Remarks on a nonlocal problem involving the Dirichlet energy

Remarks on a Nonlocal Problem Involving the Dirichlet Energy.

M. CHIPOT(*) - V. VALENTE(**) - G. VERGARA CAFFARELLI(***)

ABSTRACT- We introduce a class of nonlinear parabolic problems of nonlocal type depending on the Dirichlet integral and study the questions of existence, uniqueness and asymptotic behaviour for the solution.

1. Introduction.

Let V be a bounded, smooth open subset of Rⁿ, nF1 . For f4f(x)L²(V) ,

(1.1)

u0H0¹(V) , (1.2)

we would like to consider the problem of finding u4u(x,t) solution to

. /

´

u_t2a

u

V

N˜uN²dx

v

u(x, 0 )4u₀(x) in V, u40

in V3R¹, on ¯V3R¹. (1.3)

Here a4a(s) is a continuous function such that 0EmGa(s)GM. (1.4)

(*) Indirizzo dell’A.: Institute of mathematics, Winterthurerstr. 190, CH- 8057 Zürich, Switzerland.

(**) Indirizzo dell’A.: Istituto per le Applicazioni del Calcolo, C.N.R., Viale del Policlinico, 137, I-00161 Roma, Italy.

(***) Indirizzo dell’A.: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma «La Sapienza», via A. Scarpa 16, I-00161 Roma, Italy.

for some positive constantsmandMand¯Vdenotes the boundary ofV. Nonlinear nonlocal problems present several interesting features, see for instance [9], [6], [7], [8], [2], [3], [4], [10], [16] for various issues and applications. In particular several equilibria (up to a continuum) can appear. This makes the study of the asymptotic behaviour of these prob- lems interesting and challenging. In addition to the usual properties of nonlocal problems, as we will see, (1.3) admits a Lyapunov function. This will allow us to describe some of its asymptotic behaviour.

The paper is divided as follow. In the next section we prove existence and uniqueness of a solution to (1.3). In section 3 we study the corre- sponding stationary problem. Finally in the last section we give some re- sults regarding the asymptotic behaviour of (1.3).

2. Existence and uniqueness.

We will need the following Lemma.

LEMMA 2.1. Let T be a positive number and suppose that a_n is a uniformly bounded sequence of functions such that

aⁿ(t)Ka^Q(t) a.e. t( 0 , T).

(2.1)

Suppose also that

u0ⁿKu0Q, fⁿKf^Q in L²(V) . (2.2)

If uⁿ is the solution of the problem

uⁿL²(0,T;H₀¹(V))OC([0,T];L²(V)), u_tⁿL²(0,T;H²¹(V)), d

dt(uⁿ,v)1aⁿ(t)

V

˜uⁿ˜v(x)dx4(fⁿ,v), in D8(0,T) (vH₀¹(V), uⁿ(Q, 0)4u₀ⁿ, (2.3)

then

uⁿKu^Q in L²( 0 ,T;H0¹(V) ) (2.4)

where u^Q is the solution to

u^QL²(0,T;H₀¹(V))OC([0,T];L²(V)), u_t^QL²(0,T;H²¹(V)), d

dt(u^Q,v)1a^Q(t)

V

˜u^Q˜v(x)dx4(f^Q,v), in D₈(0,T) (vH0¹(V), u^Q(Q, 0)4u₀^Q. (2.5)

PROOF. The existence and uniqueness of a solution to (2.3), (2.5) fol- lows by a well known result of J. L. Lions (see [11], [5]). We have denot- ed by ( , ) the usual scalar product inL²(V). We refer to [11], [5] for no- tation. By difference from (2.3), (2.5) we get

d

dt(uⁿ2u^Q,v)1aⁿ(t)

V

˜(uⁿ2u^Q)˜v(x)dx4(fⁿ2f^Q,v)1

1(a^Q(t)2aⁿ(t) )

V

˜u^Q˜v(x)dx, in D₈^{( 0 , T)} (vH₀¹(V).

Takingv4(uⁿ2u^Q) it comes by the Poincaré inequality for some con- stant c

1 2

d

dtNuⁿ2u^QN²²1aⁿ

N

N

V

˜u^Q˜(uⁿ2u^Q)(x)dxG GcNfⁿ2f^QN²

N

N

N

N

N

N

(NN² denotes the usual L²(V)-norm, NN the Euclidean norm). Using the Young inequality we get

1 2

d

dtNuⁿ2u^QN22

1m

N

N

Ge

N

N

N

N

2 this leads to d

dtNuⁿ2u^QN²²1m

N

N

GC]Nfⁿ2f^QN²²1 Naⁿ2a^QN²

N

N

Integrating between 0 ,T we derive easily m

0 T

N

N

G Nu₀ⁿ2u₀^QN²²1CTNfⁿ2f^QN²²1C

0 T

Naⁿ2a^QN²

N

N

THEOREM 2.1. For any TD0 there exists a weak solution to (1.3), i.e. there exists u such that

(2.6)

uL²(0,T;H₀¹(V))OC([0,T];L²(V)), u_tL²(0,T;H²¹(V)), d

dt(u,v)1a

u

V

N˜uN²dx

v

V

˜u˜v(x)dx4(f,v), in D₈^(0,T) (vH₀¹(V), u(Q, 0)4u₀. PROOF. Let us set

B4 ]vL²( 0 , T; H0¹(V) ); NwN^{L2( 0 ,T;H}0¹(V) )GC(

whereCis a constant that we will fix later on. Letwbe fixed inB. There exists a unique u solution to

(2.7)

uL²(0,T;H₀¹(V))OC([0,T];L²(V)), u_tL²(0,T;H²¹(V)), d

dt(u,v)1a

u

V

N˜wN²dx

v

V

˜u˜v(x)dx4(f,v), in D₈^(0,T) (vH₀¹(V), u(Q, 0)4u₀. We want show that the mapping

wKu4R(w)

admits a fixed point. For that, taking v4u in (2.7), we get 1

2 d

dtNuN²²1a(

N

N

N

N

N

N

by the Poincaré inequality for some constant c. Using (1.4) we get 1

2 d

dtNuN²²1m

N

N

2mNfN²²1m

2

N

N

d

dtNuN²²1m

N

N

(2.8) NuN²²1m

0 T

N

N

1 Nu₀N22

¨

¨NuN^{2L2( 0 ,T;H}0¹(V) )GT c²

m²NfN²²1 1

mNu₀N²²uC². We have defined C by the above inequality. Note that we choose also

NuN^{L2( 0 ,T;H}0¹(V) )4

{

0 T

N

N

}

With C defined in (2.8) we see that wKu4R(w) maps B into B. We claim now that R is continuous from B into B. Indeed, consider a se- quence wⁿB such that

wⁿKw^Q in B

Denote byuⁿthe solution to (2.7) wherewis replaced bywⁿand byu^Q the solution to (2.7) where w is replaced by w^Q. We have, for a subsequence,

N

N

2K0 a.e. t( 0 , T) which implies

N

N

N

N

a(

N

N

2)Ka(

N

N

2) a.e. t( 0 , T).

Applying the Lemma 2.1 we conclude that

u^nkKu^Q in L²( 0 , T; H₀¹(V) )

and the continuity of the mapRfollows sinceu^Qis the only possible lim- it ofuⁿ. In order to apply the Schauder fixed point theorem we now just need to prove that R(B) is relatively compact in B. For that we select aⁿC^Q( [ 0 ,T] ), fⁿD^{(V), u}0n

D(V) such that m

2 GaⁿG2M, aⁿKa(

N

N

2) a.e. t

fⁿKf in L²(V), u0n

Ku0 in H0¹(V)

and we denote byuⁿthe solution to (2.3).uⁿis very smooth and in partic- ular since uⁿ(x,t)N¯V40 for any t, we have

u_tⁿH₀¹(V)(t.

This implies also, by the second equation of (2.3) that it holds DuⁿH₀¹(V) (t.

Thus taking v4Duⁿ in (2.3) we get 1

2 d

dt

N

N

¨ 1 2

d

dt

N

N

21 m

2 NDuⁿN²²G NfⁿN²²

m 1 m

4 NDuⁿN22. It follows easily by integration that

1

2

N

N

21m

4

0 T

NDuⁿN²²dtGTNfⁿN²² m 11

2

N

N

0 T

NDuⁿN²²dtGC

for some other constant independent of nand w. Using the Lemma we obtain

0 T

NDuN22dtGC (2.9)

From (1.3) we derive also

NutN²²GC(NDuN²²1 NfN²²) , and

Nu_tN^{L2( 0 ,T;L2(V) )}GC. (2.10)

The estimates (2.9)-(2.10) insure thatR(B) is relatively compact inB(see [15]) and this concludes the proof. r

To establish uniqueness, we suppose now that ais a Lipschitz conti- nuous function that is to say for some K it holds that

Na(z)2a(z8)N GKNz2z8N (z,z8R. (2.11)

THEOREM 2.2. Under the assumptions of Theorem2.1 and if (2.11) holds there is a unique weak solution to the problem (1.3) i.e. to (2.6).

PROOF. If u is solution to (2.6) we set a(t)4

0 t

a

u

V

N˜u(x,s)N²dx

v

0 t

a(

N

N

It is clear that a is a one-to-one mapping from ( 0 ,1Q) into itself.

Set

w(x,a(t) )4u(x,t).

(2.13)

Due to (1.4), both a and a²¹ are Lipschitz continuous so that w(x,t)4u(x,a²¹(t) )H¹( 0 ,T;H0¹(V),H²¹(V) ).

(2.14)

(See [11], [5] for a definition of this space). If u is solution to (2.6) and

WD^{( 0 , T), vH}0¹(V), we have (2.15) 2

0 T

V

u(x,t)v(x)W8(t)dx dt1

0 T

a(t)

V

˜u(x,t)˜v(x)W(t)dx dt4

4

0 T

V

f(x)v(x)W(t)dx dt, where we have set

a(t)4a(

N

N

(2.16)

This equality holds true for any WH¹( 0 , T). If WD^{( 0 ,} a(T) ), then W(a(t) )H¹( 0 , T) and using this function in (2.14) we obtain

(2.17) 2

0 T

V

]w(x,a(t) )v(x)W8(a(t) )1

1˜w(x,a(t) )˜v(x)W(a(t) )(a(t)dx dt4

0 T

V

f(x)v(x)W(a(t) )a(t) a(

N

N

dx dt. Then, we use the following lemma of change of variables in the integrals:

LEMMA 2.2. Let

fL_loc¹ (R) , aL^Q(R) , 0EmGaGM, a(t)4

0 t

a(s)ds, (2.18)

then it holds that

0 T

f(a(t) )a(t)dt4

0 a(T)

f(t)dt. (2.19)

Assuming this Lemma proved we get for WD^{( 0 ,a(T) ), vH}0¹(V):

(2.20) 2

0 a(T)

V

]w(x,t)v(x)W8(t)1˜w(x,t)˜v(x)W(t)(dx dt4

4

0 a(T)

V

f(x)v(x)W(t) a(

N

N

2)^{dx dt,}

i.e w(x,t) is solution to

wL²(0,a(T);H0¹(V))OC([0,a(T)];L²(V)), wtL²(0,a(T);H²¹(V)), (2.21) d

dt(w,v)1

V

˜w˜v(x)dx4

(f,v) a(

N

N

2) ^,

in D8( 0 , a(T) ) (vH0¹(V), w(Q, 0 )4u0. Consider thenu1and u2two solutions of (2.6). Thenw1andw2defined by

w_i(x,a(t) )4u_i(x,t), i41 , 2 , (2.22)

are solution to (2.21). Takingv4w₁2w₂in (2.21) written for w₁ andw₂ we get easily by subtraction

1 2

d

dtNw₁2w₂N22

1

N

N

4(f,w₁2w₂)

{

_N

_N

2 2) ²

1 a(

N

N

2)

}

Note thata_i(T) are not necessarily the same, but they can be taken arbi- trarily like T. Recalling (1.4) we get

1 2

d

dtNw₁2w₂N²²1

N

N

G NfN²Nw₁2w₂N²

{

^N

^N

^N

^N

}

From (2.11) we deduce

N

N

N

N

N

N

N

N

GK

N

N

N

N

1 2

d

dtNw₁2w₂N²²1

N

N

m² Nw12w2N²K

N

N

N

N

Using the Young inequality we get 1

2 d

dtNw₁2w₂N²²1

N

N

2

N

N

2m⁴K²

N

N

d

dtNw₁2w₂N²²GC(t)Nw₁2w₂N²²

where C(t)L¹( 0 ,T). From the Gronwall inequality we obtain that w14w2 and the uniqueness of u follows. r

Let us now turn to the proof of the Lemma which is more or less clas- sical but that we establish for the reader convenience.

PROOF OF THE LEMMA 2.2. Let fC(R), a_nC(R) such that m

2 Ga_nG2M a_nKa a.e. on ( 0 ,T).

One has clearly with an4

s

0 t

a_n(s)ds

0 T

f(a_n(t) )a_n(t)dt4

0 a_n(T)

f(t)dt. (2.23)

Passing to the limit inn we see that (2.19) holds for fC(R). Let f_n C(R) such that f_nKf in L_loc¹ (R). We have

0 T

f_n(a(t) )a(t)dt4

0 a(T)

f_n(t)dt. (2.24)

We claim thatf_n(a(t) )a(t) is a Cauchy sequence inL¹( 0 ,T). This follows indeed from

0 T

Nf_n(a(t) )a(t)2f_m(a(t) )a(t)Ndt4

0 a(T)

Nf_n2f_mNdsGe

for n,m large enough. Thus, there exists a function gL¹( 0 ,T) such that

f_n(a(t) )a(t)Kg(t)

in L¹( 0 , T). Up to a subsequence one can assume that f_n(s)Kf(s) on [ 0 ,a(T) ]0N

whereNis a set of measure 0. Sincea²¹is Lipschitz continuous,a²¹(N) is also of measure 0 and

f_n(a(t) )Kf(a(t) ) on [ 0 ,T]0a²¹(N).

Thusg(t)4f(a(t) )a(t)a.e. and passing to the limit in (2.24) the proof of the lemma is complete. r

3. Steady states.

In this section we study the stationary solutions to (1.3). These steady states can be obtained as minimizers of some energy or as critical points. Let E(u) be the energy defined by

E(u)41 2A

u

V

N˜uN²dx

v

(3.1) where

A(s)4

0 s

a(z)dz. (3.2)

We have:

THEOREM 3.1. The functional E(u) admits a global minimizer in H₀¹(V).

PROOF. First, we see thatEis coercive and bounded from below. In- deed, from the Poincaré inequality we have

N(f,u)N G NfN²NuN²GcNfN²

N

N

and hence

E(u)41 2A

u

V

N˜uN²dx

v

^N

^N

^N

^N

(3.3)

This shows the coerciveness of E. E is also bounded from below since

m

2

N

N

22cNfN²

N

N

LetunH0¹(V) be a minimizing sequence ofE. Due to (3.3)unis bound- ed in H₀¹(V) and we can assume that for some u_QH₀¹(V) we have

u_nu_Q in H₀¹(V), u_nKu_Q in L²(V).

Due to the weak lower semicontinuity of the norm we deduce lim

N

N

N

N

Consider a subsequence u_nk such that lim

N

N

24lim

k

N

N

inf

H₀¹(V)

E(u)4lim

k E(u_nk)4 1

2

0 limNN˜u_nNN2

2

a(s)ds2(f,u_Q)F

F 1

2

0 NN˜u_QNN2

2

a(s)ds2(f,u_Q)4E(u_Q).

Thusu_Q is a minimizer of Eon H₀¹(V). Note that for any sequence u_n satisfying

u_nu_Q in H0¹(V), we have shown that it holds that

limE(u_n)41

2

0 limNN˜u_nNN2

2

a(s)ds2(f,u_Q)FE(u_Q),

i.e. E is weakly lower semicontinuous on H₀¹(V). This completes the proof of the theorem. r

As we will see belowEmight have several minimizers. First note that if u_Q is a (local) minimizer of E on H0¹(V) then u_Q is a solution of

. /

´

2a

u

V

N˜uN²dx

v

u40

in V, on ¯V. (3.5)

Indeed if u_Q is a (local) minimizer it holds that (Euler equation) d

dlE(u_Q1lv)N^l4040 (vH₀¹(V).

This leads to d

dl

{

0 N^N˜(uQ1lv)NN2

2

a(s)ds2(f,u_Q1lv)

} _N

¨ 1

2a(

N

N

V

˜u_Q˜v(x)dx2(f,v)40 , (vH₀¹(V), i.e. u_Q is a solution to (3.5) and a stationary point.

Regarding stationary points we have:

THEOREM 3.2. The mapping uKl(u)4

s

VN˜uN²dx is a one-to-one mapping from the set of weak solution to(3.5)onto the set of solutions to

a(m)²m4l(W)4

V

N˜WN²dx (3.6)

where W is the weak solution to ./

´

2DW4f W40

in V, on ¯V. (3.7)

PROOF. Let u be a solution of (3.5), we have

a(l(u) )u4W ¨ l(a(l(u) )u)4l(W) ¨ a²(l(u) )l(u)4l(W)

andl(u) is a solution to (3.6). This shows thatl goes from the set of sol- utions to (3.5) into the set of solutions to (3.6). Let nowmbe a solution to (3.6), denoting by u the solution to

2a(m)Du4f in V, u40 on ¯V, one has

a(m)u4W ¨ l(a(m)u)4l(W)

¨ a(m)²l(u)4l(W)4a(m)²m ¨ l(u)4m

anduis a solution to (3.5). Now if we havel(u₁)4l(u₂) then a(l(u₁) )4 4a(l(u2) ) and u14u2. This completes the proof of the theorem. r

REMARK 3.1. The stationary points are determined (through (3.6)) by the solutions to

a(m)4

o

Of course we can selectaso that this equation admits 1 , 2 ,Racontinu- um of solutions. The critical points ofEare not all global minimizers. In- deed we have:

THEOREM 3.3 (comparison of energies). Let u₁, u₂be two solutions of (3.5) corresponding to the solutions m1, m2 of (3.8). Assume that

a(m)D

o

o

o

then it holds that

E(u₂)DE(u₁) (resp . E(u₂)EE(u₁),E(u₂)4E(u₁) ) PROOF. Recall that u_i is solution to

2a(m_i)Du_i4f in V, u40 on ¯V, i.e. we have

mi4

N

N

Thus it comes

E(u_i)41 2

0 mi

a(s)ds2(f,u_i)4

4 1 2

0 mi

a(s)ds2

g

h

0 mi

a(s)ds2 l(W) a(m_i) , since (f,W)4(2DW,W)4l(W). By subtraction we obtain in the first case above

E(u₂)2E(u₁)4 1 2

m₁ m2

a(s)ds1 l(W)

a(m₁)2 l(W) a(m₂)4 1

2

m₁ m2

a(s)ds1

1

k

k

m1 m2

o

k

k

k

k

k

k

REMARK 3.2. The functional E(u) is not convex and might have as shown by the above theorem several global minimizers.

4. Asymptotic behaviour.

Let us denote by H¹( 0 , T; L²(V) ) the space

H¹( 0 , T;L²(V) )4]vL²( 0 ,T; L²(V) );v_tL²( 0 , T; L²(V) )(. (4.1)

LEMMA 4.1. For uH¹( 0 ,T; L²(V) )OC( ( 0 ,T);H₀¹(V)OH²(V) ) it holds that

s t

V

]

N

N

(

(4.2)

PROOF. By density (see [11], [5]) we can assume that u C¹( ( 0 ,T);L²(V) ) and then thatuC¹( ( 0 , T); H₀¹(V)OH²(V) ) (see [1] p. 74 for a similar argument). Then for such a u we have

V

]

N

N

(

V

a(

N

N

4 d dt

1 2

V

A(

N

N

V

fudx4 d

dtE(u(t) ) whereEis defined by (3.1). The result follows by integrating between s and t. r

LEMMA 4.2. Let u(Q,t) be the unique solution to the problem (1.3) or (2.6) then when tK Q there exists a subsequence t_k such that

u(Q,t_k)u_Q in H₀¹(V) where u_Q is a stationary point.

PROOF. Using the equation (1.3) and the Lemma 4.1 we have (recall also (2.10))

2

s t

V

ut2dx dt4E(u(t) )2E(u(s) ).

(4.3)

It follows that E(u(t) ) is nonincreasing and there exists E_Q such that lim

tK QE(u(t) )4E_Q. (4.4)

Passing to the limit in (4.3) we have

0 Q

Nu_tN²²dt4E(u(s) )2E_Q, (4.5)

and from (3.4)

N

N

24

V

N˜u(x,t)N²dxGconst .4C1 (tFs. (4.6)

From (4.5) we have

limNutN²²40 as tK Q

so there is a subsequence which we denote by t_k such that Nu_t(Q,t_k)N²²K0 as t_kK Q.

(4.7)

We also have from (4.6) that

N

N

hence passing eventually to a subsequence if u_k4u(Q,t_k) we have u_ku_Q weakly in H₀¹(V),

u_kKu_Q strongly in L²(V).

Now, we want to prove thatu_Qis a stationary point i.e. is a solution to (3.5). Since E is nonincreasing

E(u(Q,t_k) )41 2A

u

V

N˜u_kN²dx

v

¨ 1 2A

u

V

N˜ukN²dx

v

and recalling that A is invertible

V

N˜u_kN²dxKl_Q4A²¹( 2E_Q12(f,u_Q) ).

(4.8)

Taking v4u and W solution of (3.7) in (2.6) we get 1

2 d

dtNuN²²1a(

N

N

N

N

dt(u,W)1a(

N

N

)

4

4

2

4

From (4.8) we deduce that 1

2 d

dtNu(t_k)N²²4(u_t(Q,t_k),u_k)4

4 2a(

N

N

N

N

Since u is bounded in L²(V) and (4.7) holds we get (f,u_Q)4a(l_Q)l_Q, l(W)4a(l_Q)(f,u_Q)

¨ l(W)4a²(l_Q)l_Q

i.e. l_Q is a solution to (3.6). For any v belonging to H₀¹(V) it holds that (ut(Q,tk),v)1a

u

V

N˜ukN²dx

v

From (4.7), (4.8) passing to the limit we get

a(l_Q)(˜u_Q,˜v)4(f,v) (vH0¹(V)

and we obtain that u_Q is a stationary point – see Theorem 3.2. r REMARK 4.1. The convergence is in fact strong in H₀¹(V) since

V

N˜u_kN²dxKl_Q4

V

N˜u_QN²dx kK Q, see (4.8).

We can then show:

THEOREM 4.1. Let u be the unique solution to the problem (1.3) then, if E admits a unique stationary point u_Q, it holds that

u(Q,t)Ku_Q in H₀¹(V) when tK Q.

PROOF. Since u(t) is uniformly bounded in H₀¹(V) for some subse- quence it holds that

u(Q,tk)v_Q in H01(V)2weak . By Lemma 4.2 – see also Remark 4.1 – we have

E(u(t) )KE(u_Q)

whereu_Qis the global minimizer ofE. By the weak lower semicontinuity of E (see Theorem 3.1) we have

E(u_Q)4 lim

tkK Q

E(u(t_k) )FE(v_Q).

Sincev_Qis a minimizer we have necessarilyv_Q4u_Q. Since this holds for any subsequence the result follows – see also the Remark 4.1.

REMARK 4.2. If E admits a unique global minimizer u_Q and if E(u₀)EE(u_i)

for any stationary pointu_icu_Q, then with the same proof as above we have

u(Q,t)Ku_Q in H₀¹(V) when tK Q.

We would like to end this section by a result of asymptotic stability.

Let c be the constant defined by the Poincaré inequality

N

N

G

N

N

We introduce the function A^(s)₄^a(s2) which, from (1.4), verifies 0EmGA^(s)GM.

Then we have

THEOREM 4.2. Assume A ^{is a C1} function and u_Q is a stationary solution to the problem (3.5) which verifies

m2 NA₈(

N

N

) _N

m

N

N

D

D

then u_Q is a locally asymptotically stable stationary solution in the sense of Lyapunov. That is to say, there exits r4r(m) such that if

N

N

N

N

N

N

m c2t

(tF0 .

PROOF. Let u_Q be the solution of the stationary problem that veri- fies (4.9). From (1.3), (3.5), (see also (2.9)-(2.10)), we have:

(4.10) (u2u_Q)_t2a

u

V

N˜uN²dx

v

y

u

V

N˜uN²dx

v

u

V

N˜u_QN²dx

v z

Putting h4u2u_Q and multiplying the above equation by Dh we get

21 2

d dt

V

N˜hN²dx2a

u

V

N˜uN²dx

v

V

NDhN²dx4

4

y

u

V

N˜uN²dx

v

u

V

N˜u_QN²dx

v z

V

Du_QDh dx.

Applying the mean value theorem we deduce (4.11) 1

2 d dt

V

N˜hN²dx1a

u

V

N˜uN²dx

v

V

NDhN²dxG

G

N

N

N

N

N

N

N

NN N

N

N

N

N

Q

N

N

^N

^N

^N

^N

^N

^N

^N

^{N N}

^N

Q

N

N

^N

^N

^N

^N

^N

^N

^N

^N

We have thus (see (3.5)) (4.12) 1

2 d

dt

N

N

1

g

^N

^N

^N

^N

^N

^N

^N

^N

h

Now, if

N

N

m2

N

N

N

N

N

N

N

N

mNfN²DmD0 and we have

d

dt

g _N

N

m c2t

h

This completes the proof. r

REMARK 4.3. If Ais a Lipschitz continuous function with Lipschitz constant K and

m2K c

mNfN²DmD0 (4.14)

then the stationary problem has a unique solution and hence for this sol- ution we have global asymptotic stability. Indeed, ifu_Q¹ and u_Q² are two solutions of the problem (3.5), repeating the steps of the proof of Theo- rem 4.2 we get (see (4.12))

g

h

(4.15)

which leads to u_Q¹4u_Q².

Acknowledgements. The first author would like to acknowledge the support of the Swiss National Science foundation under the contract J20-67618.02. The work of the second author has been supported by the European Community Program «Improving the Human Research Po- tential and Socio-Economic Knowledge Base» under the contract HPRN-CT-2002-00284. The work of the third author has been supported by Progetto Cofinanziato 2002 «Modelli Matematici per la Scienza dei Materiali».

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Manoscritto pervenuto in redazione il 28 marzo 2003.