Global Solution of a Degenerate Evolution System in ... fileGlobal Solution of a Degenerate Evolution System in Population Dynamics 1Abdullah Mohammed and 2Messaoud Souilah 1USTHB

Global Solution of a Degenerate Evolution System in

Population Dynamics 1Abdullah Mohammed and 2Messaoud Souilah

1USTHB,

El Alia Bab Ezzouar, Algiers,

Algeria.

[email protected]

2USTHB,

El Alia Bab Ezzouar, Algiers,

Algeria,

[email protected]

Abstract We study the global existence of solutions to a degenerate reaction

diffusion system intervening in population dynamics (SI model) or ecology

(predator prey model). Perhaps except the Lyapunov technique the

classical methods to study the global existence of its solution don’t apply.

We use here an ancient method due to [1] for studying the local solution

which we perform to obtain the global solution.

Index Terms:Global existence, reaction diffusion systems, population

dynamics, SI, SIR, SIER models, self diffusion.

International Journal of Pure and Applied MathematicsVolume 117 No. 20 2017, 485-497ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

485

1. Local Existence of the Solution

Consider the following model (1) of two species (or two chemical substances in

reaction) living in a bounded domain of 1,2,3=, NNR of spatial densities

),(),,( xtvxtu at time 0.t Note by n is the unit outward normal to .

xt

xt

xt

xt

vvuun

v

n

u

vuvvdv

vuvudu

t

t

0,=,

0,>,

0,>,

0,>,

=0,.,=0,.

0==

=

=

00

2

1

(1)

0>, 21 dd are the diffusion coefficients and 2.,

is the spatial Laplacian operator .=2

2

1= i

N

i x

0>, are intra and inter specific ecological constants.

Let’s define

vuvvug

vuvvuf

=,

=, (2)

To achieve the theorem guaranteeing the existence and uniqueness of the local

solution vu; to our problem (1) in Sobolev spaces, we use the work of [1]

where we implement five conditions (Q1)-(Q5). We have first to write the the

auxiliary quasilinear equations

,,=;=,T

t vuUUFUUtAU (3)

so we have

,0

0==,

1

2

1

1

vd

udUAUtA (4)

with

,1,

1,=

22

2

22

1

vvdvug

uudvufUF

where

.,0max= uu

International Journal of Pure and Applied Mathematics Special Issue

486

The condition 1Q

It requires the choice of two spaces 10 , XX where .01 XX ° We choose

2

0

22,

1 ,=,,= RR pp LXWX (5)

and define for ps 1/1>

on0=:= ,,

n

uWuW psps

B (6)

The Condition 2Q

It is based on the Theorem 11.6 of [3]. We have the interpolation space between

pL and pW 2,

2

1,,

2

1=,,

==2,

2

12,

,2

pp

pp

p

WL

WLWX (7)

According to the Propostion 2.1 of [2], there exists a constant 0>C such that

for 10 we have the interpolation relation

.,

1

,

pW

pL

pW

C

(8)

The Condition 3Q

For UtAUA ,= defined above, if ,),(= 21

Tyyy we have

,0

0=

2

22

2

11

yd

ydYA (9)

We have to prove that for 1<<0 we have

01

,1 ,,0,,, XXVTCytAyt L

(10)

where V is an open set of .X

We will choose 2,2 ,=

RpWV and we define the normed space

XSC ,, where XSC ,,L means

.,,,,,, VUxsxsxxssCxsxs ''

V

'

U

'

x

''

LL (11)

For TTyyyyyy 2

2

2

12

1

2

1

11 ,=,,= ,2

R we have


487

,1

,,

21

2

2

2

12,2,

2

2

2

12

2

2

2

112211

LLLp

BW

pB

W

pL

pL

pL

yyyyvuC

vyyd

uyydUYtAYtA

(12)

where we used LW p °,2 under the condition ,Npp >2>2 then

.,, 212,,221

0,

12211

ttyyCytAytA pWXX

RL

(13)

The Condition 4Q

To apply the fourth condition we use the Theorem 6.6 of [5], so we put

vdudUtA

xtaTT

1

12

1

11

2121

2

22

2

11

=,

,=,,=

>,=<,,,

We have the condition

0,>>,,,,< xta (14)

We have to prove that for TT (0,0)),(= 21 and ,(0,0)),(= 21

TT

,2

R then

0>>=,,,,< 2

2

2

2

1

12

2

1

2

1

1

11 ddxta (15)

because 0==n

v

n

u

on . By the Theorem 6.6 of [2], the operator A

satisfies the fourth condition.

The Condition 5Q

We use the Propostion 15.6 of [3] where

.1,

1,=

22

2

22

1

vvdvug

uudvufUF

(16)

So there exists 1<0 satisfying p

N 12<0 and

.,,,0, 2,2,21 RR ppn WWTCF (17)

The condition )( 5Q remains then to check the two conditions

(I) We note that F is of the form


488

,1,

1,=,,,

2

2

2

2221

2

1

2

1121

dg

dfxtF

GTxt ,,0,,

where G is an open set NG 22RR

and , at the forme

,,,,,,=,,= 2

1

21

1

121

TNNT we have

,,0, 22 NGTCF R (18)

So we get

.,0,, 2 RGTCgf (19)

(II) F is not depending of xt, and satisfies

.0,,,0=,0,0, 222 TxtxtF N RRR (20)

Theorem 1 Suppose that pWvu 2,

00 , are positive and .<< pN There

exists 0>T such that the system (1) has one and only one solution

.,0,,0,,0,, 12, ppp

B LTCLTCWTCvu (21)

2. Global Existence of the Solution

In this section, we will prove global solution following Herbert Amann works

for local solution which we perform to obtain the global solution. We use the

proof idea to obtain the relation

0,,:0> ,2,2 tCvCuC pB

Wp

W .

First, we have to prove that the solution is positive by proving that there are two

positive number 21,kk satisfying 1ku 0> and 2kv 0.> Then, we use

Youkana’s method to prove that the solution satisfies ,Cu PL as well as

Cv PL . Finally, we get the required result by using the Therem in page 154

of [1].

i For 0, 00 vu we have 0,, vu for 0.t

ii For

0u we have ,

u and for 0>0 kv we have 0.>kv

Positivity

Let ,0,max= vv multiply the second equation from (1) v and integrate

on

vdxvvddxvvvddxvdt

d

1

1

22

1

21

(22)


489

,=22dxvdxvu

From the Sobolev injection /2>,2, NpLW p ° and 0=. vv , we have

,:0> 2, Tp

WT CuC

(24)

thus

dxvudxvuL

22

(25)

,

2

2, dxvuC pWT

where 0>TC depending only of T and . From (24),(25) we can rewrite (22)

on the following form

.22dxvCdvv

dt

dT

(26)

Using the Gronwall relation on (26), we have

.2

0

2dxvdvv

(27)

We have 0=0,max= 00 vv because 00 v then 0v if 00 v .

By multiplying the second equation of (1) by ,u we get similarly the

positivity of 0u for 0.0 u

We need the following lemma to prove global existence.

Lemma 2 For 0>, 00 kvu

we have ., kvu

Proof. To prove this lemma, replace u by

u in tu and u then multiply

the first equation in (1) by

u we obtain

,

22

1

1

2

dxuvdxuuddxudt

d

(28)

and from the condition i we have 0,u 0,v then (28) will be in the form

0,

2

dxudt

d

(29)


490

by solving this differential equation, we obtain

,

2

0

2

dxudxu

(30)

as ,0

u then 0,=0

u this leads us to that 0,=

u therefore

.if 0

uu (31)

Similarly, we prove that 0>kv , by multiplying the second equation of (1) by

,

kv we get kv as long as kv 0 .

To prove that pL

u and pL

v are bounded, we use the method of [8],[9],[11].

Theorem 3 For ,, 00 pLvu there exists ,,:0> CvCuC pL

pL

where

C is not depending of t and 0.>p

Proof. Step 1 : .Cu pL Multiply the first equation of (1) by 1pu and

integrate over , then

.=1 121

1 dxvuvudxuuddxupdt

d ppp

(32)

Since

u from (31), we have 0,1

dxuvu p

then

0. dxudt

d p (33)

By solving this differential inequation for 0t and 0,>p we obtain

pL

pL

uu 0 and for p

.0 LL

uu (34)

Step 2 : .Cv pL We use Haraux–Youkana technical [8] and the decreasing

functional dxeuu v 21 in 0.t From the boundry conditions of

Neumann

,212

2 vdxevdxeudxvvddxedt

d vvv

(35)

similarly, from (1) and the boundry conditions , we have


491

,211

211

211

1

2

2

2

1

2

1

1

2212

2

21

1

2

dxeueuuvr

dxeueuuvur

vueuvdudr

dxuuvevdr

dxuueddxeuudt

d

vv

vv

v

v

vv

(36)

Using Young inequality, we get

vdxueuvdud v

211

2

1

1

dxuued v 21

12

(37)

,

21

8

2

1

221

2

1

1

1

2

dxveu

uvdud

d

v

Again by Young inequality, we have

dxve

u

uvdud

ddxeuu

dt

d vv 2

1

221

2

1

1

1

22 21

8

dxuuvved v 222

2 (38)

dxeueuuvu vv 212

.212 dxeueuuv vv

The relation (38) becomes

dxve

u

uvdu

kd

d

ddxeuu

dt

d vv 2

1

2

1

2

1

1

22

1

1

22 21

18

dxeuuuvu v 212 (39)

.212 dxuuuve v

Summing, (35) and (39) to obtain

dxvvddxeuudt

d v 212

2

21

dxve

u

uvdu

kd

d

d

v 2

1

2

1

2

1

1

22

1

1

2 211

8

(40)

.12223 dxuuuuuuve v

From (31),(34) , we have


492

,211

211

212

0

1

01

22

1

1

1

2

2

1

2

1

2

1

1

22

1

dxvevuukd

d

k

d

dxveu

uvdu

kd

d

v

LL

v

(41)

Choosing

,2118=

2

0

1

1

01

22

11

11

LLuu

kd

dkd

(42)

and using (41) then (40) becomes

dxeuudt

d v 21 (43)

,1223 dxuuuuuve v

Also using (31),(34) then (40) takes the form

dxeuudt

d v 21 (44)

dxuuuuve

LLLL

v

0

2

0

3

0

,12

dxve v

By choosing

,

12

23

LLLLuuuu

then

0.1 2 dxeuudt

d v

Therefore

,11 02

00

2 dxeuudxeuuvv

(45)

For 1,p we have

,1 02

00 dxeuup

vv

p

pL

(46)


493

because 0., tevp

vp

Integrating the two sides over , we get

.dxedxvp

vp

p

and applying (45) to get (46) . To complete the proof, we use the inequality

16.38 of [6] to obtain the global solution by applying the Theorem in page 154

of [1]. To this end, we start by rewriting (1) in the following form

,=

=1

2

1

1

vvuvuvvdv

uvuvuuudu

t

t

(47)

where 0> is sufficiently large. From the local solution we have the regularity

.(,0,, 2, pWTCvu

Putting

,

,,,,=,

,,,,=,

,=,

,=,

2

1

1

2

1

1

xtvxtvxtvxtuxtF

xtuxtvxtvxtuxtF

xtvdxtQ

xtudxtQ

v

u

(48)

we have

.,,=

,,=

2

1

xtFvvxtQv

xtFuuxtQu

vt

ut

(49)

From (31) we have 0,>

1

1

dQu 0.>1

2

kdQv

Since is sufficiently large, we can apply the inequality 16.38 of [6] to the

system (49). Therefore

,

00

00

20

0

10

0

dsFestuAeCvA

dsFestuAeCuA

pL

stt

pL

v

t

pL

v

pL

stt

pL

u

t

pL

u

(50)

1,<<0 0,>0,> C .=,= vvuu QtQtA From [7] in Theorem

1.6.1, we have <,2 p

u WtAD for 0t , and according to [5] , page 114,

,= 2, p

Bu WtAD 0.t (49) becomes

dsFestvCev

dsFestuCeu

pL

stt

pW

tp

W

pL

stt

pW

tp

W

20

2,0,2

10

2,0,2

(51)


494

Using the Holder inequality, we have

Lp

LL

pL

pL

pL

pL

LLp

LL

pL

pL

pL

pL

vvuC

vvuvtF

uvvuC

uvuvtF

2

1

(52)

From (34),(46), there exists C 0>

., 21

CFCF pL

pL

(53)

and

.11

0

dsse st

(54)

We use (53),(54) in (51) which becomes

.1

11

,2,0,2

1

,2,0,2

CveCv

CueCu

pW

tp

W

pW

tp

W

(55)

In this step we will prove the same result in pW 2, as that for 0,>>1 we get

(50), furthermore

.

000

000

2

1

00

1

1

00

dsFAestvAecvA

dsFAestuAecuA

pL

v

stt

pLv

tp

Lv

pL

u

stt

pLu

tp

Lu

(56)

Always according to [5] , page 114, ,= 2, p

Bu WtAD 0.t From the

above, (56) becomes

.

,

,22

1

02,02,

,21

1

02,02,

dsFestvecv

dsFestuecu

pW

stt

pW

tp

W

pW

stt

pW

tp

W

(57)

We have

.

,

,2,2,22

,2,2,2,21

pW

pW

pW

pW

pW

pW

pW

uvvF

uvuvF

(58)

As

,,2,2,

pW

pW

pW

vuuv the inequalities (55) give (58) for

0,>>1 so there exists 0>C such that

., 21

CFCF pL

pL

(59)

Using (54) and (59), in (46) for 0,>>1 there exists 0>, CC such that


495

.2,02,

2,02,

CveCv

CueCu

pW

tp

W

pW

tp

W

We can now apply the Theorem page 154 of [1] to get the global solution vu,

of (1) for 0t . This is summurized in the following theorem.

Theorem 4 Let pWvu 2,

00 , are positive initial conditions and .<< pN

There exists for all 0t one and only one solution pWvu 2,, of the system

(1).

References [1] Amman H., Quasi linear Parabolic Systems under Nonlinear Boundary

Conditions, Trans. Amer. Math. Soc., 293 (1) (1986).

[2] Amman H., Global existence for semi linear parabolic systems in Nonlinear Analysis: A Collection of Papers in Honour of Erich H. Rothe (L. Cesari, R. Kannan and H. R. Weinberger, eds.), Academic Press, New York 360 (1985), 47-84.

[3] Amman H., Existence and regularity for semi linear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa. C. Sci., 4 (11) (1984), 593-676.

[4] Amman H., Dynamic theory of quasi linear parabolic systems III, Global existence, Mathematics Zeitschrift 202 (1989), 211-250.

[5] Amman H., On abstract parabolic fundamental solutions, J. Math. Soc. Japan 39 (1), 1987.

[6] Friedman A., Partial Differential Equations, Holt, Reinhart and Winston, New York, (1969).

[7] Henry D., Geometric theory of semi linear parabolic equations, Springer-Verlag, Lecture Notes in Mathematics 840 (1981).

[8] Haraux A., Youkana A., On a Result of K Masuda concerning reaction-diffusion equations, Töhoku Math. J., 40 (1988), 159-163.

[9] Youkana A., Salem A., Global Existence of Solutions for Some Coupled Systems of Reaction-Diffusion, (2011).

[10] Melkemi L., Mokrane A.Z., Youkana A., On the uniform bounded ness of the solutions of systems of reaction-diffusion equations, Electronic Journal of Qualitative Theory of Differential Equations 24 (2005), 1-10.

[11] Melkemi L., Mokrane A.Z., Youkana A., Bounded ness and Large-Time Behavior Results for a Diffusive Epidemic Model, Journal of Applied Mathematics, (2007).

[12] Kouachi S., Yong K.E., Rana Y., Parshadz D., Global existence for a

strongly coupled reaction diffusion system, (2010).


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Documents

Global Solution of a Degenerate Evolution System in ... fileGlobal Solution of a Degenerate Evolution System in Population Dynamics 1Abdullah Mohammed and 2Messaoud Souilah 1USTHB