Local exis ence and uniqueness o egula
solu ions in a model o issue in asion by
solid umou s
C is ian Mo ales-Rod igo
Ins i u e o Applied Ma hema ics and Mechanics,
Facul y o In o ma ics, Ma hema ics and Mechanics,
Wa saw Uni e si y, ul. Banacha 2, 02-097 Wa saw, Poland
c is ianma ema [email protected]
Abs ac
In his pape we conside a nonlinea sys em o di e en ial equa ions a ising in
umou in asion which has been p oposed in [1]. The sys em consis s o wo PDEs
desc ibing he e olu ion o umou cells and p o eases and an ODE which models he
concen a ion o he ex acellula ma ix. We p o e local exis ence and uniqueness
o solu ions in he class o H¨olde spaces. The p oo o local exis ence is done by
Schaude ’s ixed poin heo em and o he uniqueness we use an idea om [2].
Keywo ds: Hap o axis; Tumou in asion o issue; Reac ion-di usion equa ions;
Uniqueness.
AMS Subjec Classi ica ion: 35K45, 35K57, 92C17
1 In oduc ion
The mos dange ous ea u e o malignan umou and he main cause o cance deceases
is he abili y o me as asize. Me as asis is he o ma ion o a seconda y umou oci a a
si e discon inuous om he p ima y umou . Two main p ocesses ha e o be aken in o
accoun du ing he me as asis.
The i s one is called angiogenesis. Tumou cells esponse o hypoxia by sec e ing umou
angiogenic ac o s (TAFs) which induce o he endo helial cells in a nea by essel o p o-
li e a e and mig a e chemo ac ically owa ds he umou .
The o he impo an p ocess occu ing du ing me as asis is he in asion. Tumou cells
on con ac wi h ex acellula ma ix (ECM) induce he p oduc ion o some p o eoly ic
enzymes, such as me allo-p o eases (MMPs) and se ine-p o eases. MMPs diges he ECM
and his enables he cance cells o mig a e h ough he issue.
In o de o unde s and be e he mechanisms leading o angiogenesis and in asion,
se e al models we e p oposed. Fo he a ea ela ed o angiogenesis we jus e e o he
ecen e iew pape [3] and he e e ences he ein. Conce ning umou in asion modelling
we b ie ly ecapi ula e some pape s.
2C. Mo ales-Rod igo
In [4] he au ho s p oposed a model o in asion. In his model he di usion o he
umou cells was neglec ed. They p o ided a a elling wa e analysis o his model, ind-
ing a singula ba ie which jus can be c ossed by he slowes membe o he amily o
a elling wa es connec ing he s eady-s a es. La e , in [5] he same sys em is s udied bu ,
by con as wi h [4] whe e jus egula a elling wa es we e ounded, he au ho s showed
a elling shock wa es which jump o e he singula ba ie . In [6] basing on expe imen al
da a, he au ho s alida e a model o in asion o he ib osa coma cell line HT1080. They
showed ha collagen concen a ion in luences he p oli e a ion o HT1080 in a biphasic
manne . Recen ly, in [7] he au ho examined he ole o he u okinase plasminogen sys-
em in cance in asion, showing how his sys em in luences he mig a o y p ope ies o
he cance cells.
In his pape we will conside a model o issue in asion ha has been p oposed by
Chaplain and Ande son in he ecen e iew book abou cance modelling [1]. They con-
side ed he ollowing a iables and ac s.
Cance Cells, n(x, ): The mo emen o cance cells is supposed o be by a andom mo ili y
and hap o axis i.e. up o he spa ial g adien s in he ex acellula ma ix.
Ex acellula Ma ix, (x, ): The ma ix is jus deg aded by he p o eases p oduced by
he umou .
P o eases, m(x, ): Fac o s in luencing he p o ease concen a ion a e assumed o be di -
usion, p oduc ion and na u al decay.
As a esul , he model eads as
∂n
∂ =
andom mo ili y
z}|{
dn∆n
hap o axis
z }| {
−γ∇·(n∇ ) in Ω ×(0, T ),
∂
∂ =
deg ada ion
z}|{
−ηm in Ω ×(0, T),
∂m
∂ =
andom mo ili y
z }| {
dm∆m
decay
z}|{
−αm
p oduc ion
z}|{
+βn in Ω ×(0, T ),
(1.1)
whe e dn, dm, α, β, γ and ηa e posi i e cons an s. Finally, deno ing by ν he uni ex e io
ec o o ∂Ω, he model is supplemen ed wi h no- lux bounda y condi ions on ∂Ω
∂n
∂ν −n∂
∂ν = 0 on ∂Ω×(0, T ),
∂m
∂ν = 0 on ∂Ω×(0, T ),
(1.2)
and he ini ial condi ions
n(x, 0) = n0(x) in Ω,
(x, 0) = 0(x) in Ω,
m(x, 0) = m0(x) in Ω.
(1.3)
Local exis ence and uniqueness in a model o issue in asion 3
In wha ollows and in o de o simpli y he o mulas we will suppose dn=dm=η=α=
β=γ= 1. Le us poin a ha ou calcula ions can be epea ed wi hou any p oblem o
gene al posi i e cons an s.
This pape is o ganized as ollows. In sec ion 2 we de ine he space in which is ou solu ion.
In sec ion 3 we p o e he exis ence and uniqueness o local-in- ime solu ion in such space.
2 No a ions
In his pape Ω ⊂IRNis an open, connec ed se wi h egula bounda y. QT= Ω ×(0, T )
is a cylinde o IRN+1. We conside he Banach space o H¨olde con inuous unc ions
Hk+α,(k+α)/2(QT) whe e k≥0 is an in ege and α∈(0,1). The associa e no m o his
space is gi en by
|u|k+α
QT:= huik+α
x,QT+hui(k+α)/2
,QT+
k
X
j=0huij
QT,
whe e
huij
QT:= X
2 +s=k
max
QT|D
Ds
xu|QT,
huik+α
x,QT:= X
2 +s=khD
Ds
xuiα
x,QT,
huik+α
,QT:= X
0<α+k−2 −s<2hD
Ds
sui(α+k−2 −s)/2
,QT,
and
huiα
x,QT:= sup
(x, ),(x0, )∈QT
|x−x0|≤ρ0
|u(x, )−u(x0, )|
|x−x0|α,0< α < 1,
hui ,QT:= sup
(x, ),(x, 0)∈QT
| − 0|≤ρ0
|u(x, )−u(x, 0)|
| − 0|α,0< α < 1.
The no m in he space Lp(Ω), 1 ≤p≤ ∞ is deno ed by k ·kp. The no m associa ed o
he classical Sobole spaces W1,p(Ω) will be deno ed by k·k1,p. Finally, he no m in he
space L∞(QT) is deno ed by k·k∞,QT.
3 Local exis ence and uniqueness o egula solu ions
Fi s o all we de ine a new a iable q=e− n, hen ou sys em is ans o med in o
∂q
∂ −∆q−∇q·∇ =−q =qm in Ω ×(0, T ),
∂
∂ =−m in Ω ×(0, T),
∂m
∂ = ∆m−m+qe in Ω ×(0, T ),
(3.1)
4C. Mo ales-Rod igo
wi h a new bounda y
∂q
∂ν = 0 on ∂Ω×(0, T ),
∂m
∂ν = 0 on ∂Ω×(0, T ).
(3.2)
Ac ually, his change o a iable has been p oposed in ano he pape s be o e as [8] and [9].
The main ad an age o his change is ha he i s equa ion o he sys em is in di e gence
o m.
In ou p oo , based on a ixed poin a gumen , he ollowing lemma will be equi ed.
Lemma 3.1. Le N≤3. Gi en such ha ∈Hα,α/2(QT), x∈Hα,α/2(QT) hen he
p oblem
∂q
∂ −∆q−∇q·∇ =qm in Ω×(0, T),
∂m
∂ = ∆m−m+e qin Ω×(0, T ),
(3.3)
wi h Neumann bounda y condi ions and egula ini ial da a admi s a unique egula solu-
ion (m, q)∈(H2+α,1+α/2(QT))2. Mo eo e , i q0,m0≥0 hen q(x, ), m(x, )≥0 o all
(x, )∈QT.
P oo . Conside he space o unc ions
X=C([0, T]; L2(Ω)).
We de ine he ope a o F:X→Xsuch ha F(q) = mwhe e mis he unique solu ion
o he linea equa ion
∂m
∂ = ∆m−m+e qin Ω ×0, T),
∂m
∂ν = 0 on ∂Ω×(0, T ),
m(x, 0) = m0in Ω.
(3.4)
On mul iplying (3.4) by mand in eg a ing in QTwe ob ain
∂
2∂ ZT
0kmk2
2+ZT
0k∇mk2
2+ZT
0kmk2
2=ZT
0ZΩ
e qm (3.5)
Applying H¨olde ’s inequali y and Young’s inequali y o he igh -hand-side o (3.5)
∂
2∂ ZT
0kmk2
2+ZT
0k∇mk2
2+µ1−1
2αke k∞,QT¶ZT
0kmk2
2≤α
2ke k∞,QTZT
0kqk2
2.(3.6)
Choosing α > 0 la ge enough in (3.6) and in eg a ing on he ime in e al [0, T] we ge
km(T)k2
2≤ km0k2
2+Tαke k∞,QTZT
0kqk2
2.(3.7)
Local exis ence and uniqueness in a model o issue in asion 5
Now, we de ine he linea ope a o G:X→Xsuch ha o each z G(z) is he unique
solu ion o
∂q
∂ −∆q−∇q·∇ =qz in Ω ×0, T),
∂q
∂ν = 0 on ∂Ω×(0, T ),
q(x, 0) = q0in Ω.
(3.8)
I is easy o see ha q∈Xis a ixed poin o H=G◦F hen is a weak solu ion o (3.3).
Taking z=F(q) = mand mul iplying (3.8) by qwe ob ain, a e in eg a ing in space.
d
2d kqk2
2+k∇qk2
2=ZΩ
q2m +ZΩ
q∇q·∇ . (3.9)
F om he Sobole inequali y kqk3≤Ckqk1/2
1,2kqk1/2
2, (N≤3), H¨olde ’s inequali y and
Young’s inequali y we in e
d
2d kqk2
2+k∇qk2
2≤ k k∞,QTkqk3kqk6kmk2+k∇ k∞,QTµα
2kqk2
2+1
2αk∇qk2
2¶≤
≤Ck k∞,QTkqk3/2
1,2kqk1/2
2kmk2+k∇ k∞,QTµα
2kqk2
2+1
2αk∇qk2
2¶≤
≤ k k∞,QT(α0kqk2
1,2+Cα0kqk2
2kmk4
2) + k∇ k∞,QTµα
2kqk2
2+1
2αk∇qk2
2¶.
Choosing α > 0 la ge enough and α0>0 small enough hen
d
d kqk2
2≤(2α0k k∞,QT+ 2Cα0kmk4
2+αk∇ k∞,QT)kqk2
2:= β( )kqk2
2.
I we choose qsuch ha kqkC([0,T];L2(Ω)) <kq0k2+ 1 = R hen, hanks o he es ima e
(3.7) β( )≤M∀ ∈(0, T), o ha
kq( )k2
2≤ kq0k2
2exp( M),∀ ∈(0, T ).
Clea ly, choosing Tsmall enough ollows ha kH(q) = qkC([0,T ];L2(Ω)) ∈BR. Fo ha ,
H:BR→BR. Now, we a e going o p o e ha His a con ac i e ope a o . Gi en
q1, q2∈BR, hen F(q1)−F(q2) = m1−m2sa is ies he equa ion
z −∆z+z=e (q1−q2) in Ω ×(0, T ),
∂z
∂ν = 0 on ∂Ω×(0, T ),
z(x, 0) = 0 in Ω.
(3.10)
On mul iplying (3.10) by m1−m2and in eg a ing in Ω we ob ain,
∂
∂ km1−m2k2
2≤Ckq1−q2k2
2,
6C. Mo ales-Rod igo
o ha ,
km1−m2kC([0,T];L2(Ω)) ≤C√Tkq1−q2kC([0,T ];L2(Ω)).(3.11)
We ha e ha H(q1)−H(q2) = q1−q2sol es he equa ion
z −∆z−∇z·∇ = m1z+ q2(m1−m2) in Ω ×(0, T),
∂z
∂ν = 0 on ∂Ω×(0, T ),
z(x, 0) = 0 in Ω.
(3.12)
Mul iplying (3.12) by q1−q2and in eg a ing in Ω gi es us
∂
2∂ kq1−q2k2
2+k∇(q1−q2)k2
2=
=ZΩ
(q1−q2)∇ ·∇(q1−q2) + ZΩ
m1(q1−q2)2+ZΩ
q2(m1−m2)(q1−q2)≤
≤ k∇ k∞,QT(²k∇(q1−q2)k2
2+C²kq1−q2k2
2)+
+k k∞,QT(α0k∇(q1−q2)k2
2+ (α0+Cα0km1k4
2)kq1−q2k2
2)+
k k∞,QT(²0(kq1−q2k2
2+k∇(q1−q2)k2
2) + C²0kq2k2
3km1−m2k2
2).
Choosing α0, ², ²0posi i e and small enough we in e
∂
∂ kq1−q2k2
2≤α( )km1−m2k2
2+βkq1−q2k2
2,
whe e βis a posi i e cons an and α( ) = C²0k k∞,QTkq2k2
3. Then,
∂
∂ ³e−β kq1−q2k2
2´≤α( )e−β km1−m2k2
2.(3.13)
F om he Sobole ’s inequali y kq2k2
3≤Ckq2k1/2
1,2kq2k1/2
2and aking in accoun ha q2∈
C([0, T]; L2(Ω)) ∩L2(0, T ;H1(Ω)) we ge
ZT
0
α(s)ds ≤M. (3.14)
Finally hanks o (3.14) and (3.11) we ob ain
kH(q1)−H(q2)kC([0,T];L2(Ω)) ≤CpMeβT pTkq1−q2kC([0,T];L2(Ω)).
Choosing T≤Tsmall enough, H:BR→BRis con ac i e and om Banach’s ixed poin
heo em we in e ha p oblem (3.3) ha e a unique solu ion in he space
C([0, T]; L2(Ω)) ∩L2(0, T ;H1(Ω)).
Local exis ence and uniqueness in a model o issue in asion 7
Since N≤3 hen he unc ion q∈L2(0, T;L6(Ω)). We begin an i e a i e a gumen ha
will p o ided us egula iza ion o ou solu ion. Le p= 6, mul iplying by pmp−1 he second
equa ion o (3.3) and in eg a ing by pa s, gi es us
∂
∂ kmkp
p+4(p−1)
pk∇(mp/2)k2
2+pkmkp
p=pZΩ
e qmp−1≤
≤pke k∞,QTkqkpkmp/2k2kmk
p−2
2
p≤pke k∞,QT(Cαkqk2
pkmkp−2
p+αkmp/2k2
2).
Choosing α > 0 small enough he ollowing es ima e ollows
∂
∂ kmkp
p≤pMkqk2
pkmkp−2
p.
F om his di e en ial inequali y we in e
km( )k2
p≤ km0k2
p+Z
0
p2
2Mkq(s)k2
pds, (3.15)
o all ∈[0, T]. Since q∈L2(0, T;Lp(Ω)) hen he in eg al e m on he igh -hand-side
o (3.15) is ini e. The e o e m∈L∞(0, T;L6(Ω)).
On mul iplying he i s equa ion (3.3) by pqp−1, hen a e in eg a ing by pa s, we ob ain
∂
∂ kqkp
p+4(p−1)
pk∇(qp/2)k2
2=pZΩ
qp−1∇u·∇ +pZΩ
qpm . (3.16)
Now, we a e going o ind he p ope bounds o he wo in eg als on he igh -hand-side.
pZΩ
qp−1∇q·∇ ≤ k∇ k∞,QT(C²kqp/2k2
2+²p2kqp
2−1∇qk2
2)
=k∇ k∞,QT(C²kqkp
p+ 4²k∇(qp/2)k2
2).
(3.17)
Taking in accoun ha m∈L∞(0, T;L6(Ω)) we ge o p= 6 ha
pZΩ
qpm ≤pk k∞,QTkqp/2k2kqp/2k3kmk6
≤pk k∞,QTkqp/2k3/2
2kqp/2k1/2
1,2kmk6
≤pk k∞,QTkqp/2k2kqp/2k1,2kmk6
≤ k k∞,QT(²0kqp/2k2
1,2+p2C²0kqp/2k2
2kmk2
6)
≤ k k∞,QT(²0kqp/2k2
1,2+p2MC²0kqp/2k2
2).
(3.18)
Choosing ², ²0small enough and pu ing he es ima es (3.17), (3.18) in (3.16) we ob ain
∂
∂ kqkp
p+k∇(qp/2)k2
2≤Cp2kqkp
p.
Easily a e in eg a ing on [s, ]⊂[0, T], implies
kq( )kp
p+Z
sk∇(qp/2)k2
2≤exp(Cp2( −s))kq(s)kp
p.(3.19)
8C. Mo ales-Rod igo
F om (3.19) and ollowing he same a gumen as in [10, p. 1197] we can p o e ha
q∈L∞(QT). Howe e , o comple eness we p esen i he e.
Conside any ∈(0, T ]. Fo simplici y =T, al hough his a gumen emains ue o
e e y ∈(0, T]. Take 0∈(T−1, T), σ= 3. De ine pm= 6σmand δm= (T− 0)σ−2m−1.
Obse e ha p2
mδm=c. Now, conside he in e als Im= [T−σδm, T −δm]. We de ine
he sequence Nm= supτ∈[ m,T]kq(τ)kpmwhe e m∈Imwill be de e mined la e . I we
apply (3.19) wi h s= m+1 and τ∈[ m+1, T ] hen
Nm+1 = sup
τ∈[ m+1,T]kq(τ)kpm+1 ≤(exp(Cp2
m+1σδm+1))1/pm+1 kq( m+1)kpm+1 (3.20)
We ha e o de e mine kq( m+1)kpm+1 . Thanks o he Sobole ’s embedding,
kq( m+1)kpm
pm+1 =kqpm/2( m+1)k2
6≤M(kq( m+1)kpm
pm+k∇(qpm/2( m+1))k2
2) (3.21)
We a e going o de e mine k∇(qpm/2( m+1))k2
2. Applying (3.19) o s= mand =
T−δm+1 (so Im+1 ⊂[s, ]) we ge
in
τ∈Im+1 k∇(qpm/2(τ))k2
2≤ |Im+1|−1exp(Cp2
m(T−δm+1 − m))kq( m)kpm
pm.
Choosing m+1 =τwe ob ain he es ima e we we e looking o . Since |Im+1|<1 hen
wi h a simila a gumen we can es ima e kq( m+1)kpm
pm. Pu ing his es ima e in (3.21) we
ge
kq( m+1)kpm
pm+1 ≤2σ2(σ3−1)−1δ−1
mexp(Cp2
m(1 −σ−2)δm)kq( m)kpm
pm
≤Cδ−1
mNpm
m
(3.22)
Thanks o (3.22) and aking in accoun ha p2
mδm=C1, we ob ain om (3.20)
Nm+1 ≤(exp(M)1/σCδ−1
m)1/pmNm
≤Ãm
Y
i=0
(C2σ2i)1/pi!N0=zmN0
Clea ly zmis ini e o all mbecause ln zm=Pm
i=0 1
6σi(ln C2+2iln σ) whe e σ > 1. Finally,
kq(T)k∞≤sup
m≥1
Nm≤C3N0≤C3sup
τ∈[0,T]kq(τ)k6<∞.
Repea ing he same a gumen o mwe ge he same egula i y. Now, he egula i y can
be imp o ed hanks o [11, Chap e 3, Theo em 10.1] and [11, Chap e 3, Theo em 12.1] ,
he i s one gi es us q, m ∈Hα,α/2(QT) and hen we can apply he second one ob aining
q, m ∈H2+α,1+α/2(QT). Since m is bounded in L∞(QT) hen, om maximum p inciple
o pa abolic equa ions we ge he posi i i y o q. Now, om he posi i i y o qwe can
in e , hanks o he maximum p inciple, he posi i i y o m.
Theo em 3.2. I he ini ial condi ion (1.3) a e egula hen he p oblem gi en by (1.1)
wi h he bounda y condi ion (1.2) and ini ial condi ion (1.3) espec i ely, has a unique
local solu ion in he space (H2+α,1+α/2(QT))3.
Local exis ence and uniqueness in a model o issue in asion 9
P oo . We de ine he ollowing ball in H={ : ∈Hα,α/2(QT)∧ x∈Hα,α/2(QT)}
Bδ( 0) = {u:|u− 0|α
QT< δ ∧|(u− 0)x|α
QT< δ}.
Now, we de ine he ope a o K:Bδ( 0)→H.K( ) is he unique solu ion o he o dina y
di e en ial equa ion
∂
∂ =−m , (x, 0) = 0(x),
whe e mis gi en as he solu ion o he second equa ion in (3.3) wi h = . Fo simpli y
he calculus, we conside 0= 1, he same calculus can be done wi h a gene al 0. We
ha e,
(x, ) = 1 + Z
0−m(x, s) (x, s)ds = 1 + Z
0−m(x, s)e−Rs
0m(x,θ)dθ
By de ini ion | −1|α
QT=h −1iα
x,QT+h −1iα/2
,QT+ maxQT| −1|.
h −1iα
x,QT:= sup
(x, ),(x0, )∈QT
|x−x0|≤ρ0
|R
0−m(x, s)e−Rs
0m(x,θ)dθ +R
0−m(x0, s)e−Rs
0m(x0,θ)dθ|
|x−x0|α≤
≤sup
(x, ),(x0, )∈QT
|x−x0|≤ρ0
kmk∞,QT³R
0ds Rs
0|m(x0, θ)−m(x, θ)|dθ´+R
0|m(x, s)−m(x0, s)|ds
|x−x0|α≤
≤T2kmk∞,QThmiα
x,QT+Thmiα
x,QT
h −1iα/2
,QT:= sup
(x, ),(x, 0)∈QT
| − 0|≤ρ0
|R
0−m(x, s)e−Rs
0m(x,θ)dθ +R 0
0−m(x, s)e−Rs
0m(x,θ)dθ|
| − 0|α/2≤
≤sup
(x, ),(x0, )∈QT
| − 0|≤ρ0
|R 0
m(x, s)e−Rs
0m(x,θ)dθ|
| − 0|α/2≤ | − 0|1−α/2kmk∞,QT
max
QT| −1| ≤ Tkmk∞,QT
Also by de ini ion |( −1)x|α
QT=h xiα
x,QT+h xiα/2
,QT+ maxQT| x|
( −1)x=Z
0
e−Rs
0m(x,θ)dθ µ−mx(x, s) + m(x, s)Zs
0
mx(x, θ)dθ¶
Le deno e ax=e−Rs
0m(x,θ)dθ,bx=m(x, s), cx=Rs
0mx(x, θ)dθ and dx=mx(x, s) hen
h( −1)xiα
x,QT:= sup
(x, ),(x0, )∈QT
|x−x0|≤ρ0
|R
0ax(−dx+bxcx)−ax0(−dx0+bx0cx0)ds|
|x−x0|α≤
