Global Existence of Classical Solutions to a Cancer Invasion Model

Global Existence of Classical Solutions to a

Cancer Invasion Model

Khadijeh Baghaei1*, Mohammad Bagher Ghaemi1, Mahmoud Hesaaraki2 1_{Department of Mathematics, Iran University of Science and Technology, Tehran, Iran}

2_{Department of mathematics, Sharif University of Technology, Tehran, Iran} Email: *_{[email protected]}

Received January 25,2012; revised March 8, 2012; accepted March 15, 2012

ABSTRACT

This paper deals with a chemotaxis-haptotaxis model of cancer invasion of tissue. The model consists of three reaction- diffusion-taxis partial differential equations describing interactions between cancer cells, matrix degrading enzymes, and the host tissue. The equation for cell density includes two bounded nonlinear density-dependent chemotactic and haptotactic sensitivity functions. In the absence of logistic damping, we prove the global existence of a unique classical solution to this model by some delicate a priori estimate techniques.

Keywords: Cancer Invasion Model; Chemotaxis; Haptotaxis; Global Existence

1. Introduction

Cancer invasion is associated with the degradation of the extra cellular matrix (ECM), which is degraded by matrix degrading enzymes (MDEs) secreted by tumor cells. The degradation creates spatial gradients which direct the migration of invasive cells either via chemotaxis (cellular locomotion directed in response to a concentration gra-dient of the diffusible MDE) or via haptotaxis (cellular locomotion directed in response to a concentration gra-dient of adhesive molecules along the ECM). Chaplain and Lolas [1] proposed a PDE model of cancer invasion of tissue, which considers the competition between the following several biological mechanisms: random diffu-sion, chemotaxis, haptotaxis and logistic growth.

Actually, cancer invasion is a very complex process which involves many various biological mechanisms. In fact, a variety of mathematical models have been devel-oped for various aspects of cancer invasion, and various attempts to give more biologically relevant models have been made by different people (see [2], for instance). Gatenby and Gawlinski [3] useda reaction-diffusion popu- lation competition model to study how the tumor invades the surrounding normal tissue or ECM. They suggested that tumor cells createan acidic environment that is toxic to normal tissue, and the high acidity gives rise to the death of the normal tissue, which provides space for tu- mor cells to proliferate and invade into the surrounding tissue. In contrast to the acid-invasion mechanism, Pe- rumpanani and Byrne [4] found that the ECM heteroge-

neity affects suchinvasion. They proposed a model under the assumptions that the ECM is degraded by proteases. The proliferation of tumor cells and the remodeling of the ECM are taken into account in the Chaplain and Lo-las model. Recently, Gerisch and Chaplain [5] devel- oped a novel non-local model which incorporates cell- cell adhesion and cell matrix adhesion, playing important roles in the tumor invasion process.

Very recently, Szymańka et al. [6] proposed a non-

local model which focuses on the role of non-local ki- netic terms modeling competition for space and degrada- tion; Szymańka et al. [7] also discussed the influence of

heat shock proteins on cancer invasion of tissue. The analytical results on various models of cancer invasion are mathematically interesting. Walker and Webb [8] proved the global existence solutions to the Chaplain and Anderson’s model [9]. Walker [10] also established the global existence of solutions to an age and spatially- structured haptotaxis model, which can be regarded as an extension of the Chaplain and Anderson’s model [9]. Mar- ciniak-Czochra and Ptashnyk recently [11] proved the uniform boundedness of solutions to the haptotaxis model [9]. Szymańka et al. [6] proved the global existence of

solutions to their non-local model.

are haptotaxis only models. However, Chaplain and Lolas’ model [1] is a parabolic-ODE-parabolic-chemotaxis-hapto- taxis system. The global existence and uniqueness of classical solutions to this model has been proved for

0

 (where  is the growth rate of cancer cells) in one space dimension (see [15]), for 0 in two space dimensions (see [16]) and for large  in three space dimensions (see [15]).We should note that the global existence is still open for small 0 in three space dimensions for the parabolic-ODE-parabolic chemotaxis- haptotaxis system and the parabolic-ODE-elliptic chemo- taxis-haptotaxis system.

Recently, in addition to global existence and uniqueness, the uniform-in-time boundedness of solutions to a sim- plified parabolic-ODE-elliptic-chemotaxis-haptotaxis sys- tem has been proved for 0 in two space dimensions and for large  in three space dimensions (see [17]).

This paper tries to analytically study a mathematical model of cancer invasion with 0. When 0, the solution Chaplain and Lolas’ model can blow up in finite time (see Section 6, [15]). However, it is obvious that the blow-up of cancer cell density in finite time is biologi- cally irrelevant. Hence, we need to deal with the follow- ing problem: how to reasonably modify the Chaplain and Lolas’ model [1] to obtain the global existence, which is the cancer of the present paper.

This paper extends Chaplain and Lolas’ model to a parabolic-parabolic-parabolic chemotaxis-haptotaxis sys- tem, and we study the global existence and boundedness of solutions to this model. This paper organized as follows: Section 2 describes the model. Section 3 proves the local existence and uniqueness of solutions. Section 4 estab- lishes some a priori estimates and proves the global exis- tence.

2. Mathematical Model

The mathematical model of cancer invasion is involved in the following three physical variables: cancer cell den- sity c x t



,



, ECM density v x t

 

, and MDE concentra- tion u x t

 

, .

The equations describing the dynamics of each vari-able read as follows:





 



 





1 random motion _randommotion

2 haptotaxis

. ,

c

c

D V

t

V c

c c

v 

 

   



 







u 

 





(1)



 

random motion _proteolysis

v v v

v

D F u

t 



 

  (2)

  

production _decay diffusion

,

u u

D u c

t

where D D Dc, v, u, , , and    are assumed to be posi-tive constants and V c1

 

and are the density-de-

pendent chemotactic and haptotactic sensitivity functions, respectively.

 

2

V c

In Equation (1), the migration of cancer cells is as-sumed to be governed by random motion, chemotaxis and haptotaxis. In Equation (2) is assumed that ECM has random motion and its degradation by MDEs upon con-tact; for simplicity, we assume that no remodeling of the ECM takes place, as done in [15,18]. Since random mo-tion ECM is so small hence we assume that Dv is small

positive constant. In Equation (3), the MDE concentra-tion is assumed to be influenced by diffusion, producconcentra-tion and decay; specifically, MDE is produced by cancer cells, diffuses throughout ECM, and undergoes decay through simple degradation. We shall consider the system (1)-(3) in a bounded domain _ in_d



d 2 or 3 ._



For any 0  T we set





Ω_T  Ω 0 t T ,





ΩT Ω 0 t T .

     

To close the system of equations, we need to impose boundary and initial conditions.

Boundary conditions:

The boundary conditions are represented by the fol-lowing equalities:

0 on ΩT,

c u v

n n n

 _ _ _{ }

   (4)

where n is the our ward normal vector to ∂Ω.

Initial conditions: We prescribe the initial data

 

   

 

 

 

0 0

, 0 , , 0 ,

, 0 , Ω

c x c x v x v x

u x u x x

 

 

0

(5)

Throughout this paper we will assume that

 

1





_0,





_,

 

_0,

 

_{0 0,}

i i

V c C  V c  Vi  (6)

 

i

V c is Lipschitz continuous, (7) where i = 1, 2 and

 

1





_0,





_and

 

_0.

F u C  F u  (8)

In Chaplain and Lolas’ original model [1], it is assumed that V c₁

 

c V c, ₂

 

c and F u

 

u (where  , and  are some positive constants). For this choice of

   

1 ,V c2

V c and F u

 

, although the assumptions (6)- (8) are satisfied but we would like to slightly modify the choice of V c V c₁

   

, ₂ a

 

u such that the modi- fied model has a unique global solution. To this end, in addition to the assumptions (6)-(8), we will assume that

ndF

 

 

1 and 2 are bounded for any 0,

V c V c c (9)

u

 

 _{   }

For example, we may take ₁

 

1

, 1

c V c

c  



 

2

2

1

c V c

c  

 and

 

1 3

u F u

u  

 ( are

small positive constants ). Clearly

1, and2 3

  

c

 

1

V c  _as

 

10,V c2 c



 

as 3

F u u 

as and

For this choiceof

20



0.

 V c V₁

   

, ₂ c , andF u

 

of

, the assumptions (6)-(10) are satisfied. An-other choice ₁

 

c an

 

c satisfying (9) is that

for _{which has a clear}

biologically relevant interpretation: the cancer cells stop to accumulate at a given point of the tumor tissue after their density attains a maximal density . A similar as- sumption for a prey taxis sensitivity function was made in [19].

V

 

 

1 0 and

V c  V c 

d V₂ 

2 0 m,

c c c

m

In next section we will prove the local existence and uniqueness of a solution for the system (1)-(5) by a fixed point argument.

3. Local Existence and Uniqueness

Throughout this paper we assume that

 

 

 

     

 

0 0 0

2

2

0 0 0

0 0 0

0, 0, 0,

Ω ,0 1,

, , ,

( ) ( ) ( )

0 on .

c x v x u x

C

c x v x u x C

c x v x u x

n n n











  

   

 

  

   

   Ω

(11)

For brevity we set



, ,

U c v u



(12) For notations’ convenience, in what follows we denote various constants which are independent of T by A0,

whereas we denote various constants which depend on T by A .The constants A0 and A may be different from line

to line.

In the following, under the assumptions (6)-(8) and (11), we shall prove that the system (1)-(5) has a unique local (in time) smooth solution.

Theorem 3.1. Under the assumptions (6)-(8), there ex-

istsa unique solution 2 ,1 2



_Ω

T

UC  



of the system (1)-(5) for some small T 0 which depends on

 

.,0 _C2 _{ Ω} .

U _

Proof. We shall prove the local existence by a fixed point argument. We introduce the Banach space X of the

vector function U (defined in (12)) with norm

 

1 , 2_Ω (0 1)

T C

U  U   T



:



,

M

X  UX U M

where

 

_{ }

 

_{ }

 

 

2 2

2

2

0 _Ω 0 _Ω

0 Ω

Ω

:

1

(.,0) 1

C C

C

C

M c x v x

u

U

 





 





 

 

 

,

M UX

Given any we define a corresponding.

Function U FU by



and a subset

( , , )

U c v u , where U

sat-isfies the equations

 

0 inΩ ,

v T

v

D v F u v t



   

 (13)

0 on ΩT,

v n

 _{ }

 (14)

 

,0 0

 

inΩ,

v x v x (15)

inΩ ,

u T

u

D u u c

t  

 _{   }

 (16)

0 on ΩT,

u n



 

 (17)

 

,0 0

 

inΩ,

u x u x (18)





1 , , inΩ ,

c T

c

D c h c u v t

 _{  }

 (19)

0 on ΩT,

c n



 

 (20)

 

,0 0

 

inΩ,

c x c x (21)

where







 



 





 

 

 

 





1 1

2

1 2

1 2

, ,

h c u v V c u

V c v

V c u V c v

V c u V c v c

 

 

 

   

   

    

 

    

(22)

We first consider the linear parabolic (13)-(15). By (8), (11) and the parabolic Schauder theory (for example, see [20]) there exists a unique solution v, and

 

 

 

2 ,1 2ΩT 0 ,0 2 Ω 0

C C

v   A v x  AM (23)

Similarly, from UX_M,

(16)-(1

(11) and the parabolic Schau-

der theory, problem 8) has a unique solution u

satisfying

 



 

   



2 ,1 2Ω 0 2 Ω , 2Ω

0

,0

T T

C C C

u A u x c

A M

     

    



 

(24)

 

2

V c are Lipschitz continuous, we have

  , 2

1 Ω 0 ,

h   A M

T

C (25)

Hence, by Schauder theory as before, the problem (19)- (21) admits a unique solution c satisfying

 

 

_{   }





2 ,1 2

, 2 2

Ω

0 ,0 Ω 1 Ω

T

T C

C

C 0

c

A c x  h   A M

   (26)

onclude from (23), (24) and (26) that

   

We c

 

2 ,1 2_Ω 0 .

U A M

T

C  (27)

By direct calculations, we obtain

 

 

_{ }

 

_{ }

1 , 2

1 , 2

Ω

0 _Ω

T

T

C C

A T U

 

 









If we further take T

suffi-, ,0

U x t U x

where 

 

T 0 ifT 0.

small, then by (27) ciently

 

_{ }

 

_{ }

 

 

_{ }

 

_{ }

 

_{ }

 

_{ }

 

 

_{ }

1 , 2 1

1 , 2

1 2 ,1 2

1

1

Ω Ω

Ω

0

Ω Ω

0

Ω

Ω

,0

, ,0

,0

,0

,0 1

.

T

T

T

C C

C

C C

C C

U x

U x t U x

U x A T U

U x A T M

U x

M

  

 

  









 



  





 

 

 

 

 ,

,

U x t 

Hence UXM, i.e. F maps XM into itself. We

that

next show F is contract mapping. Take

1, 2in ,

a ion

M

U U X and set U₁FU₁ and U₂FU₂. set-ting

1 2

U U

  

We derive from (13) that





_

_

 





 

 



1 2 1 2

1 1 2 2 2 1 ,

v

U U D v v

t

F u v v v F u F u

     



   

1 2

v v

 

where

 

 





_{ }

  , 2

, 2

2 2 1 _Ω

0 1 2 Ω

0 .

T

T C

C

v F u F u

A U U

A

 

 

 

 



Hence, since



v1v2



_t00, Schauder theory yields

  2 ,1 2

1 2 Ω 0 .

v A

T C

v   



ilarly, we derive from (16) and

 (28)

Sim



u1u2



_t00,

that

   

 

2 ,1 2 , 2

, 2

1 2 Ω 0 1 2 Ω

0 1 2 Ω

0

T T

T

C C

C

u u A c c

A U U

A

   

 



 

  

 



(29)

We next turn to the equation for c1c2:



c1 c2



_

_

D c c

 

1 2 2

c h

t    

 (30)

where

  





 

 



  





 

 



 





 





 

 





 





 





 

 





2

h   _{1 1 1 2 2 1 1 1 2}

2 1 1 2 2 2 1 2 2

1 1

1 1

1 1

1 2 1 2 1 2

1 2 2 2

1 1 1 2 1 2 1 2

1 2 2 2

2 2

2 2

V V

V

V c u u u V c V c

V c v v v V c V c

c c u u c u c c

c c u c

c c v v c v c c

c c v

V

V V

V V c

 

 

 



 



  

   

      

 

 

 

 

 

   

       

  



 

 

Noting V1

 

c and V2

 

c are Lipschitz continuous

and using (6), (7), (27), (28 nd (29), we have ) a

  , 2

2 C ΩT 0 . h   A



c1c2



_t00, By Schauder theory, since

  , 2

2C  Ω A0

  2 ,1 2

1 2 C ΩT 0 T .

c c   A h   (31)

Combining this with (28) and (29), we get

  2 ,1 2

1 2 _{C Ω} 0 .

U U   A

T



 

 



U1U2



 

x,0 0

Nothing and proceeding as

be-fore, we have

 

 

_{ }

 

 

_{ }

1 , 2

2 ,1 2Ω

1 , 2

1 2 _Ω 0 1 2

0 0 1 2 Ω

T C T

T

C

C

U U  A T U U

A T A T U U

 

 

 



  

 



 

   (32)

Taking T small such that ₀

 

1

2

A T  we conclude

from (32) that F is a contraction in XM.

fi

By the contrac- tion mapping theorem F has a unique xed point U in

M

X which is the unique solution of (1)-(5).

es and Global Existence

4. A Priori Estimat

e To continue the local solution established in the abov section to all t > 0 we need to establish some a priori

es-timates. Throughout this section, in addition to the as-sumptions (6)-(8) and (11) we assume that the assump-tions (9) and (10) hold.

Noting V₁

 

0 V₂

 

0 0, c₀

 

x 0, v₀

 

x 0 and

 

0 x 0, easily

ollowing lemma.

Lemma 4.1. Assume that





u and using the maximum principle, we

f prove the

 

2,1

, , Ω is

a solution to (1)-(5), then T

0, 0,

c v 33)

Lemma 4.2. Assume that 0.

u (





2,1

 

U c 

2

p ,

, ,v u C ΩT is

we have a solution to (1)-(5) then for

 Ω p

T L

c A, (34)

 

2,1 0, (35)

p T

W

v _ A

 

2,1_Ω .

p T

W

u A (36)

Proof. For p2, we derive from









 





 













 

2 1 2

Ω Ω Ω

2 1 Ω 2 2 Ω 2 2 2 Ω 2 0 _Ω 2 2 Ω d d 1 d 1 d 4 1 d 1 d

1 d .

p p p c p p

c c c c x

p p c V c c u x

p p c V c c v x

p D

c x

p

A p p c c u x

p p c V c c v x

                              











(37)

We now consider the integral x. By

Equation (3) and the parabolic

d p_d p _{d 1}

t c

c x p x p p D

t







  



2

Ω d

p

c   c v u



p

L estimate [20] we have

 



   



 

0 0 pΩ ,

T

L

A A c

 



In particular,

2,1_{Ω 0 0} 2_Ω

Ω

p

p T p T

W W L

u A u  c

(38)

  p Ω 0 0 p Ω

T T

t L A A cL

   (39)

Multiplying Equation (3) by

Ω p T L u u 1 p

c  and in , we obtain

d d

p

t

And there for, by Young’s inequality and estimate (39), we have

tegrating in

Ωt



1



t t

p D 2

0Ω 0Ω

1 1

0Ω 0Ω

d d

d d d d

p u

t t

p p

t

c c u x t c x

c u x t c u x t          





















 

 

2 0

0Ω 0Ω

0 0

0Ω 0Ω 0Ω

1

t t

p d d d d

d d d d d d .

p p

u

t t t

p

p p

t

D c c u x t A p c x t

u x t u x t A p c x t A

_{  }    











 







(40)

Also, by Equation (2) and the parabolic _Lp _estimate

we have

   

2,1_{Ω 0 0} 2_{Ω 0}

p T p

W W

v A v A (41)

In particular,

 Ω 0. p

T L

v A

  (42)

By (9), Young’s inequality and estimate (42)

 









2 2

2 0

0Ω 0Ω

c d d d

p p

c V  c v x tA c   c v









2 2

2 2

0

0Ω 0Ω

2 2

0 2

0Ω 0Ω

2 2 2

0Ω 0Ω

0 0

0Ω

d

d d ( ) d d

4

d d , d d

4

d d d d

, d d

t t t t p p t t p p t t p p t p x t

c c x t A c v x t

c x t A p c x t

p

v x t c x t

p

A p c x t A

              





























      (43)

Integrating with respect to t on both sides (37), noting

(11) and using estimates (40) and (43) and taking su

 fficiently small, we obtain



_

_





 

 

 

0 Ω 0 0 0Ω 0 0 0Ω d d ,

d d .

c

t p

t p

c x s

A p c A p

A p c x t A p     

 

 

 



Gronwall’s lemma yields

(44)

Now, by (39) and (44) we have

2 2 Ω

4 1

d t

p p p

c x D

p    





 

0Ω . T A p  d d p c x t





 

 

2,1_Ω p T W

u A p . (45)

This completes the proof of lemma

In the following result we obtaina better bound of c, a

4.2.

L—bound. Let p > 1 and define B:  _c

main

 

 

D I, with

do 2 _{Ω : 0}

p

c

D B c W

n  on   _    _ 

Ω For 

 each

0,

 define the sectorial operator B (see [21]) and

 

:

X D B with the norm X Lp.

ma 4 3

c  B c

Lem . . Let 21,



ve

then for t



t T0, where

0 0

t  , we ha

( ) A. (46)

X

c t



Proof. We have that

 

0

c t e c

 







 



( ) 1 2 0 d tB t t s B

e V c u V c v

 

 







 



0

( )

1 2

0

d tB

X X

t t s B

X

c t e c

e V c u V c v c

 



 



 



 



_

_       



s 

(47) By [21] (Theorem 1.4.3) and (11)

 

0 0 0 pΩ 0

tB t t

L X

e c A t e c A t e



 

 _   _   ₍₄₈

and, by (34),

)

 

_

_

 

 





 

Ω

,

p

s L t s

c

A t s e 





  

   (49)



. Moreover, by [22] (Lemma 2.1), (9),

(35) an ain

0

t s B t

X

e c A t s e



  _{ }  

where Q



0,1

d (36), we obt

 







 



 







 



 

 

 



 

( )

1 2

( )

0 1 2

1 ( 2

( )( )

X

t

A t s e





   _{ }

 _    )

0 1 2 _(Ω)

1

2 ,

p t s B

t s

s

L

t s

e V c u V c v

A e V c u V c v

V c u V c v

A t s  e

 

 

 

 

 

    

       

 

 

 _       _

  

 _   

(50)

where

X_

1

0 such that 1.

2 

   

   

Inserting (48)-(50) into (47) and noting 1 1

2

   

and  1, we obtain

 

  



1₂ ( )





t s  



0

0

d

t X

t

t s A t e



0

c t

A  t s    e  t s e

 _

_

_     

s

A t



  

 



 

 





for all .

Th of lemma 4.3

Lemma 4.4. We have that





0,



(0 0 )

t t T  t T

is completes the proof .

 

_{ Ω} for all



0,



.

L

c t  A t T (51)

Proof. Let P d,2 ,1 . since 2 p

 

 _{ }



d d

p

 

 we have

by

 [21] (Theorem 1.6.1) that

 

Ω

X_c

Thanks to lemma 4.3 we have that

 

_{L  Ω}

 

0 0

c t  A t for 0.tt 

Moreover, the local existence Theorem yields

 

_{L  Ω} 0

c t  A

Therefore

 

_(Ω) for all



0,



.

L

c t  A t T

This completes the proof of Lemma 4.4.

We ha at

Lemma 4.5. ve th

  2,1_Ω

p T

W

c A (52)

Proof. By the Sobolev embedding theorem (see [20],

(L tly large, then

(3

emma 3.3, p. 80)), if we take p sufficien

5) and (36) yields

  2 _Ω ,

T A

, 2_{Ω 0,} ,

T

C C

v   A u  

  

an

 (53)

d therefore

 Ω_T 0,  Ω_T ,

L L

v  A u  A

    (54)

By (7) and (51), we have

 

 

 

_{ }

Ω

1 2

Ω ,

L

L

V c _ A V c A



  . (55)

Now, Equation (1) can be rewritten as

for tt0.

 

 





 

 

1 2

1 2

.

C u V c v c

t

V c u V c v



 

c

D c V c

 _{ }

      



    

where

 

 

 

1 2

Ω

L

V c u V c v A

 



 _{ }  _{  (56)}

 

 

_{ }

1 2 p_Ω

T L

V c u V c v A

     (57)

), (35), (36), (53) and (55). These, along with (11) and the parabolic

By (9

p

L estimate, yield the es

Lemma 4.6. Assume that

timate (52).



_{, ,}



2,1

 



T U c v u C is

a solution to (1)-(5), then

  2 ,1 2_Ω

T C 

Proof. By (52) nd the Sobolev embedding theor .

U A (58)

a em

(taking p large),

  , 2

C

c   _Ω .

T A

 (59)

Also, (35), (36) and the Sobolev em (ta

bedding theorem king p large) yield

   ΩT

, 2_{Ω 0}, , 2 .

T

C A uC  A

v     (60)

Now, from (3), (11), (59) and the parabolic Schauder estimates we have

  2 ,1 2_Ω

T C

u   A (61)

Also, the parabolic Schauder estimates yield

 

2 ,1 2Ω 0.

C

v   A (62) T

London/Boca Raton, 2003, pp. 267-297.

[10] C. Walker, “Global Existence for an Age and Spatially Structured Haptotaxis Model w

 

 

 

 

 

_

, 2

, 2

1 2

Ω

1 2

C C

V c u V c v _{ }

 

  _{ }  _

Hence, by the parabolic Schauder estimates, we obtain

A



 ΩT

T

V c u V c v A

     _{ary Conditions,”} ith Nonlinear Age-Bound-

European Journal of Applied Mathema- tics, Vol. 19, No. 2, 2008, pp. 113-147.

doi:10.1137/060655122

 

2 ,1 2_Ω .

c A

T

C  (

This completes the proof of Lemma 4.6

With a priori estimate (58), we can extend the local classical solution established in Theorem 3.1 to all as

Theorem 4.7. There exists a uniq

63) [11] A. Marciniak-Czochra and M. Ptashnyk, “Boundedness of Solutions of a Haptotaxis Modelsmodel,” Mathematical Model and Methods in Applied Sciences, Vol. 20, No. 3, 2010, pp. 449-476. doi:10.1142/S0218202

.

510004301 s-Rodrigo,” Global Solutions and

av-

icroscopic and Non-Local

0

t , [12] G. Litcanu and C. Morale

Asymptotic Behavior for a Parabolic Degenerate Coupled System Arising from Biology,” Nonlinear Analysis, Vol. 72, No. 1, 2010, pp .77-98.

[13] G. Litcanu and C. Morales-Rodrigo, “Asymptotic Beh

done in [15]. Namely we have

ue global solution

 

2 ,1 2 _Ω T

UC   of the system (1

Invas Network and H

pp. 399-439. doi:10.3934/nhm.2006.1.399

)-(5) for any given

T 0.

ior of Global Solutions to a Model of Cell Invasion,” Mathematical Model and Methods in Applied Sciences, Vol. 20, No. 9, 2009, pp. 1721-1758.

[14] M. Lachowiz, “Towards M

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