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110

Remark 4.4 In particular, if

and P

are strongly monotone in M, namely

(A(fi) - Pi{ri)){r

- r

) > m

\ri - r

Vr-j,

e R and i = 1,2, (18)

Wiere m

is a positive constant, then the uniqueness of a solution of (5) is a direct appli-

cation of

[16,

Theorem 3.1]. The main point of the proof is to use the estimate

/ (uj. - u

•

\7{F

- u

))dx

(Q),

< mi|v|

(

|ui -u

|i!i(n)|ui

(19)

where mi is a positive constant so that

IVF-^l^^a < mi\F-

(= mi\z\

.) Vz e V*.

In fact, for two solutions «! and u

, we observe that

{u[(t) - u'

(t),r,)

+ (F(f3(

(t)) - p(u

(t))),

)

(ui{t) - u

{t))(v{t)

•

Vri)dx = 0,

VT?

e V, a.e. t € [0,T].

Now, take i

?_1

(ui(i)

—

(t)) as n and use (18) and (19). Then

~K(t) - «2(t)|

^ + (mo ~ S)\ui(t) - u

(t)\

LHn)

-^MCHQJSM*)

" u

(t)\

fora.e.te [0,T],

where 8 is an arbitrary positive number. Choosing m

/2 as S and applying the Gronwall's

inequality, we infer that Uj = u

. •

5 Application to Czochralski crystal growth process

This work is motivated by modeling the Czochralski process of crystal growth. The

Czochralski pulling method is widely used for the production of a column of single silicon

crystal from the melt. The idea for pulling method due to Czochralski is quite simple.

A crucible, equipped with a heater, contains the melt substance and a pul-rod with seed

crystal, which moves vertically and rotates flexibly, is positioned above the crucible. The

rod is dipped into the melt, and then lifted slowly with an appropriate speed so that a

meniscus surface is formed below the crystal seed and the melt attached to the crystal

solidifies continuously. By controlling some thermal situations in the process, one obtains

the growth of a single crystal column with a desired radius as well as a desired growth

pattern of the solid-liquid interface and temperature pattern in the crystal in order to

improve the crystal quality (see [3, 5, 6, 7, 13]).

In such a model of crystal growth, if we assume that the motion of the material

(solid-liquid) domain is prescribed and a 3-dimensional convective vector field, which is

caused by the deformation of the material domain, is prescribed too, at least satisfying

condition (A2), then our approach seems applicable to the Czochralski process. In this

111

case,

we consider two equations in the crucible. One of them is a degenerate parabolic

equation which describe the solid-liquid phase transition in the material region. The other

is an usual heat equation in the gas region. These two equations are combined by the

transmission condition on the common boundary.

In this case the transmission condition implies that the temperature is possibly dis-

continuous on the solid-gas and liquid-gas interfaces. This is one of the important points

which should be furthermore discussed from the mechanical point of view.

References

[1] H. Attouch and A. Damlamian, Application des methodes de convexite et monotonie

a l'etude de certaines equations quasi lineaires, Proc. Royal Soc. Edinburgh, 97A

(1977),

107-129.

[2] H. Brezis, Operateurs Maximaux Monotones et Semi-groupes de contractions dans

les espaces de Hilbert. Math. Studies 5, North-Holland, Amsterdam, 1973.

[3] A. B. Crowley, Mathematical modelling of heat flow in Czochralski crystal pulling.

IMA J. Appl. Math., 30 (1983), 173-189.

[4] A. Damlamian, Some results on the multi-phase Stefan problem, Comm. Part.

Diff.

Eq., 2 (1977), 1017-1044.

[5] E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems

with change of phase. Arch. Rat. Mech. Anal., 123 (1993), 99-116.

[6] T. Fukao, N. Kenmochi and I. Pawlow, Two phase Stefan problems in non-cylindrical

domains. In preparation.

[7] T. Fukao, N. Kenmochi and I. Pawlow, Transmission problems arising in Czochral-

ski process of crystal growth. In Mathematical Aspects of Modeling Structure For-

mation Phenomena, GAKUTO Intern. Ser. Math. Sci. Appl., Vol.17, pp.228-243

Gakkotosho, Tokyo, 2001.

[8] A. Ito, N. Yamazaki and N. Kenmochi, Attractors of nonlinear evolution systems gen-

erated by time-dependent subdifferentials in Hilbert spaces. In Dynamical Systems

and Differential Equations,

Vol.1,

ed. W. Chen and S. Hu, pp.327-349 Southwest

Missouri State Univ., Springfield, 1998.

[9] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent con-

straints and applications. Bull. Fac. Education Chiba Univ., 30 (1981), 1-87.

[10] N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with

time-dependent constraints. Nonlinear Analysis, 10 (1986), 1181-1202.

[11] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear

equations of

parabolic

type. Transl. Math. Monographs, Amer. Math. Soc, 23, 1968.

[12] M. Niezgodka and I. Pawlow, A generalized Stefan problem in several spaces variables.

Appl. Math. Optim., 9 (1983), 193-224.

[13] I. Pawlow, Three-phase boundary Czochralski model. In Mathematical Aspects of

Modeling Structure Formation Phenomena, GAKUTO Intern. Ser. Math. Sci. Appl.,

17,

pp.203-227 Gakkotosho, Tokyo, 2001.

[14] J. F. Rodrigues, Variational methods in the Stefan problem. In Phase Transitions

and Hysteresis. Lecture Notes Math. Vol.1584, pp.147-212 Springer-Verlag, 1994.

[15] J. F. Rodrigues and Yi-Fahuai, On a two-phase continuous casting Stefan problem

with nonlinear flux. European J. Appl. Math., 1 (1990), 259-278.

[16] K. Shirakawa, A. Ito, N. Yamazaki and N. Kenmochi, Asymptotic stability for evo-

lution equations governed by subdifferentials. In Recent Developments in Domain

Decomposition Methods and Flow Problems, GAKUTO Inter. Ser. Math. Sci. Appl.,

11,

pp.287-310 Gakkotosho, Tokyo, 1998.

Existence of solutions of a segregation model arising in

population dynamics

Gonzalo Galiano

Maria L. Garzon

Departamento de Matematicas,

Universidad de Oviedo,

33007 Oviedo, Spain,

Email: galiano@orion.ciencias .uniovi.es,

maria@or ion. ciencias. uniovi.es

Ansgar Jungel

Fachbereich Mathematik und Statistik,

Universitat Konstanz,

78457 Konstanz, Germany,

Email: juengel@fmi.uni-konstanz.de

Abstract

A strongly coupled cross-diffusion model for two competing species is analyzed.

The unknowns are the population densities of the species. An existence proof for

the evolution problem in one space dimension is sketched. The proof is based on

a symmetrization of the problem via an exponential transformation of variables

and the use of a new entropy functional. The solutions can be shown to be non-

negative, by means of the exponential transformation of variables. Moreover, nec-

essary conditions for segregation phenomena for the steady-state problem are given

and numerically illustrated.

1 Introduction

For the time evolution of two competing species with homogeneous population density,

usually the Lotka-Volterra differential equations are used as an appropriate mathemati-

cal model. In the case of non-homogeneous densities, diffusion effects have to be taken

into account leading to reaction-diffusion equations. Shigesada et al. proposed in their

pioneering work [27] to introduce further so-called cross-diffusion terms modeling inter-

specific influence of the species. Denoting by rii the population density of the i-th species

(i = 1,2) and by J; the corresponding population flows, the time-dependent equations

113

114

can be written as

+ div Jx = gi{n

), (1)

9jn

+ div J

("1,^2), (2)

Ji = -V ((ci + a

ni + ai

)ni) +

diriiVU,

(3)

h = -V ((c

+ oci\n

+ a22n

) +

VU,

(4)

in the bounded domain

C R

(d > 1) with time t > 0. Here, £/ = U(x) is the (given)

environmental potential, modeling areas where the environmental conditions are more or

less favorable [24, 27]. The diffusion coefficients c* and a*, are non-negative, and dj£l

(i,j = 1, 2). The source terms are in Lotka-Volterra form:

9i{ni,n

) = (fli-7n«i-7i2«2)ni, (5)

02 ("l, n

) = {R

721711

- 722"2)«2, (6)

where i?, > 0 is the intrinsic growth rate of the i-th species (i = 1,2), 711 > 0 and

722 > 0 are the coefficients of intra-specific competition, and 712 > 0 and 721 > 0 are

those of interspecific competition.

The above system of equations is completed with mixed Dirichlet-Neumann boundary

conditions and initial conditions:

n-i

n-D,i

on To x (0,00), Jj

•

v = 0 onf^x (0,00), (7)

ni(-,0) = n

,j in fl, i = 1,2, (8)

where v denotes the exterior unit normal to d£l. This means that the population density

is fixed at a part of the domain boundary (due to emigration and immigration processes),

whereas no flux boundary conditions are prescribed at the remaining boundary parts.

Eqs.

(l)-(4) contain various types of reaction-diffusion models. Indeed, in the case

OLij

= 0 for i, j = 1,2, they reduce to the drift-diffusion equations, which has been studied

in various fields of application, e.g. electro-chemistry,

[1,

3], biophysics [7] or semiconductor

theory [23]. When c\ = c

= 0 and «i2 = a

i = 0, Eqs. (l)-(4) are of degenerate type.

These types of problems arise, for instance, in porous media flow [17], oil-recovery [8],

plasma physics [16] or semiconductor theory [15]. In chemotaxis, related models appear

[9,

25].

For Q12 > 0 and a

i > 0, the problem becomes strongly coupled with full diffusion

matrix

Ci +

20:1171!

+ Qi

012^1

A(ni,n

) = ,

+ 2a

+ a

ini

Nonlinear problems of this kind are quite difficult to deal with since the usual idea to apply

maximum principle arguments to get a priori estimates cannot be used here. Furthermore,

the diffusion matrix is not symmetric.

Up to now, only partial results are available in the literature concerning the well-

posedness of the problem. We summarize some of the available results for the time-

dependent equations (see [29] for a review) and refer to [20, 21] for the stationary problem.

We also refer to [6, 13, 18] for related results on various cross-diffusion models. Global

115

existence of solutions and their qualitive behavior for a

= a

i = 0 have been

proved in, e.g., [2, 22, 26]. In this case, Eq. (2) is only weakly coupled. For sufficiently

small cross-diffusion parameters a

> 0 and a

> 0 (or equivalently, "small" initial

data) and vanishing self-diffusion coefficients an = a

= 0, Deuring proved the global

existence of solutions [5]. For the case Ci = c

, a global existence result in one space

dimension has been obtained by Kim [19], Furthermore, under the condition

8«n > a

, 8a

> a

, (9)

Yagi [30] has shown the global existence of solutions in two space dimensions assuming

= a

\. A global existence result for weak solutions in any space dimension under

condition (9) can be found in [10]. Condition (9) can be easily understood by observing

that in this case, the diffusion matrix is positive definite:

A(n

)£ > min{ci,c

}|£|

for all £ e R

hence yielding an elliptic operator. If the condition (9) does not hold, there are choices

of c

7ij

> 0 for which the matrix A(ni, n

) is not positive definite, and it is therefore

unclear if the problem (l)-(8) can be solved for these data. More recently, Ichikawa and

Yamada [14] have improved the results of Yagi, replacing condition (9) by

64a

> a

or 64a

= a

i > 0. (10)

They use the same techniques as Yagi combined with suitable energy estimates. From the

view-point of mathematical biology, conditions like (9) and (10) mean that self-diffusion

or diffusion is dominant over cross-diffusion.

The aim of this paper is to show how the existence of solutions of problem (l)-(8)

can be obtained, without assuming conditions like (9) or (10). In fact, these conditions

are just technical restrictions, needed in [30, 14], since the existence of solutions of the

steady-state problem can be proved without this condition (see [20]).

More precisely, we are able to show that for any

Cj,

a; > 0 and in one space dimension

d = 1, there exists a weak solution U\,u

to (l)-(4), (7)-(8) such that U\ and u

are non-

negative. We stress the fact that the non-negativity property is obtained without the use

of the maximum principle. The idea of the proof is as follows: The system (l)-(4) is first

symmetrized via an exponential transformation of variables. A priori estimates are then

derived by using a new entropy functional yielding H

bounds which are independent of

the solutions. The non-negativity property is obtained from the embedding i3"

(f2) <-)•

L°°(fi),

which holds only in one space dimension, and the transformation of variables.

We sketch the proof in Section 2. For the detailed

proof,

we refer to [11].

Before we state the results and sketch the method of

proof,

we perform (for a smoother

presentation) the following change of unknowns:

Ui = a

ini, u

= ot\

, and q = —VU.

We assume that a

> 0 and a

i > 0 which is no restriction since if a.\

= 0 or a

\ = 0,

at least one of the equations (1), (2) is weakly coupled, and the results of [26] apply. Eqs.

116

(l)-(8) can be reformulated as

Ui - div(cjVu, + 2a;i4;Vuj + V(uiu

) +

diU

{

= fi(ui, u

), (11)

= u

Dti

onr

x(0,T), (12)

+ 2a

+ V{u

)+d

q)-v =

onT

(0,T),

(13)

u(-,0) =

infi, i = 1,2, (14)

where T > 0, u

i = a

^, u

D}2

= ai

Dt2

, «? = a2i«o,i, "2 =

i2«o,2 and a

= 0:11/021,

= a

/ai

. The source terms are given by fi{u\,u

) = {Ri

—

Ai"i

—

U2)

i, with

/?K

= 7ii/«2i, /?2i = 72i/ai2, i = l,2.

2 Existence of solutions of problem (11)-(14)

In this section we assume:

(Al) n C R is a bounded interval, dQ = T

, and T

III.

(A2) u° € L°°(n) satisfies

> 7 > 0 in 0, u

,i = const. > 0 on T

,i = 1,2.

(A3) a

Ci > 0, di € R (i = 1,2) and 9 e L

(f2 x (0,T)).

(A4) /; :

[0,

oo)

—>

R (i = 1,2) is continuous and it holds for all s, a > 0, p,q > 0:

fi(s,a)<C

fi(8,a)log- + Ms,<T)\og-<C

, and h(s,o~)

(---)+

h(s,o) (- - ±) < C

P 1 \p s) \q a)

for some d, C

{p,q),

{p,q)

> 0.

We observe that the Lotka-Volterra source terms satisfy condition (A4) if

/3a

> 0, i = 1, 2

and £12 = hi > 0.

The main result is the following theorem.

Theorem 1 Let T > 0. Under assumptions (Al)-(A^) there exists a weak solution

(ui,«

) 0/ (ll)-(U) satisfying u

€ L

(0,T; ff

(n)) n W^O.T; tf

(^)) and

ui{x,t),

{x,t) > 0 for {x,t) 6 O x (0,T).

The proof consists of several steps.

Step 1. We work with unknowns which symmetrize the elliptic operator. Introduce

w = (wi,w

) by defining u

= e

and set b{w) = (&i(u>),&

H) =

With the diffusion coefficients

au(w) = c

{

+ 2a

- +

m+w

*, i = 1,2, a

(w) = a

i(w) =

Wl+W2

Eqs.

(11)-(14) are formally equivalent to

bi(w)-d\v lj2

ij(

)

+ d

Mw)q) =Fi(w), infix(0,T), (15)

I ^2 Oij(w)Vw.,- + dibi(w)q J

•

v = 0 on T

x (0,T), (16)

w = u>

onr

x(0,T), (17)

w(0) = w° in fi, (18)

where F^w) = /

), w

D>i

= log(«

Dii

), and w° = log(uj), i = 1,2.

Step 2. In order to solve the above problem, we introduce a semi-discrete problem. Let

e N and let r = T/N be the time step. Given w

, approximating w in the interval

((k

—

2)T,

—

l)r], we are seeking solutions w

of the elliptic problem

k(w

)

-uw

-i) _

div

/£

{wk)Vw

+dA{w

Fi(w

}

(19)

^ a^iu^Vur) + dibi{w

\-v =

onT

(20)

= w

on T

, (21)

for

= 1,..., N. Here we defined q

= \

(ife

1)T

<?(•,

£)<&•

Step 3. The idea of obtaining appropriate a priori bounds is to use an entropy functional.

We introduce the discrete entropy (for k = 0,..., N)

=Vi+

ar)

where a

—

2min{ci,c

}, f]

is the discrete "physical" entropy

= X) /

(

iK)K

ti»D,0

h{w

)

bi{w

))dx

and

is another discrete entropy-type functional:

= E [(^-^ -

,i))dx

In the original variables, r]

can be written as f(ui(logUi

—

logup^)

—

+ up^dx, which

justifies the notion of "entropy".

We can prove the following entropy-type estimate, which holds in any space dimension.

118

Lemma 1 Let w

€ i7

(Q;R

) be a weak solution of (19)-(21). Then there exists a

constant C > 0 such that for any k = 1,..., N and any T > 0,

J2 f (iH^I

a|Ve

\Ve

" A dx < n^

+ CT.

(22)

The key of the proof of this lemma is to use (wf

—

v>D,i)

a(6j(—IBJ)

—

6j(—«);)) €

HQ(£1

FJV)

= {t)£ i7

(Q) :

= 0 on To} as a test function in the weak formulation of

(19)-(21) (see [11] for the details).

Step 4. We use the Lax-Milgram's lemma together with the Leray-Schauder's fixed-point

theorem to prove the existence of weak solutions of the semi-discrete problem (see [11]).

Lemma 2 Let

K;*

-1

G L°°(fl;R

), k > 1. Then there exists a weak solution w

€

H^Q-R

) of (19)-(21).

We notice that, since the solution satisfies w

e i7

(fi;R

) «-> L°°(Q;R

) in one space

dimension, the unknowns u

= exp(iwf) are well defined and elements of

(Q).

Hence

) = (e

, e"*

), k = 1,..., N, is a solution of the discrete problem corresponding to

(11)-(14).

Step 5. We define the piecewise constant functions u/

) by

{T)

{x, t) = w

{x) if {x, t) G Q x ((fc -

1)T,

fcr]

and g(

' in a similar way. Observe that assumption (A3) implies

(T)

->? in L

) as r -> 0. (23)

An immediate consequence of Lemma 1 is the following result:

Corollary 1 Let r > 0. Then we have ||?/

'||L<>°(o,:r;i,i(n)) < C and

J2 (a||e

M/2

||^(o,r;^(n)) +

ill

effi,M

(o,T

;

ifHn))) < C,

iti/iere C > 0 is independent of T, a = 2min{ci,C2}, and

»>

(T)

(t) =E / (^(^

!r)

-«'fl)-''i(^

) + &i(«'i>)+aft(«'

lT)

)-^

+^,0)W^-

We also need an estimate for the discrete time derivative. For this we define

ftW(.

)

fl_I (p(w

) - biw"-

)) +

b(w

t > 0,

and introduce a

, the shift operator:

(T)

{-,t) = w

if t e {{k-i)T,kr], k =

l,...,N.

Then we have (see [11])

119

Lemma

3 It

holds

||6(u/M)

KoM^h^Ty) <

CT,

Hft6W||

(

o,T

II^IILW;^))

where

does

not

depend

on T and V*

(HQ(Q U IV))*

the

dual

ofV =

Hg(fl

Step

already have

all the

necessary estimates

pass

the

limit

—>

Since

the

embedding

^!)

<->

L°°(Q)

compact

one

space dimension,

we can

apply Aubin's

lemma [28]

view

the

uniform bounds

Lemma

obtain,

up to

subse-

quence which

is not

relabeled,

->•

weakly

(0, T;

V), (24)

> ->•

weakly

(0,T;

{n)),

(25)

->•

strongly

(0,T; L°°(n)),

(26)

6(U/

)

->• u weakly in L

(0, T;

H^Q)).

(27)

By Lemma

3 we

have,

as r

—>•

||6M -

&(™

(T)

)||LI(O,T

;

< ||6(t«W) - 6(a

(T)

)|UHo,r

;

v) -> 0,

and hence

z = u.

Finally, using

the

above estimates

and

convergences,

we are

able

prove that

bi{w

{T)

)

-»

strongly

i =

1,2.

(28)

Now

we can let r -» 0 in the

weak formulation

(19),

i = 1,2,

which reads

for

0eL°°(O,T; (W^iil))*)

/" (d

(T)

,<f)dt

(

.+ Ve"

T)+,

)

•

V^ctedt

-di f

q<-

'>-V<f>dxdt+

[

fi{e

^,e

^)cj)dxdt.

JQT

In view

(24)-(28),

(23)

and

Assumption

(A4)

obtain

{dtu

4>)dt+

'Vu

V{u

U2))-V4>dxdt

—

u,q

-V<pdxdt

Ji{u\,U2)<l>dxdt,

JQT

i.e.

u =

(«i,U2)

is a

weak solution

(11)-(12). Moreover,

the

initial condition

(14)

satisfied

in the

sense

consequence

the

above proof

obtain

positivity-preserving numerical

scheme. This scheme will

employed

for

numerical simulations

the evolution problem

future work.

For

numerical stationary solutions,

refer

to, e.g., [10].