Bemelmans J. et. al. (editor) Elliptic and Parabolic Problems: Rolduc and Gaeta 2001

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480

Moreover, suppose that

, %eL

(0,T;ff),

\M\LH0,T;H)

t<Pl\\

*(0,T;H)

+ \\

P\ \\

lA(0,T;X)

2\\^(0,T;V)

for a given R > 0. TTien i/

(<^I,CI,ID

,£I)

and

(</S2,C2,

^2,^2) ire couples of solutions to

(17) related to the initial data </9o,i, c^j and

</3o,2,

Co,2 respectively, we have

\\<Pl

P2\\

«.(

,T;V)nL

(0,T;rr

(n)) + ll

_ C

V\L°°(0,T;V')nL2(0,T;V)

< k

(||</3

,l - (^0,2||y + ||C0,1 - CoAv) '

7n particular, the solution

(<p,

VJ,£)

provided by Theorem 2.1 is unique.

Remark 2.3 We note that (24)i is surely satisfied if (20) holds.

Theorem 2.4 Let (4), (14)-(15), (18) and (22)

hold.

Let us assume that

e H

(ft), <9

= 0, on T, 3£

£ V : £

) a.e. in ft. (25)

Then,

there exists

€ (0,T] such that the solution (ip,c,w,£) given by Theorem 2.1

fulfills the additional regularity

c € H

(0,

; V) n i°°

(0,

; ff

(ft)), , .

»er

(o,

;

y) n

(O,

(ft)).

(M)

Moreover, if (22)\ holds, we can choose

= T.

Remark 2.5 Thanks to the first of

(26)\,

under suitable hypotheses on

<po

(e.g. ipo €

(ft) and d

ifo = 0, on T), further

parabolic

regularity could be deduced also for

differentiating (17)\ and multiplying the result by

<p,

Btp. We do not enter into the

details, since the procedure is standard.

Theorem 2.6 Let (4), (14)-(15), (18) and

(22)%

hold for two pairs of initial data y?o,i,

(fyi and

>po,2,

also satisfying (23). Let also To G (0,T] and suppose (y>i,Ci,u)i,£i)

and

(v?2,C2,

^21^2) is a couple of solutions to (17) related to the different initial data and

fulfilling

, Bn, 8

e L

(0,T

; H), Vw

e L

(0,T

(ft)),

llyi|li

(0,ro;ff) + ll

*Vlllt

(0,To;ff) +

PI\\L*{0,TO;H)

+ ll

\\L*(0,T

;H)

(27)

+ \\dtCi\\

Li{oa

+ IIBdll^jo.r^H) + \\^

2\\mo,T

;LHn)) < R,

for a given R > 0. Then we have

IIVl

f2\\L^(0,To;V)nL^(0,T

]H^(n)) + ll

l ~~

2|lz,»«(0,To;V')nL

(0,To;^)

< k

(||ipo,i - Vo,21|v + l|co,i - co,2|lv) •

In particular, the solution (p,c,w,£) given by Theorem 2.1 is unique on

[0,To].

(28)

Remark 2.7 As before, (27) are not satisfied in the given regularity setting. However, if

(20) and (25) are assumed, then (27) do hold at least up to a suitably small final time T

481

3 Proofs of the Theorems

Proof of Theorem 2.2. Set (ip,c,w,£) := {<Pi,ci,Wi,£i)

—

(ip

,C2,W2,(,2)-

Then, writing

(17) for the two solutions and substracting the results, we obtain

tp + Btp = F

{ip{)

- F

(<p

)

(

) - F

(<p

))

+ cF

(<p

(29)

Doing the same with (17)2,3, we have, respectively

c +B

w + (B

-B

= 0, . .

w = Be + £ + 7 (ci) - 7 (c

) + g (fi) - g (p

)

•

Then, we test (29) by if + Bip and integrate over

(0,

t) for t < T. By (14)

lj2

, (15) and

Young's inequality, we easily infer

\ llv Wily + II V^li

2(Qt)

+ ^ l|B</>ll£»

,) < 2 IIVo.1 " Vo,a||v + k

\\<fi\\l

HQt)

+ fc

\\c\\

LHQt)

(31)

Indeed, (15) guarantees that

||CI||

„(QX

< 1.

Now, we test (30)

by c (which is in V

by (23) and (17)e) and (30)i by A/^c =: u,

substract the results and integrate over (0,t). Noting that two terms cancel and using

Young's inequality and the monotonicity of

f3,

we get

/ (d

u),u)dt+ / (fi^)-/j,(<f

))Vw

- Vudxdtdx + \\Vc"

Jo „ J oj n

LHQt) (32)

(Qt)

*• "

lli

(Qt) •

< k

\\u>\\

+ k lie"

Let us deal with the first two terms I\ and h on the left hand side of the preceding

inequality. For the first one, by (5), (8) and (22)

, we have

\( n{^{t))\Vu{t)\

dx-

-[ fi&

0tl

)\Vu(0)\

+1 [ [ v'(<Pi)dm\Vu\

dxdt (33)

J oi n

> f |V«(i)|

-f

IKi-coX'-f/

/ \dm\\Vu\

dxdt.

Then, using (11) and a Galiardo-Nirenberg inequality [5, p. 125], we infer

/ / \dtipiWVu\Uxdt

J oJ a

<k f |^i|||Vu||*

t(n)

«tt

< k j \d

{\\u\\%\

IVu|

1/2

+ IVu|

) dt (34)

< kj

\dt

\Vu\

dt +

kj^\d

\Vu\

(|c|

3/2

+ ||Hl5p

(

n) |Vn|

3/2

) dt

< f \d

\\Vu\

dt + k\\c\\

+k f (Ift^il

+ l^l

) |V«|

dt.

Jo Jo

482

The second term

(32) gives, thanks

to the

continuous embedding

(Q)

L°°

(CI)

\h\

f IML~(n) IVtual |

V«|

dt < | J* (|^|

\B<p\

)

dt + kf

\Vw

\Vu\

dt.

(35)

Now, take

the

sum

(31)

and

(32)

and

note that (10) can

applied with

c in

place

(, so

that

the

-norms

of c on the

right hand sides

(31)

and

(32)

are

controlled

taking

opportune 6. Taking advantage

(33)-(35) and recalling assumptions (24),

conclude applying Gronwall's Lemma

in the

form

of,

e.g. [3, Lemme A.4,

156],

to the

function

i->

||<^

(t)||J.

|Vu(i)|

•

Proof

Theorem 2.4. We derive new

priori bounds for the solutions

(17).

We point

out that,

this level, some passages are formal. Anyway, they could

made rigorous

performing the estimates

an approximating level

(cf. e.g.

the Faedo-Galerkin argument

of [4, Sec. 3]). Then, differentiate (17)3 with respect

time. This gives

£,

+ 7'

(c)

c +

(<p)

dtip.

(36)

Now test (17)2

by d

w and (36) by d

c and

integrate over (0,i),

t < T.

Noting that

two terms cancel

and

using (6), (14)i

, (22)

and the

monotonicity

/3,

we get

\\dtVc\\l

2Kt)

^\Vw(t)\

<^\Vw(0)\

J ((\dM +

c\)

\Vw\

+ k

\\d

<p\\

LHQt)

+ k

\\d

c\\

LHQt]

) dxdt.

Of course,

the

main trouble

given

by the

integral term

J on the

right hand side.

Using Sobolev's embeddings,

(11) and the

Gagliardo-Nirenberg inequalities, we infer

< kj

(\\dM\

+ \\d

c\\

)\\Vw\\

L12/Hn)

k f

(\\dM\v

\\d

c\\

)

(\Vwf

\\w\\U

\Vw\

)

•J

(\\9M\v

+ \\d

c\\

)

(\Vwf

||H|$

(n)

+ |VH

)

•

(\\dM\y

+ \\d

c\\

)

(38)

x (|VH

3/2

1/2

+ \Vw\

(l +

\\<p\\

HHn)

llcll^p,,))

dt := J2 Jj,

3=1

where

the

integrals

Jj are

defined

this way

:= k [

\\dM\v\VM

i/2

\dtc\

1/2

dt,

:= k f

\\d

c\\

\Vw\V

1/2

dt,

:= kj

||3^||

|MI^

(

n))^

J* •= */

||^|lv|Vw|

||c|

kJj\d

c\\

\Vw\

(l + M

H2m

)dt,

:= kj

||a

(

c||

|VH

(fl)

"*>

IIH

(SJ)

^*-

483

Now we estimate the Jj's. The term J

does not give troubles. Instead, Jj, J

, J4, J5,

owing to the third of (21), are bounded as follows:

k [ \Vw\

\\d

p\\y

dt + k f \d

dt,

J Q

J 0

h < 6 [ \\d

dt + k

f \Vwfdt,

Jo J 0

-9 ° t (39)

Ji < kj

\Vw\

\\c\\*

H2{Q)

+ kJ \Vw\

\\dM\fdt,

h < S f \\d

dt + k

[ \Vw\

dt.

Jo Jo

The estimation of the latter term J

, instead, is more complicated and we proceed

similarly as in [1, p. 497]. More precisely, we collect in the next computation also the first

term resulting from J

on the right hand side of

(39)3.

By elliptic regularity :

+ k [ \Vw\

\\c\\

HHn)

t t

(40)

< S I

\\d

c\\

dt + ks f (|VH

(|c|

+ |Bc|

) + \Vw\

(|c|

+ |Bc|

)) dt.

Jo Jo

Then, we multiply (17)3 by Be. Owing to the monotonicity of

/3,

to (14)^2, and to the

second of (16)2, we easily infer, a.e. in (0,T)

\Bc\

< k (1 + |V«f + |Vc|

) < k (1 + |Vu;|

)

•

(41)

Then

Je + k J \Vw\

\\c\\

H2(n)

dt<6 J \\d

c\\l dt + ks j

{\Vw\

+ \Vw\

)dt. (42)

Finally, we have to bound the norm of w (0) in (37). Writing (17)3 for t = 0 (this could

be made rigorous working on the discrete scheme) and recalling (25), we see that

\Vw (0)|

< k (l + |h||^

(n)

Uof

+ \\<p

) < k. (43)

Now, let us take (38)-(43) into account and go back to (37). If (22)

holds, the situation

is simple since J

= J4 = J5 = Je = 0. Actually, we can apply (10) with

c, w, tp in place of £, u, v and z, (44)

respectively. Thus choosing 8 suitably small and recalling (21), we see that Gronwall's

lemma applies to the function t i-> \Vw(t)\

. This yields Vui € L°°(0,T;H) and also

to the first of (26)i for T

= T. The second of (26)i is a consequence of (41). Now the

first of (26)2 would easily follow once we show that w 6 L°° (0,T

; H). Due to Neumann

conditions, this is not obvious and has to be proved by suitably adapting the argument

of e.g. [2, Subsec. 5.3]. At this point, also the second of (26)2 is proved since, looking

484

back

(17)2,

we can

exploit

(11)

with

the

choices

(44) and

take advantage

of the

third

(21) and of the

second

(26)i.

Instead,

(22) does

not

hold,

deduce from (37)-(43) that

liavc|k

|v™(t)|

k +

kj*

(/|VH

rfi,

where

/ := (l +

||%>||y

)

1/(0,

T),

for

some

p > 1.

Then, using

generalized

Gronwall's lemma

(e.g.

slightly modified version

[6, Thm. 7.1]

works

for

our

case),

we obtain

the

same relations

before, holding

now up

to a

final time To

(0,

T].

Proof

Theorem

2.6.

It is

very similar

the

proof

Theorem

2.2.

Then,

just

give

the

highlights. Proceeding

as in

that proof

(but

choosing

now

t 6

(0,T

j),

we see

that

the

last term

in (33) now

depends also

Ci. However,

can

still

estimated

(34) by

taking

the

contribution

into account. Thus,

the

last term

(34)

will

now depend

on all the

norms

(27)2 but the

latter.

The bound

(35)

needs

modified more carefully. Actually,

we now

have

/ (/•'(VijCi)

-M(V2,C

))

VM;

•

V« dxdt

J \M\

°°(n)\V™2\\Vu\dt

kJ ||c||

(f!) l|Vw

3(fj) |Vu|

dt.

Thus,

the

first term

can now be

bounded

before, while

the

second

one is

less than

IIL

T-V)

^«

Jo II Vw2||

3/m I

V«|

dt, so

that,

account

of the

last

(27)2,

further

application

Gronwall's Lemma [3, Lemma A.4,

p. 156]

permits

conclude.

•

References

[1]

J. W.

Barrett

and

J. F.

Blowey, Finite element approximation

the

Cahn-Hilliard

equation with concentration dependent mobility. Math. Comp.,

(1999), 487-517.

[2]

Bonetti,

Dreyer,

and G.

Schimperna, Global solution

viscous Cahn-Hilliard

equation

for

tin-lead alloys with mechanical stresses. Submitted.

[3]

Brezis, Operateurs maximaux monotones

semi-groupes

contractions dans

les

espaces

Hilbert. North-Holland Math. Studies,

North-Holland, Amsterdam, 1973.

[4]

Kessler,

J.-F.

Scheid,

Schimperna

and U.

Stefanelli. Study

of a

system

for

the

isothermal separation

components

binary alloy with change

phase. Paper

preparation.

[5]

Nirenberg,

elliptic partial differential equations.

Ann.

Scuola Norm.

Sup.

Pisa

(3),

(1959), 115-162.

[6]

Sassetti

and A.

Tarsia,

nonlinear vibrating-string equation.

Ann. Mat.

Pura

Appl.

(4), 161

(1992),

1-42.

Bifurcation in population dynamics

Kenichiro Umezu

Faculty of Engineering, Maebashi Institute of Technology

460 Kamisadori, Maebashi 371-0816, Japan

Email : ken@maebashi-it.ac.jp

Abstract

This paper is devoted to the study of a local bifurcation problem for a nonlinear

elliptic boundary value problem arising in population dynamics, having nonlinear

boundary conditions. Necessary and sufficient conditions for the existence of bifur-

cation points are studied. Ecological interpretations of our results are also given.

1 Introduction

Let D be a bounded domain of B.

, N > 2, with smooth boundary dD. In this paper we

consider the following nonlinear elliptic boundary value problem:

-AM

= \(m(x)

—

au)u, in D, — = b(x)g(u), on dD, (1)

where A is a positive parameter, m e C

(D) is a real-valued Holder continuous function

with exponent 0 < 6 < 1 on the closure D of D, and satisfies : m(xo) > 0, at some point

€ D, a is a given positive constant, b 6 C

1+e

(dD), b > 0 and b ^ 0 on dD, g is a

real-valued sufficiently smooth function near t = 0 which satisfies g(0) = 0, and n is the

unit exterior normal to dD.

It is well-known (cf. [3, 4]) that the equation :

—AM

= \(m(x)

—

au)u, in D, describes

the steady state for the population density of some species which diffuses at the rate

1/A, where m(x) denotes its growth or decay rates at x G D, meaning that the regions

{x € D : m(x) > 0} and {x e D : m(x) < 0} are respectively favorable and unfavorable

for the species, and a represents the crowding effect! The nonlinear boundary condition

du/drx = b(x)g{u) on dD means that the rate of the flux of population into the region D

is governed in a nonlinear way by b(x)g(u) on the boundary dD. From this viewpoint,

the existence of positive solutions is of ecological interest. Here a solution u 6 C

(D) of

(1) is called positive if u > 0 on D. We assume here for g that

g(t) is non-negative for all t > 0 close to the origin. (2)

Ecologically speaking, assumption (2) means that only the influx of population can

occur at the boundary, if its density is sufficiently small there.

485

486

In the Neumann case, that is g = 0, Hess proved in [6] the existence, uniqueness and

asymptotic stability of positive solutions and also determined the qualitative behavior of

the unique positive solution when parameter

runs, especially when

J. 0. By

fJ.i{\),

denote the unique principal eigenvalue of the eigenvalue problem

—A<p —

\m(x)if

—

fJ.{\)<p,

in D, — = 0, on dD, (3)

where principal eigenvalues mean eigenvalues with corresponding positive eigenfunctions.

Brown and Lin proved in [2] that there exists a unique and positive Ai(m) such that

(Ai(m)) = 0, provided

mdx

< 0.

Our consideration starts with the following result.

Theorem 1.1 (Hess [6]) Let g = 0. Assume first that J

mdx > 0. Then there exists

a unique positive solution u\ €

2+e

(D)

of (1), for every

> 0, which is globally asymp-

totically stable, and the trivial solution u = 0 is unstable for all X > 0. Moreover, we

have

m(x)dx

u\-

a\D\

_ —>0, asXIO,

(D)

where \D\ is the volume of D.

Assume now that J„ mdx < 0. Then there exists a positive solution ux €

2+e

(D)

(1) if and only if

\\{m),

u\ being unique, globally asymptotically stable and such that

A||

2+8(D)

—* °>

as A

i M

Moreover, the trivial solution u = 0 is unstable for any A > Ai(m) and globally

asymptotically stable for each 0 <

< Ai(m).

Some ecological interpretation of Theorem 1.1 can be given concerning the qualitative

behavior of the globally asymptotically stable, non-negative solution U\ when A J. 0. If

mdx < 0, then Ux is identically equal to zero for 0 < A < Ai(m). This would say that

the species becomes extinct if he diffuses at the so large rate. If f

mdx = 0, then U\

tends to zero as A J. 0, which means that the steady state for the population becomes

small gradually and extinct asymptotically as the diffusion rate goes to infiriity. In the

case J

mdx > 0, Ux keeps positive even when A J. 0. This implies that the species is

persistent even if the diffusion rate goes to infinity. To sum up, if the food environment

becomes rich, then the more the species diffuses, the more we find out clearly that it is

favorable for him.

In the case of the nonlinear boundary condition, the uniqueness of positive solutions

does not necessarily hold (see Amann [1, Theorem 2.6] and Pao [7, Theorem 4.6.3] for

criterions for the uniqueness). This suggests that it is quite difficult to obtain always a

globally asymptotically stable solution for (1). For this reason we restrict our attention

to the problem how the stable steady state behaves as

J. 0 for the corresponding initial-

boundary value problem

— =

—At;

+ m(x)v

—

(0,

oo) x D,

v(0,x) = u

(z)>0 inD, (4)

— = b(x)g(v) on (0, oo) x dD,

487

with sufficiently small initial data u

^ 0. For this problem, the following bifurcation

and stability problems arise naturally. Our main interest of this paper is to study the

problem of bifurcation to the right at the origin for (1) of positive solutions, meaning

the problem whether there exist sequences {A^} and

{WA^}

of positive solutions of (1)

for A = \j, such that A.,- J. 0 and

HMAJICCD)

-» 0 as j -» oo, or not, and additionally

to investigate the stability of the bifurcation positive solutions. Our results under the

nonlinear boundary condition are compared with Theorem 1.1 in the framework of the

local bifurcation problem.

In the work [9], we study the behavior of the minimal positive solution of (1) when

| 0, under some growth condition on g, mainly for the case J

mdx > 0 by the method of

super- and sub-solutions, in which there is the corresponding result for the case J

mdx =

0 that is rather weak. In the present work, we focus on the case f

mdx = 0 and on the

existence of the local bifurcation branch, which represents the minimal positive solution,

within the scope mentioned above. In the present work, our approach which relies on

bifurcation theory, is different from that used in [9].

2 Main results

Before stating our main results, we remark that the condition g'(0) = 0 is necessary for

the origin (X,u) = (0,0) to be a bifurcation point to the right for (1). Indeed, if

is a

positive solution of (1) for

= Xj such that A., J. 0 and

H^j

Hcrrn)

~~*

0> then we can verify

that the function Vj := Uj/llujllpfm tends to some v

e C

(D) which satisfies

-Av

= 0, in D, -^ = b(x)g'(0)v

, on dD.

Now we can formulate our main results. First we have the following result for the case

mdx > 0.

Theorem 2.1 Suppose that J

mdx > 0. // condition (2) is fulfilled, then the origin

(A,u) = (0,0) is not a bifurcation point to the right for (1).

Next we have the following result for the case J

mdx < 0.

Theorem 2.2 Suppose that J

mdx < 0. // there exists an integer k > 2 such that

g'(0)

= ---=g

(0)=0 and ff

(0)>0,

then the origin

(A,

u) = (0,0) is a bifurcation point to the right for (1). More precisely, the

set of all non-trivial solutions u of (1) in

> 0 and

||W||

both small enough consists

exactly of a unique smooth branch of positive solutions emanating from the origin.

Remark 2.3 In the case J

mdx < 0 the origin

(A,

u) = (0,0) is not a bifurcation point

to the right for (1) provided g(t) = 0 for all t > 0 close to the origin, see Theorem 1.1.

This indicates that Theorem 2.2 would be optimal under the additional assumption that g

is analytic at t = 0.

488

Finally we state our results for the case J

mdx = 0. Let 70 €

2+e

(D)

be the unique

solution of the Neumann boundary value problem

—A70 = m(x), in D, -^— = 0, on dD, /

= 0.

on J

Theorem 2.4 Suppose that J

mdx = 0. //

«/(0)=

"(0) = 0, <T(0)>0, (5)

b{X)da>V0

'"(*)5

?dx>

(6)

where da is the surface element of dD. Then the origin (A,u) = (0,0) is not a bifurcation

point to the right for (1). More precisely, there is no non-trivial solution u of (1) for any

> 0 and |Mlc(5) both sufficiently small.

On the other hand, we have the following.

Theorem 2.5 Suppose that f

mdx = 0, and suppose condition (5) holds. If

b(x)da <

VQ,

(7)

JdL

then the origin (A,u) = (0,0) is a bifurcation point to the right for (1). More precisely,

for any

> 0 small, there exists a minimal positive solution ux €

2+e

(D)

of (1), which

satisfies that

||"A|IC

(O) —•

0 as

J. 0. Additionally if

is analytic att = 0, then the set

of all non-trivial solutions u of (1) in

> 0 and ||u||c(i5) both sufficiently small, consists

exactly of two smooth branches of positive solutions emanating from the origin.

The following two results are complementary to Theorems 2.4 and 2.5.

Theorem 2.6 Suppose that J

mdx = 0, and that g'(0) = 0. Additionally if g"(0) > 0,

then the origin (\,u) = (0,0) is not a bifurcation point to the right for (1).

Theorem 2.7 Suppose that

mdx

= 0, and that condition (2) holds. Ifg'{0) = g"{0) =

g'"(0)

= 0, then the origin (A,u) = (0,0) is a bifurcation point to the right for (1). More

precisely, problem (1) has a minimal positive solution ux G

2+e

(D)

for any

> 0 small

such that ||uA|lc

(5) -^ 0 as

| 0.

3 Proofs of our main results

In this section we give a sketch of the proofs of our main results. Theorem 2.1 is proved

having a priori lower bounds by a positive constant for positive solutions of (1), which is

a direct consequence of [9, Theorem 3]. For the case f

mdx < 0, we can apply the well-

known local bifurcation theory for simple eigenvalues due to Crandall and Rabinowitz [5,

Theorem 1.7] and then Theorem 2.2 follows.

489

Now we prove Theorems 2.4 and 2.5. It seems to be difficult to apply to the case

mdx

= 0 the local bifurcation theory due to [5, Theorem 1.7], where the so-called

transversarity condition given as J

mdx

=/=•

0 breaks down. To overcome the difficulty, we

use the Lyapunov and Schmidt type of procedure in order to reduce (1) to a bifurcation

equation in R

as follows.

Assume that g(0) = g'(0) = 0. We see that C

2+e

(D) and C

(D) are decomposed

uniquely by R respectively as

2+e

(D) = R ffi X

, . with X

= lue C

2+e

(D) : I udx = o| ,

C(D) = R © Y

, with Y

= Iv G C

2+e

(S) : J vdx = o\ .

As well, the projector Q of C

(D) onto Y

is introduced as

Qv = v--^- fvdx, veC

(D).

The projector Q and Green's identity :

[ -Audx = - J -£-do, u e C

(D),

JdD

are combined to prove that if u € C

2+6

{D) satisfies (1) then

-Au

+ j--7 b(x)g{a + u

)da = \Qf(x,a + u

), in D,

l-^l J BD

—^- = b(x)g(a + u

), ondD,

$(A,a,u

) :=

/ f(x,a + u

)dx + b(x)g(a + u

)da = 0,

J D J 3D

where u = a®u

eH®X

and f(x, u) = (m(x)

—

au)u. Next we define the nonlinear

mapping T : R x R x X

—>

Z\, where

/(</>,

V) e C(D) x C

1+e

{dD) : f

<f>dx+

f i/ida = o| ,

T(A, a, u

) = ( -Au

+ Tjj; H

)9(a + u

)dx - XQf (x, a + u

) ,

-^ - 6(s)s(a + u

]