he ajec o ies
o he wo semig oups coincide, and his will be enough o deduce
he exponen ial s abili y o 𝑆(𝑡) om ha o 𝑇(𝑡) (Figu e 1).
6The Double Viscoelas ic Sys em
We now u n ou a en ion o sys em (3.4), whe e he con olu ion
(o memo y) ke nels 𝜇and
𝜇a eassumed obenonnega i e,
noninc easing, absolu ely con inuous, and summable unc ion
on ℝ+,o o almass
∫∞
0
𝜇(𝑠)𝑑𝑠 =𝜅>0and ∫∞
0
𝜇(𝑠)𝑑𝑠 =
𝜅>0.
A pa icula example o ke nels o his kind is gi en by Equa-
ion (3.5), bu o cou se much mo e gene al ke nels a e possible,
e en exhibi ing an in eg able singula i y abou ze o. The main
ea u e o Equa ion (3.4) is ha he con olu ion in eg als ac
on he a iables 𝑢and 𝑤 o all imes up o he ac ual ime 𝑡.
The e o e, being he ini ial ime con en ionally se a 𝑡=0, he
unc ions 𝑢and 𝑤 o nega i e imes a e assumed o be p esc ibed
ini ialda a, whichneed no sol e heequa ion, bu a he desc ibe
he pas his o y o he sys em. As sugges ed by he seminal
wo k [26](seealso[27, 28]), one way o handle Equa ion (3.4)is
o in oduce o 𝑡≥0and 𝑠>0 he auxilia y a iables
𝜂𝑡(𝑠) =𝑢(𝑡) −𝑢(𝑡 −𝑠) and 𝜉𝑡(𝑠) =𝑤(𝑡) −𝑤(𝑡 −𝑠),
whe e we omi ed he dependence on 𝑥. Acco dingly, besides
imposing he homogeneous Di ichle bounda y condi ions on
𝑢,𝑤, 𝜂, 𝜉, we supplemen ed Equa ion (3.4) wi h he ini ial
condi ions
⎧
⎪
⎨
⎪
⎩
𝑢(0) =𝑢0,
𝑢(0) =𝑢1,
𝜂0(𝑠) =𝜂0(𝑠), ⎧
⎪
⎨
⎪
⎩
𝑤(0) =𝑤0,
𝑤(0) =𝑤1,
𝜉0(𝑠) =𝜉0(𝑠),
(6.1)
whe e 𝑢0,𝑢
1,𝜂
0,𝑤
0,𝑤
1,𝜉
0a e assigned da a. Thus, sys em (3.4)
can be ew i en in he o m
⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝑢−𝜚𝑢𝑥𝑥 −∫∞
0
𝜇(𝑠)𝜂𝑥𝑥(𝑠)𝑑𝑠 =𝑝
𝑤𝑥,
𝑤−
𝜚𝑤𝑥𝑥 −∫∞
0
𝜇(𝑠)𝜉𝑥𝑥(𝑠)𝑑𝑠 =
𝑝
𝑢𝑥,
(6.2)
whe e
𝜂𝑡(𝑠) ={𝑢(𝑡) −𝑢(𝑡 −𝑠) 0 <𝑠≤𝑡,
𝜂0(𝑠 −𝑡) +𝑢(𝑡) −𝑢0𝑠>𝑡, (6.3)
and
𝜉𝑡(𝑠) ={𝑤(𝑡) −𝑤(𝑡 −𝑠) 0 <𝑠≤𝑡,
𝜉0(𝑠 −𝑡) +𝑤(𝑡) −𝑤0𝑠>𝑡. (6.4)
In o de o se his p oblem in he co ec unc ional amewo k,
we in oduce he weigh ed 𝐿2-spaces o 𝐻1- alued unc ions
=𝐿2
𝜇(ℝ+;𝐻1)and
=𝐿2
𝜇(ℝ+;𝐻1),
no med by
‖𝜂‖=(∫∞
0
𝜇(𝑠)‖𝜂(𝑠)‖2
1)1
2
and
‖𝜉‖
=(∫∞
0
𝜇(𝑠)‖𝜉(𝑠)‖2
1)1
2
.
Then, de ining he phase space
=𝐻1×𝐻××𝐻1×𝐻×
,
endowed wi h he no m
‖(𝑢, 𝑢∗,𝜂,𝑤,𝑤∗,𝜉)‖2
=𝜚‖𝑢‖2
1+‖𝑢∗‖2+‖𝜂‖2
+𝑝
𝑝[
𝜚‖𝑤‖2
1+‖𝑤∗‖2+‖𝜉‖2
],
we iew Equa ions (6.2)–(6.4) as an e olu ion sys em in .
Ac ually, deno ing by 𝖳 he in ini esimal gene a o o he igh -
ansla ion semig oup on , ha is, he linea ope a o
[𝖳𝜂](𝑠) =−𝜂′(𝑠) wi h dom(𝖳) ={𝜂∈∶𝜂
′∈,𝜂(0)=0},
and by
𝖳 he analogous one on
, i is eadily seen ha Equa-
ions (6.3)and(6.4) a e he mild solu ions (in he sense o [29,
30]) o he e olu ion equa ions on and
, espec i ely,
𝜂=𝖳𝜂 +
𝑢and
𝜉=
𝖳𝜉 +
𝑤.
5o 10
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Acco dingly, Equa ions (6.2)–(6.4) can be gi en he equi alen
o m
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
𝑢−𝜚𝑢𝑥𝑥 −∫∞
0
𝜇(𝑠)𝜂𝑥𝑥(𝑠)𝑑𝑠 =𝑝
𝑤𝑥,
𝜂=𝖳𝜂 +
𝑢,
𝑤−
𝜚𝑤𝑥𝑥 −∫∞
0
𝜇(𝑠)𝜉𝑥𝑥(𝑠)𝑑𝑠 =
𝑝
𝑢𝑥,
𝜉=
𝖳𝜉 +
𝑤.
(6.5)
Sys em (6.5) can be shown o gene a e a s ongly con inuous
semig oup o linea con ac ions 𝑇(𝑡) ∶ →. Hence, o e e y
ini ial da um 𝒗0=(𝑢0,𝑢
1,𝜂
0,𝑤
0,𝑤
1,𝜉
0)∈, i s unique solu ion
a ime 𝑡is gi en by
𝒗(𝑡) =𝑇(𝑡)𝒗0,
wi h co esponding ene gy
𝖤𝖵(𝑡) =1
2‖𝑇(𝑡)𝒗0‖2
.(6.6)
The p oo o his ac can be done along he same lines o [28],
whe e he case o a single iscoelas ici y equa ion is ea ed,
ac ually o a much mo e gene al class o memo y ke nels,
possibly exhibi ing discon inui ies. The key s ep is p o ing he
ene gy equali y, alid o all egula ini ial da a (in pa icula , wi h
𝜂0and 𝜉0in he domains o 𝖳and
𝖳)
𝑑
𝑑𝑡𝖤𝖵−1
2∫∞
0
𝜇′(𝑠)‖𝜂(𝑠)‖2
1𝑑𝑠 −𝑝
2
𝑝∫∞
0
𝜇′(𝑠)‖𝜉(𝑠)‖2
1𝑑𝑠 =0.
(6.7)
Such an iden i y ollows by a mul iplica ion o Equa ion (6.5)by
𝒗(𝑡) in he phase space .
7 Exponen ial Decay o he Double Viscoelas ic
Sys em
The nex s ep is showing ha he semig oup 𝑇(𝑡) gene a ed by
sys em (6.5) is exponen ially s able. To his end, we need some
u he assump ions on he memo y ke nels: as commonly done
in he li e a u e, we equi e ha
𝜇′(𝑠) +𝛿𝜇(𝑠) ≤0and
𝜇′(𝑠) +𝛿
𝜇(𝑠) ≤0,
o some 𝛿>0and almos e e y 𝑠>0. This yields he con ols
−∫∞
0
𝜇′(𝑠)‖𝜂(𝑠)‖2
1𝑑𝑠 ≥𝛿‖𝜂‖2
and
−∫∞
0
𝜇′(𝑠)‖𝜉(𝑠)‖2
1𝑑𝑠 ≥𝛿‖𝜉‖2
.(7.1)
The esul eads as ollows.
Theo em 7.1. The e exis cons an s 𝑄≥1and 𝜔>0such ha
he exponen ial es ima e
‖𝑇(𝑡)𝒗0‖≤𝑄‖𝒗0‖𝑒−𝜔𝑡
holds o e e y 𝑡≥0and e e y 𝒗0∈.
Fo simplici y, we shall assume he ea e ha bo h 𝜇and
𝜇a e
bounded abou ze o, namely,
𝜇(0) =lim
𝑠→0 𝜇(𝑠) ∈ (0,∞)and
𝜇(0) =lim
𝑠→0
𝜇(𝑠) ∈ (0, ∞).
The gene al case can be handled by in oducing a sui able cu -
o (see [28] o mo e de ails). Fo egula ini ial da a 𝒗0=
(𝑢0,𝑢
1,𝜂
0,𝑤
0,𝑤
1,𝜉
0),wi h𝜂0∈dom(𝖳)and 𝜉0∈dom(
𝖳),le
𝒗(𝑡) =𝑇(𝑡)𝒗0=(𝑢(𝑡),
𝑢(𝑡),𝜂𝑡, 𝑤(𝑡),
𝑤(𝑡),𝜉𝑡)
be he co esponding ( egula ) solu ion o Equa ion (6.5). We
in oduce he auxilia y unc ional
Φ(𝑡) =−
1
𝜅∫∞
0
𝜇(𝑠)⟨𝜂𝑡(𝑠),
𝑢(𝑡)⟩𝑑𝑠 −𝑝
𝜅
𝑝∫∞
0
𝜇(𝑠)⟨𝜉𝑡(𝑠),
𝑤(𝑡)⟩𝑑𝑠.
By di ec calcula ions, we ind he iden i y
𝑑
𝑑𝑡 Φ+‖
𝑢‖2+𝑝
𝑝‖
𝑤‖2=−1
𝜅∫∞
0
𝜇(𝑠)⟨𝖳𝜂(𝑠),
𝑢⟩𝑑𝑠 −𝑝
𝜅
𝑝∫∞
0
𝜇(𝑠)⟨
𝖳𝜉(𝑠),
𝑤⟩𝑑𝑠
(7.2)
+1
𝜅∫∞
0
𝜇(𝑠)(∫∞
0
𝜇(𝜎)⟨𝜂(𝑠),𝜂(𝜎)⟩1𝑑𝜎)𝑑𝑠
+𝑝
𝜅
𝑝∫∞
0
𝜇(𝑠)(∫∞
0
𝜇(𝜎)⟨𝜉(𝑠),𝜉(𝜎)⟩1𝑑𝜎)𝑑𝑠
+𝜚
𝜅∫∞
0
𝜇(𝑠)⟨𝜂(𝑠),𝑢⟩1𝑑𝑠 +
𝜚𝑝
𝜅
𝑝∫∞
0
𝜇(𝑠)⟨𝜉(𝑠),𝑤⟩1𝑑𝑠
+𝑝
𝜅∫∞
0
𝜇(𝑠)⟨𝜂𝑥(𝑠),
𝑤⟩𝑑𝑠 +𝑝
𝜅∫∞
0
𝜇(𝑠)⟨𝜉𝑥(𝑠),
𝑢⟩𝑑𝑠.
In eg a ing by pa s in 𝑠( he bounda y e ms anish, see [28]), we
in e ha
−1
𝜅∫∞
0
𝜇(𝑠)⟨𝖳𝜂(𝑠),
𝑢⟩𝑑𝑠 =−
1
𝜅∫∞
0
𝜇′(𝑠)⟨𝜂(𝑠),
𝑢⟩𝑑𝑠
≤1
4‖
𝑢‖2+1
𝜅2(∫∞
0
−𝜇′(𝑠)‖𝜂(𝑠)‖𝑑𝑠)2
≤1
4‖
𝑢‖2−𝜇(0)
𝜅2∫∞
0
𝜇′(𝑠)‖𝜂(𝑠)‖2
1𝑑𝑠,
whe e he las bound ollows om he Hölde and he Poinca é
inequali ies, which will epea edly used he ea e , o en wi hou
men ion. Simila ly,
−𝑝
𝜅
𝑝∫∞
0
𝜇(𝑠)⟨
𝖳𝜉(𝑠),
𝑤⟩𝑑𝑠 ≤𝑝
4
𝑝‖
𝑤‖2−
𝜇(0)𝑝
𝜅2
𝑝∫∞
0
𝜇′(𝑠)‖𝜉(𝑠)‖2
1𝑑𝑠.
Mo eo e
1
𝜅∫∞
0
𝜇(𝑠)(∫∞
0
𝜇(𝜎)⟨𝜂(𝑠),𝜂(𝜎)⟩1𝑑𝜎)𝑑𝑠
≤1
𝜅(∫∞
0
𝜇(𝑠)‖𝜂(𝑠)‖1𝑑𝑠)2≤‖𝜂‖2
,
and, by he same oken,
𝑝
𝜅
𝑝∫∞
0
𝜇(𝑠)(∫∞
0
𝜇(𝜎)⟨𝜉(𝑠),𝜉(𝜎)⟩1𝑑𝜎)𝑑𝑠 ≤𝑝
𝑝‖𝜉‖2
.
6o 10 S udies in Applied Ma hema ics,2025
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Nex , i is eadily seen ha
𝜚
𝜅∫∞
0
𝜇(𝑠)⟨𝜂(𝑠),𝑢⟩1𝑑𝑠 +
𝜚𝑝
𝜅
𝑝∫∞
0
𝜇(𝑠)⟨𝜉(𝑠),𝑤⟩1𝑑𝑠
≤𝐶‖𝑢‖1‖𝜂‖+𝐶‖𝑤‖1‖𝜉‖
.
He e and in he sequel o his p oo , 𝐶>0s ands o a gene ic
s uc u al cons an , independen o he ini ial da a, which may
a y e en wi hin he same line. Finally,
𝑝
𝜅∫∞
0
𝜇(𝑠)⟨𝜂𝑥(𝑠),
𝑤⟩𝑑𝑠 +𝑝
𝜅∫∞
0
𝜇(𝑠)⟨𝜉𝑥(𝑠),
𝑢⟩𝑑𝑠
≤1
4‖
𝑢‖2+𝑝
4
𝑝‖
𝑤‖2+𝐶‖𝜂‖2
+𝐶‖𝜉‖2
.
Subs i u ing all he es ima es ob ained so a in o Equa ion (7.2),
we a i e a
𝑑
𝑑𝑡Φ+1
2‖
𝑢‖2+𝑝
2
𝑝‖
𝑤‖2≤𝐶‖𝑢‖1‖𝜂‖+𝐶‖𝑤‖1‖𝜉‖
+𝐶‖𝜂‖2
+𝐶‖𝜉‖2
−𝐶∫∞
0
𝜇′(𝑠)‖𝜂(𝑠)‖2
1𝑑𝑠 −𝐶∫∞
0
𝜇′(𝑠)‖𝜉(𝑠)‖2
1𝑑𝑠. (7.3)
A his poin , we conside he u he unc ional
Ψ(𝑡) =⟨
𝑢(𝑡),𝑢(𝑡)⟩+𝑝
𝑝⟨
𝑤(𝑡),𝑤(𝑡)⟩.
A s aigh o wa d compu a ion en ails he equali y
𝑑
𝑑𝑡 Ψ+𝜚‖𝑢‖2
1+
𝜚𝑝
𝑝‖𝑤‖2
1=−
∫∞
0
𝜇(𝑠)⟨𝜂(𝑠),𝑢⟩1𝑑𝑠
−𝑝
𝑝∫∞
0
𝜇(𝑠)⟨𝜉(𝑠),𝑤⟩1𝑑𝑠
+‖
𝑢‖2+𝑝
𝑝‖
𝑤‖2−𝑝⟨
𝑤,𝑢𝑥⟩−𝑝⟨
𝑢,𝑤𝑥⟩.
I is no di icul o check ha he igh -hand side abo e is
con olled by
𝐶‖𝑢‖1‖𝜂‖+𝐶‖𝑤‖1‖𝜉‖
+𝐶‖
𝑢‖2+𝐶‖
𝑤‖2+𝐶‖
𝑤‖‖𝑢‖1+𝐶‖
𝑢‖‖𝑤‖1
≤𝜚
2‖𝑢‖2
1+
𝜚𝑝
2
𝑝‖𝑤‖2
1+𝐶‖
𝑢‖2+𝐶‖
𝑤‖2+𝐶‖𝜂‖2
+𝐶‖𝜉‖2
.
The e o e, we end up wi h
𝑑
𝑑𝑡Ψ+𝜚
2‖𝑢‖2
1+
𝜚𝑝
2
𝑝‖𝑤‖2
1≤𝐶‖
𝑢‖2+𝐶‖
𝑤‖2+𝐶‖𝜂‖2
+𝐶‖𝜉‖2
.
(7.4)
In o de o inish he p oo , we in oduce o all 𝜀∈(0,1) he
pe u bed ene gy unc ional
Λ𝜀(𝑡) =𝖤𝖵(𝑡) +𝜀3
2Φ(𝑡) +𝜀2Ψ(𝑡),
whe e 𝖤𝖵has been de ined in Equa ion (6.6). Since |Φ|≤𝐶𝖤𝖵and
|Ψ|≤𝐶𝖤𝖵,up o aking𝜀small enough we ha e
1
2𝖤𝖵≤Λ𝜀≤2𝖤𝖵.(7.5)
In addi ion, collec ing Equa ions (7.3)and(7.4) and he ene gy
iden i y (6.7), we ob ain
𝑑
𝑑𝑡Λ𝜀−(1
2−𝐶𝜀3
2)∫∞
0
𝜇′(𝑠)‖𝜂(𝑠)‖2
1𝑑𝑠
−(𝑝
2
𝑝−𝐶𝜀3
2)∫∞
0
𝜇′(𝑠)‖𝜉(𝑠)‖2
1𝑑𝑠
+1
2[𝜀2𝜚‖𝑢‖2
1+𝜀3
2‖
𝑢‖2]+𝑝
2
𝑝[𝜀2
𝜚‖𝑤‖2
1+𝜀3
2‖
𝑤‖2]
≤𝐶𝜀3
2‖𝑢‖1‖𝜂‖+𝐶𝜀3
2‖𝑤‖1‖𝜉‖
+𝐶𝜀2‖
𝑢‖2+𝐶𝜀2‖
𝑤‖2
+𝐶𝜀3
2‖𝜂‖2
+𝐶𝜀3
2‖𝜉‖2
.
Fo all 𝜀su icien ly small, an exploi a ion o Equa ion (7.1) yields
−(1
2−𝐶𝜀
3
2)∫∞
0
𝜇′(𝑠)‖𝜂(𝑠)‖2
1𝑑𝑠 −(𝑝
2
𝑝−𝐶𝜀
3
2)∫∞
0
𝜇′(𝑠)‖𝜉(𝑠)‖2
1𝑑𝑠
≥𝛿
4[‖𝜂‖2
+𝑝
𝑝‖𝜉‖2
].
Mo eo e ,
𝐶𝜀3
2‖𝑢‖1‖𝜂‖+𝐶𝜀3
2‖𝑤‖1‖𝜉‖
+𝐶𝜀2‖
𝑢‖2+𝐶𝜀2‖
𝑤‖2
+𝐶𝜀3
2‖𝜂‖2
+𝐶𝜀3
2‖𝜉‖2
≤𝐶𝜀5
2‖𝑢‖2
1+𝐶𝜀2‖
𝑢‖2+𝐶𝜀5
2‖𝑤‖2
1+𝐶𝜀2‖
𝑤‖2
+𝐶𝜀1
2‖𝜂‖2
+𝐶𝜀1
2‖𝜉‖2
≤1
4[𝜀2𝜚‖𝑢‖2
1+𝜀3
2‖
𝑢‖2]+𝑝
4
𝑝[𝜀2
𝜚‖𝑤‖2
1+𝜀3
2‖
𝑤‖2]
+𝛿
8[‖𝜂‖2
+𝑝
𝑝‖𝜉‖2
].
In conclusion, ixing 𝜀>0small enough, he e exis s a s uc u al
cons an 𝜔>0such ha
𝑑
𝑑𝑡Λ𝜀+4𝜔𝖤𝖵≤0.
In oking Equa ion (7.5) and he G onwall lemma, om he
inequali y abo e we in e ha
𝖤𝖵(𝑡) ≤4𝖤𝖵(0)e−2𝜔𝑡,
o e e y 𝑡≥0. By densi y, he conclusion holds o e e y ini ial
da a 𝒗0∈.
Rema k 7.2. Theo em 7.1 is ac ually ue o a mo e gene al class
o memo y ke nels, including ke nels exhibi ing (e en in ini ely
many) discon inui y poin s. This equi es mino changes in he
p oo , bo owing he echniques om [28].
7o 10
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8P oo o Theo em 5.1
Le us s a om sys em (2.1) wi hin assump ion (2.4). Th ough-
ou his sec ion, we will conside Equa ion (6.5) o he pa icula
choice o he pa ame e s and he memo y ke nels
𝜚=𝛾
𝛼,
𝜚=
𝛾
𝛼,𝜇(𝑠)=𝛼𝜘e−𝛼𝑠,
𝜇(𝑠) =
𝛼𝜘e−
𝛼𝑠.
We begin o es ablish a link be ween he wo sys ems.
Lemma 8.1. Le 𝒖=(𝑢,
𝑢,
𝑢,𝑤,
𝑤,
𝑤) be he solu ion
o Equa ion (2.1) co esponding o he ini ial da um 𝒖0=
(𝑢0,𝑢
1,𝑢
2,𝑤
0,𝑤
1,𝑤
2)∈, and le 𝒗0=(𝑢0,𝑢
1,𝜂
0,𝑤
0,𝑤
1,𝜉
0)∈
, ha ing se
𝜂0=1
𝜘[−𝖠−1𝑢2−𝛾
𝛼𝑢0+𝑝𝖠−1𝑤1𝑥],(8.1)
𝜉0=1
𝜘[−𝖠−1𝑤2−
𝛾
𝛼𝑤0+
𝑝𝖠−1𝑢1𝑥].(8.2)
Then 𝒗=(𝑢,
𝑢,𝜂, 𝑤,
𝑤,𝜉), wi h 𝜂and 𝜉gi en by Equa ions (6.3)
and (6.4), espec i ely, is he solu ion o Equa ion (6.5) co espond-
ing o he ini ial da um 𝒗0.
P oo . Le us deno e by 𝒘=(𝑟,
𝑟,𝜙,𝑦,
𝑦,𝜓) he unique solu ion
o Equa ion (6.5) wi h ini ial da um 𝒗0. We obse e ha
𝑟(0) =𝑢2and
𝑦(0) =𝑤2.
Indeed, since 𝒘sol es Equa ion (6.5), i s hi d componen 𝜙is
gi en by
𝜙𝑡(𝑠) ={𝑟(𝑡) −𝑟(𝑡 −𝑠) 0 <𝑠≤𝑡,
𝜂0+𝑟(𝑡) −𝑢0𝑠>𝑡,
as 𝜂0(𝑠) is cons an in 𝑠. Hence, he i s equa ion o Equa ion (6.5),
w i en in he a iable 𝑟,𝑦 in place o 𝑢, 𝑤, eads
𝑟(𝑡) −𝛽𝑟𝑥𝑥(𝑡) +𝛼𝜘∫𝑡
0
e−𝛼𝑠𝑟𝑥𝑥(𝑡 −𝑠)𝑑𝑠 =𝑝
𝑦𝑥(𝑡) +𝜘(𝜂0𝑥𝑥 −𝑢0𝑥𝑥)e−𝛼𝑡.
(8.3)
Fo 𝑡=0, we ob ain
𝑟(0) =𝛽𝑢0𝑥𝑥 +𝑝𝑤1𝑥 +𝜘(𝜂0𝑥𝑥 −𝑢0𝑥𝑥)=𝑢2,
due o Equa ion (8.1). The equali y
𝑦(0) =𝑤2is de i ed in he
same way by using Equa ion (8.2).
Nex , we show ha (𝑟,
𝑟,
𝑟,𝑦,
𝑦,
𝑦) sol es Equa ion (2.1). To his
end, we mul iply Equa ion (8.3)bye
𝛼𝑡, oge
[
𝑟(𝑡) −𝛽𝑟𝑥𝑥(𝑡)]e𝛼𝑡 +𝛼𝜘∫𝑡
0
e𝛼𝑠𝑟𝑥𝑥(𝑠)𝑑𝑠 =𝑝
𝑦𝑥(𝑡)e𝛼𝑡 +𝜘(𝜂0𝑥𝑥 −𝑢0𝑥𝑥).
Taking he ime de i a i e, and di iding he esul ing equa ion by
e𝛼𝑡, we a i e a
𝑟+𝛼
𝑟−𝛽
𝑟𝑥𝑥 −𝛾𝑟𝑥𝑥 =𝑝(𝛼
𝑦𝑥+
𝑦𝑥).
Simila ly, we ob ain he co esponding equa ion o 𝑦.By
uniqueness, we ha e he desi ed equali y 𝒘=𝒗.□
In wha ollows, 𝑀≥1will deno e a gene ic s uc u al cons an ,
independen o he ini ial da a, which may a y e en wi hin he
same line.
Lemma 8.2. Fo e e y 𝒖0and 𝒗0as in he s a emen o
Lemma 8.1,
‖𝒗0‖2
≤𝑀[‖𝑢0‖2
1+‖𝑢1‖2+‖𝑢2‖2
−1+‖𝑤0‖2
1+‖𝑤1‖2
+‖𝑤2‖2
−1]≤𝑀‖𝒖0‖2
.
P oo . The la e is an immedia e consequence o he Poinca é
inequali y. As o he i s one, i clea ly su ices o show ha
‖𝜂0‖2
+‖𝜉0‖2
≤𝑀[‖𝑢0‖2
1+‖𝑢1‖2+‖𝑢2‖2
−1+‖𝑤0‖2
1
+‖𝑤1‖2+‖𝑤2‖2
−1].
Le us ocus on 𝜂0. On accoun o Equa ion (8.1), no ing again ha
𝜂0does no depend on 𝑠,
‖𝜂0‖2
=1
𝜘‖‖‖−𝖠−1𝑢2−𝛾
𝛼𝑢0+𝑝𝖠−1𝑤1𝑥‖‖‖
2
1≤𝑀[‖𝖠−1𝑢2‖2
1
+‖𝑢0‖2
1+‖𝖠−1𝑤1𝑥‖2
1].
Bu
‖𝖠−1𝑢2‖1=‖𝑢2‖−1and ‖𝖠−1𝑤1𝑥‖1=‖𝑤1‖.
Repea ing he a gumen o 𝜉0, he claim is es ablished. □
We ha e now all he ing edien s o comple e he p oo o
Theo em 5.1. Le us choose an a bi a y, bu mo e egula , ini ial
da um o Equa ion (2.1), ha is,
𝒖0=(𝑢0,𝑢
1,𝑢
2,𝑤
0,𝑤
1,𝑤
2)∈1.
Then, conside sys em (6.5) wi h ini ial da um 𝒗0as in Lemma 8.1.
Exploi ing he exponen ial s abili y o 𝑇(𝑡), ensu ed by Theo-
em 7.1, we d aw he es ima e
‖𝑢(𝑡)‖2
1+‖𝑤(𝑡)‖2
1≤𝑀‖𝑇(𝑡)𝒗0‖2
≤𝑀‖𝒗0‖2
e−2𝜔𝑡,
and by Lemma 8.2 we in e ha
‖𝑢(𝑡)‖2
1+‖𝑤(𝑡)‖2
1≤𝑀‖𝒖0‖2
e−2𝜔𝑡.(8.4)
The nex s ep is o ake he ime de i a i e o Equa ion (2.1),
obse ing ha he unc ions
𝑣=
𝑢and 𝑧=
𝑤
con inue o sol e Equa ion (2.1), bu his ime o he ini ial da um
𝒚0=(𝑢1,𝑢
2,𝑢
3,𝑤
1,𝑤
2,𝑤
3),
whe e
𝑢3=−𝛼𝑢2+𝛽𝑢1𝑥𝑥 +𝛾𝑢0𝑥𝑥 +𝑝(𝛼𝑤1𝑥 +𝑤2𝑥),
𝑤3=−
𝛼𝑤2+
𝛽𝑤1𝑥𝑥 +
𝛾𝑤0𝑥𝑥 +
𝑝(
𝛼𝑢1𝑥 +𝑢2𝑥).
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No ice ha 𝒚0∈, due o he assumed highe egula i y o 𝒖0.
Besides, in ligh o he Poinca é inequali y,
‖𝑢3‖2
−1+‖𝑤3‖2
−1≤𝑀‖𝒖0‖2
.(8.5)
A his poin , we apply once again Lemma 8.1 wi h 𝒚0in place o
𝒖0. Acco dingly, in place o 𝒗0we will ha e
𝒛0=(𝑢1,𝑢
2,𝜂
1,𝑤
1,𝑤
2,𝜉
1),
whe e
𝜂1=1
𝜘[−𝖠−1𝑢3−𝛾
𝛼𝑢1+𝑝𝖠−1𝑤2𝑥],
𝜉1=1
𝜘[−𝖠−1𝑤3−
𝛾
𝛼𝑤1+
𝑝𝖠−1𝑢2𝑥].
Due o he i s inequali y o Lemma 8.2 oge he wi h Equa-
ion (8.5),
‖𝒛0‖2
≤𝑀[‖𝑢1‖2
1+‖𝑢2‖2+‖𝑢3‖2
−1+‖𝑤1‖2
1+‖𝑤2‖2
+‖𝑤3‖2
−1]≤𝑀‖𝒖0‖2
.
The e o e, since
𝑣=
𝑢and
𝑧=
𝑤, by a u he applica ion o
Theo em 7.1 we conclude ha
‖
𝑢(𝑡)‖2
1+‖
𝑢(𝑡)‖2+‖
𝑤(𝑡)‖2
1+‖
𝑤(𝑡)‖2≤𝑀‖𝑇(𝑡)𝒛0‖2
≤𝑀‖𝒖0‖2
e−2𝜔𝑡.
Collec ing his inequali y and Equa ion (8.4), we end up wi h he
inal es ima e
‖𝑆(𝑡)𝒖0‖≤𝑀‖𝒖0‖e−𝜔𝑡,
alid o e e y 𝒖0∈1. By densi y, such an es ima e emains
alid o e e y 𝒖0∈. This inishes he p oo .
Acknowledgmen s
L.L. has been suppo ed by he Alexande on Humbold Founda ion. F.D.
and V.P. ha e been pa ially suppo ed by he I alian MIUR-PRIN G an
2020F3NCPX “Ma hema ics o indus y 4.0 (Ma h4I4)”.
Open access publishing acili a ed by Poli ecnico di Milano, as pa o he
Wiley - CRUI-CARE ag eemen .
Da a A ailabili y S a emen
Da a sha ing is no applicable o his pape as no new da a we e c ea ed
o analyzed in his s udy.
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