An evolutionary double-well problem

www.elsevier.com/locate/anihpc

An evolutionary double-well problem

Tang Qi

, Zhang Kewei

Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom Received 24 February 2002; accepted 24 April 2002

Available online 20 December 2006

Abstract

We establish the existence theorem and study the long time behaviour of the following PDE problem:

⎧⎨

⎩

u_t−div∇W (∇u)−f (x)=0 inΩ×(0,−∞),

∇W (∇u)·n|∂Ω×(0,∞)=0, u(x,0)=u₀(x) inΩ

(0.1)

whereW is a specially given quasiconvex double-well function and f ∈L²(Ω)is a given function independent of timet. In particular, the existence theorem is established for general given source termf, the long time behaviour is analyzed under the assumption that

Ωf (x)dx=0.

The system is an evolutionary quasimonotone system. We believe that the existence of solutions established here is stronger than the usual Young Measure solution and is the first of its kind. The existence of a compactω-limit set ast→ ∞is also established under some non-restrictive conditions.

©2007 Elsevier Masson SAS. All rights reserved.

MSC:35K55; 49J45

Keywords:Quasiconvexity; Long time behaviour; Semi-flow

1. Introduction

In [8], it was established that for a general parabolic system, generated as the gradient flow of a rank-one convex energy functional, the energy at any time is less than or equal to the energy ‘smoothly sampled’ at earlier times.

In this paper, we give a concrete example to this result. Furthermore, for the given example, despite lack of higher regularity, we obtain the long time convergence of solutions. Before we start our discussion, we mention the recent interesting papers of [18] and [7]. In [18,3], the existence of Young measure solutions for non-convex elasto-dynamics was discussed. In [7], the existence of weak but non-Young measure solutions for the non-linear flow equation of the type we dealt with was discussed. The setting of [7] is abstract with strong assumptions on the quasimonotonicity of the energy potential. In our setting, we look at a concrete quasiconvex function that has double-well potential and not necessarily strongly quasimonotone. Related problems have been discussed in [13], [14] and [17]. The distinct feature

* Corresponding author.

E-mail addresses:q.tang@sussex.ac.uk (Q. Tang), k.zhang@sussex.ac.uk (K. Zhang).

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.11.002

of our paper is that our potential energy function is specifically given, with clear double-well structure [5,12] and is not necessarily strongly quasimonotone, we can obtain the existence of strong solution that has long time convergence properties. We believe that this is the first result of its kind.

It is worth pointing out that if we replace the Neumann boundary condition by Dirichlet boundary condition in (0.1), then the corresponding problem becomes strongly monotone and the analysis can be carried out along the lines of standard non-linear parabolic theory.

We conclude that the evolution of micro-structure in the particular setting of our problem settles down to the steady state over the long time period. However, it is not known if the evolutionary solution will settle down to the energy minimizers. In a separate paper (cf. [21]), we establish some examples illustrating that under some special boundary conditions, the solution of the heat-flow problem could converge to solutions at different energy levels and provide a general picture for analyzing these problems.

In [24], the steady state problem of (0.1) has been studied div∇W_λ(∇u)+f (x)=0 inΩ,

∇W_λ(∇u)·n|∂Ω=0 (1.1)

wheref is some given smooth function onΩ,Ω is a bounded open smooth subset ofRⁿ,uis a mapping fromΩ intoR^N,Nandnare any positive integers>1,W_λ:M^N×n→Ris a non-negative, quasiconvex double-well function vanishing at two matrix points. In this paper, we consider the corresponding evolutionary system (0.1).

In Section 2, we give some preliminary results and introduce our double-well model and give the detailed properties established in [24]. In Section 3, we present preliminary results and introduce the weak formulation using finite difference. In Section 4, we establish a priori estimates independent of discretization. In Section 5, we discuss how to obtain convergence using insufficient a priori estimates, the existence of solutions is established using the given f∈L²(Ω). In Section 6, we use Galerkin method to briefly discuss the existence of solutions usingf with constraint

Ωf (x)dx=0, subsequently, we establish the foundation for discussing the long time behaviour of solutions [11,4]

and established the convergence results. In Section 7, we look at a precise result on how the flow evolves ast→ ∞ in some specific circumstances.

2. Preliminaries and the model

In this section, we describe the energy densityW_λin (1.1) which we will use throughout this paper.

Definition 2.1.LetW:M^N×n→Rbe a continuous function. The following are some conditions related to weak lower semicontinuity of the integralI (u)=

ΩW (Du(x))dx:

(i) W isrank-one convexif for anyN×nmatrixP, any rank one matrixB=a⊗b=(a_ib_j)and any real number s∈ [0,1], we have

W (P+sB)sW (P +a⊗b)+(1−s)W (P ).

(ii) W isquasiconvexat the constant matrixP ∈M^N×n if for a given, non-empty bounded open set Ω⊂Rⁿ and everyφ∈W₀^1,∞(Ω, Rⁿ),

ΩW (P + ∇φ)dxW (P )meas(Ω).W is quasiconvex if it is quasiconvex at every A∈M^N×n. The class of quasiconvex functions is independent of the choice ofΩ.

It is well known that (ii) implies (i), while (i) does not imply (ii) (cf. [2,16,6,19]).

For quadratic forms onM^N×n, the two definitions (i) and (ii) in Definition 2.1 are the same.

For a continuous functionW:M^N×n→Rthat is bounded from below, we may define its quasiconvex envelop (cf. [6]) as the largest quasiconvex function less than or equal to W. More precisely, QW = sup{GW, Gquasiconvex}. Similarly, the convex envelop off isCW=sup{GW, Gconvex}.

The following is the ellipticity condition which was first introduced in [22] to prove the existence of weak solution for elliptic systems [10], and in [9] for the partial regularity property for weak solutions. From now on, we use the summation convention for repeated indices withI from 1 toN andμfrom 1 ton.

Definition 2.2. A continuous mappingB:M^N×n→M^N×n is strongly quasimonotone if for every constant matrix P ∈M^N×n, every bounded open setΩ⊂Rⁿand everyφ∈W₀^1,∞(Ω,R^N),

Ω

B_αⁱ(P+ ∇φ)D_αφⁱ(x)dxc₀

Ω

|∇φ|²dx wherec0>0 is a constant independent ofP,Ω andφ.

If under above notation assumptions, we have only

Ω

B(P+ ∇φ): ∇φdx0

then the mappingBis called quasimonotone.

When B(P )= ∇W (P ) for some scalar valued W, if B(P ) is strongly quasimonotone (respectively, quasi- monotone), we call∇Wa strongly quasimonotone (respectively, quasimonotone) gradient mapping.

It is easy to check that convex functions and rank-one convex quadratic forms give gradient quasimonotone map- pings.

Now we introduce our model double-well integrandsWλand show thatWλ=Gλ+HλwhereGλis convex and H_λis a rank-one convex quadratic form, which implies that∇W_λis quasimonotone.

LetA∈M^N×nbe a given matrix with rank(A) >1,|A| =1. LetE=span[A]andK= {−A, A},C_Edenote the convexification operation on the one-dimensional space E,P_E be the standard Euclidean projection operator onto E such that for any given arbitraryN×nmatrixX:P_E(X)=(X·A)A=(

i,jX_ijA_ij)A. Sometimes, where the circumstance is clear, we may also use the notation P_E(X) for the scalar quantity X·A. Subsequently, we have P_E⊥(X)=X−PE(X). Letλ0be the largest eigenvalue ofA^TA. From|A| =1, we know that 0< λ01. We define

W_λ(X)=C_E dist² P_E(X), K

+λP_E(X)²

+H_λ(X)=G_λ(X)+H_λ(X),

H_λ(X)=P_E⊥(X)²−λP_E(X)² (2.1)

where

0λλ^∗=1−λ₀

λ₀ . (2.2)

It is easy to check that under our assumptions, for each fixedλsatisfying 0λλ^∗, the functionH_λ(X)defined onM^N×nis a rank-one convex quadratic form, so is quasiconvex. Furthermore,∇H_λ(X)is strongly quasimonotone when 0λ < λ^∗.

If f (t ) is a continuous function on the real line, we denote by Cf (t )the convexification off. In the case of G(P_E(X)), it is a continuous function defined onE. Let g(t )=G(t A),t ∈R. We define the convexification of G(P_E(X))onEas

C_EG P_E(X)

=Cg(X·A).

It can be verified that G_λ(X)=G_λ P_E(X)

=f_λ(X·A) where

f_λ(t )=

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

(t+1)²+λt², t− 1 1+λ, λ

1+λ, |t| 1

1+λ, (t−1)²+λt², t 1

1+λ

for 0< λλ^∗. Sincef is an explicit function, we can easily verify that there is a constantC >0 such that for all 0< λλ^∗andt, s∈R,

f_λ(t )−f_λ(s)2

C f_λ(t )−f_λ(s)

(t−s). (2.3)

The functionWλdefined below are our model integrands (cf. [24]).

Remark 2.1.Our discussion also works for matricesAthat|A| =1. In this case, we need to introduceA₀=A/|A| and the corresponding projection operators will need to be modified. To avoid unnecessary complication of notation, we useAwith unit modulus.

Proposition 2.1.LetW_λbe as in(2.1)with0λλ^∗. Then

(1) Wλ(X) is quasiconvex. Furthermore, when λ =λ^∗, Wλ(X)=Qdist²(X, K) and when λ=0, Wλ(X)= Cdist²(X, K)for allX∈M^N×n.

(2) For0λτ λ^∗,W_λ(X)W_τ(X).

(3) For0< λλ^∗, λ

1+λdist²(X, K)W_λ(X)dist²(X, K) for allX∈M^N×n.

(4) For0< λ < λ^∗,∇W_λ(X)is strongly quasimonotone, while∇W_λ∗(X)is quasimonotone.

(5) For0< λλ^∗, we haveW_λ(X)=0if and only ifX=AorX= −A. Furthermore, whendist(X, K) < λ/(1+λ), W_λ(X)=dist²(X, K).

(6) There are positive constantsc₀,C₀andc₁depending on|A|andλsuch that c₀ |X|²−1

W_λ(X)C₀ |X|²+1

, ∇W_λ(X):Xc₁ |X|²−1 for0λλ^∗.

(7) ∇W_λ(X)is a global Lipschitz function:

∇W_λ(X)− ∇W_λ(Y )C₀|X−Y| for allX, Y∈M^N×n.

LetI∈M^2×2be the identity matrix, X=

X₁₁ X₁₂ X₂₁ X₂₂

∈M^2×2be an arbitrarily given matrix.

An example ofWλforK= {−^√1₂I,√¹ 2I}is W_λ(X)=f_λ

1

√2(X₁₁+X₂₂)

+H_λ(X) with 0< λ√

2−1=λ^∗. Note that in this special case,λ₀=1/√

2. If we restrict W_λ to the one-dimensional subspaceE=span[A], then

W_λ(t A)=f_λ(t )−λt²=

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

(t+1)², t 1 1+λ, λ

1+λ−λt², |t| 1 1+λ, (t−1)², t 1

1+λ. Therefore, alongE,W_λhas a double well structure.

Finally, we state the following result that is a special case of a general theorem due to J. Krestensen [15].

Proposition 2.2.SupposeΩ⊂Rⁿis bounded and smooth, andu_j→uinW^1,p(Ω)where1< p <∞. Then there are two bounded sequences(vj)inW^1,p(Ω)and(wj)inW₀^1,p(Ω)such thatuj−u0=vj+wj, and up to a subsequence

(i) ∇v_j→0almost everywhere inΩ,

(ii) w_j 0inW₀^1,p(Ω)and|∇w_j|^pis equi-integrable onΩ.

3. The discretized problem

We discretize the problem and try to solve, for anyφ∈H¹(Ω)andm=1,2, . . .,

Ω

um−um−1

t ·φdx+

Ω

∇W_λ(∇u_m): ∇φdx−

Ω

f (x)·φdx=0, u₀=u⁰.

(3.1)

Theorem 3.1.LetWλbe given as in(2.1),f be given inL²(Ω), then the following minimization problem inf

um∈H¹(Ω)

Ω

1

2t |u_m|²−2um·u_m−1

+W_λ(∇u_m)−f·u_m

dx (3.2)

admits at least one solution inH¹(Ω).

Proof. Because of the estimate in Proposition 2.1(6), the growth condition required by [1] is satisfied, we also know that by Proposition 2.1(1),W_λis quasiconvex, hence we can use the main result of [1] to conclude the existence of an energy minimizer.

Due to the growth condition satisfied byWλ, the energy minimizers lie in the spaceH¹(Ω). 2 Corollary 3.1.The energy minimizers of (3.2)solves the variational equality(3.1).

Proof. (3.1) is the Euler–Lagrange equation of (3.2). 2 4. A priori estimates independent of discretization

For technicalities, we should restrict to a bounded time interval(0, T )in the first place and then extend the result to(0,∞). To simplify the argument, we only deal with the solutions{u_m}obtained by minimization method.

Proposition 4.1.Let{um}be the sequence of solutions obtained by minimizing energies defined in(3.2), we have, for some constantCdepending only on the initial data,

sup

m

Ω

1 2

|u_m−u_m−1|²

t +W_λ(∇u_m)+f·u_m

dxC (4.1)

and

m

Ω

|u_m−u_m−1|²

(t )² tC(T +1). (4.2)

Proof. First, from the definition of the energy minimizers, we obtain, for anym,

Ω

1 2

|u_m−u_m−1|²

t +W_λ(∇u_m)+f·u_m

dx= inf

u∈H¹(Ω)

Ω

1 2

|u−u_m−1|²

t +W_λ(∇u)+f ·u

dx

Ω

1 2

|u−um−1|²

t +W_λ(∇u)+f ·u

dx|u=um−1

=

Ω

W_λ(∇u_m−1)+f ·u_m−1 dx

Ω

1 2

|u_m−1−u_m−2|²

t +W_λ(∇u)+f ·u

dx

...

Ω

Wλ(∇u0)+f·u0

dx.

This implies (4.1). Furthermore, since the energy minimizers satisfy the Euler–Lagrange equation (3.1), by choosing φ=(u_m−u_m−1)/t, we have, for some constantC

0=

m

t

Ω

um−um−1

t

²+ ∇W_λ(∇u_m): ∇um−um−1

t +f ·um−um−1

t

m

t

Ω

∇W_λ(∇u_m): ∇u_m−u_m−1

t +1

2

u_m−u_m−1 t

²

dx−CT . To study the term

m

t

Ω

∇W_λ(∇u_m): ∇u_m−u_m−1

t dx,

we introduce the following functions

UN(x, t )=um(x) whent∈(tm−1, tm], m=0, . . . , N, (4.3) YN(x, t )=t−(m−1)t

t um(x)+

1−t−(m−1)t t

um−1(x)

whent∈(t_m−1, t_m], m=1, . . . , N (4.4)

whereN=_t^T is assumed to be an integer. Then we have

m

t

Ω

∇W_λ(∇u_m): ∇u_m−u_m−1

t dx

=

m

t

Ω

∇W_λ(∇U_N): ∇um−um−1

t dx

=

m

t

Ω

∇ W_λ(∇U_N)−W_λ(∇Y_N)+W_λ(∇Y_N)

: ∇um−um−1

t dx

−

m

t

Ω

∇(um−um−1)²dx+ T 0

Ω

∇Wλ(∇YN): ∇∂Y_N

∂t dx −CT −1.

Summarizing, we easily deduce (4.2). The proof of the proposition is finished. 2 5. Convergence

We now have, for anyφ∈L²(0, T;H¹(Ω)), notice thatu0=u⁰,

m tm

tm−1

dt

Ω

u_m−u_m−1

t ·φ+ ∇W_λ(∇u_m): ∇φ−f (x)·φ

dx=0. (5.1)

Because of the estimates obtained in the previous section, we conclude that both {U_N} and{Y_N}admit common subsequences still denoted by{U_N}and{Y_N}such that (byN→ ∞we meant→0)

U_N→u and Y_N→u asN→ ∞.

The convergence is strong inL²(Ω×(0, t )), weak-∗inL^∞(0, T;L²(Ω))and weak inL²(0, T;H¹(Ω)). In addition, {YN}converges weakly inH¹(Ω×(0, T )).

Moreover, we know that

U_N−Y_NL²(0,T;H¹(Ω))→0 asN→ ∞, (5.2)

u_m−u_m−1

t =∂_tY_N on (m−1)t, mt

. (5.3)

Using the results from [14,15,23] and the discussions above, we know that U_N=Y_N+(U_N−Y_N)=u+v_N+w_N+(U_N−Y_N)

where

(i) v_Nis bounded inH¹(Ω×(0, T )),∇v_N→0 almost everywhere inΩ×(0, T ).

(ii) w_N is bounded inH₀¹(Ω×(0, T )),w_N 0 inH₀¹(Ω×(0, T ))and|∇w_N|²+ |∂_tw_N|²is equi-integrable on Ω×(0, T ).

(iii) U_N−V_Nconverges strongly inL²(0, T;H¹(Ω)).

In order to prove that there is a solution, we need to show that∇v_N+ ∇w_N tends to 0 strongly inL²(Ω×(0, T )).

To achieve this, chooseφ=v_N+w_N+(U_N−Y_N)in (5.1), we obtain T

0

dt

Ω

∂V_N

∂t · vN+wN+(UN−YN)

+ ∇Wλ(∇UN): ∇ vN+wN+(UN−YN)

−f (x)· v_N+w_N+(U_N−Y_N) dx=0.

Take the limitN→ ∞, from what we know, it is easy to conclude that

Nlim→∞

T 0

dt

Ω

∇W_λ ∇u+ ∇ v_N+w_N+(U_N−Y_N)

: ∇(v_N+w_N)dx=0. (5.4)

Using the fact that∇Wλis Lipschitz and that∇(UN−YN)converges strongly inL²(Ω×(0, T )), we have

Nlim→∞

T 0

dt

Ω

∇W_λ ∇u+ ∇(v_N+w_N)

: ∇(v_N+w_N)dx=0. (5.5)

From here, we can show that∇(v_N+w_N)→0 inL²(Ω×(0, T )):

Recall that

W_λ(X)=C_E dist² P_E(X),{−A, A}

+λP_E(X)²

+H_λ(X)

=G_λ(X)+H_λ(X),

which is the sum of a convex function of quadratic growth and a rank-one convex quadratic form. Due to convexity, we know that

∇G_λ(X)− ∇G_λ(Y )

:(X−Y )0 for allX, Y∈M^2×2.

From the property of decomposition, we know that∇u∈L²(Ω×(0, T ))and|∇w_N|²equi-integrable, so for any μ >0, there is 0< δ < μsuch that for any measurable setF ⊂Ω×(0, T )with measF < δ, we have

F

|∇u|²dxdtμ,

F

|∇w_N|²dxdtμ.

Since ∇vN →0 almost everywhere, using Egorov’s Theorem that for every δ >0, there is a measurable subset E1⊂Ω×(0, T )with measE1< δsuch that∇vN→0 uniformly on(Ω×(0, T ))\E1asj→ ∞. Therefore, we have (whereQ=Ω×(0, T ))

Q

∇Wλ(∇u+ ∇vN+ ∇wN):(∇vN+ ∇wN)dxdt

=

Q\E₁

+

E₁

∇Wλ(∇u+ ∇vN+ ∇wN):(∇vN+ ∇wN) dxdt

=A_N+B_N. We estimateANfirst

AN=

Q\E₁

∇Wλ(∇u+ ∇vN+ ∇wN):(∇vN+ ∇wN)dxdt

=

Q\E₁

∇Wλ(∇u+ ∇vN+ ∇wN)− ∇Wλ(∇u)

:(∇vN+ ∇wN)dxdt

+

Q\E1

∇W_λ(∇u):(∇v_N+ ∇w_N)dxdt

=A⁽¹⁾_N +A⁽²⁾_N . It is easy to see that

A⁽²⁾_N →0 asN→ ∞.

Using the facts that∇H_λis linear andG_λis convex, we have A⁽¹⁾_N =

Q\E₁

∇W_λ(∇u+ ∇v_N+ ∇w_N)− ∇W_λ(∇u)

:(∇v_N+ ∇w_N)dxdt

=

Q\E1

∇G_λ(∇u+ ∇v_N+ ∇w_N)− ∇G_λ(∇u)

:(∇v_N+ ∇w_N)dxdt

+

Q\E1

∇H_λ(∇u+ ∇v_N+ ∇w_N)− ∇H_λ(∇u)

:(∇v_N+ ∇w_N)dxdt

Q\E₁

∇H_λ(∇u+ ∇v_N+ ∇w_N)− ∇H_λ(∇u)

:(∇v_N+ ∇w_N)dxdt

=

Q\E₁

∇Hλ(∇u+ ∇vN+ ∇wN):(∇vN+ ∇wN)dxdt−

Q\E₁

∇Hλ(∇u):(∇vN+ ∇wN)dxdt

=

Q\E₁

∇Hλ(∇u+ ∇vN+ ∇wN): ∇vN+ ∇Hλ(∇u+ ∇vN): ∇wN

− ∇H_λ(∇u): ∇(v_N+w_N) dxdt+

Q\E1

∇H_λ(∇w_N): ∇w_Ndxdt

=C_N⁽¹⁾+C_N⁽²⁾

where

Q\E1

∇H_λ(∇u+ ∇v_N+ ∇w_N): ∇v_Ndxdt→0, asN→ ∞

because∇v_N→0 uniformly inΩ\E₁and|∇H_λ(∇u+ ∇v_N+ ∇w_N)|is bounded inL²(Q);

Nlim→∞

Q\E1

∇H_λ(∇u+ ∇v_N): ∇w_Ndxdt=0

because∇H_λ(∇u+ ∇v_N)→ ∇H_λ(Du)strongly inL²(Ω\E₁)asN→ ∞and∇w_N 0 inL²(Ω);

Nlim→∞

Q\E1

∇H_λ(∇u): ∇(v_N+w_N)dxdt=0

because∇(v_N+w_N) 0 inL²(Q). Hence C⁽¹⁾_N →0

asN→ ∞. To study the convergence ofC_N⁽²⁾, we study two cases:

Case I:λ < λ^∗. Using the fact that∇H_λ(X):X=2Hλ(X), we have C⁽²⁾_N =

Q\E₁

∇H_λ(∇w_N): ∇w_Ndxdt

=

Q\E₁

2Hλ(∇w_N)dxdt

=

Q

2Hλ(∇w_N)dxdt−

E1

2Hλ(∇w_N)dxdt

2ε0

Q

|∇w_N|²dxdt−2C0

E1

1+ |∇w_N|² dxdt

2ε0

Q

|∇w_N|²dxdt−4C0μ.

Hereε0is defined such thatλ=(1−λ0−ε0)/λ0as discussed in Proposition 2.1 andC0is as defined in Proposi- tion 2.1(6).

Consequently, we obtain A_N2ε0

Ω

|∇w_N|²dx+A⁽²⁾_N +C_N⁽¹⁾−4C0μ (5.6)

whereA⁽²⁾_N +C_N⁽¹⁾→0 asN→ ∞.

Now we estimateB_N. B_N=

E1

∇W_λ(∇u+ ∇v_N+ ∇w_N):(∇v_N+ ∇w_N)dxdt

=

E1

∇W_λ(∇u+ ∇v_N+ ∇w_N):(∇u+ ∇v_N+ ∇w_N)dxdt−

E1

∇W_λ(∇u+ ∇v_N+ ∇w_N): ∇udxdt

c₁

E1

∇(u+v_N+w_N)²−1

−C₁

E1

1+∇(u+v_N+w_N)|∇u|dxdt

c₁ 2

E1

|∇v_N|²dxdt−c₁

E1

|∇u| + |∇w_N|2

dxdt−μ

−C₁

Ω

1+ |∇u+ ∇v_N+ ∇w_N|2

dxdt

E1

|∇u|²dxdt

c₁ 2

E₁

|∇v_N|²dxdt−C₄√ μ.

This leads to

Q

∇W_λ ∇u+ ∇(v_N+w_N)

: ∇(v_N+w_N)dxdt

2ε0

Q

|∇w_N|²dxdt+D_N−4C0μ+c1

2

E1

|∇v_N|²dxdt−C₄√

μ, (5.7)

where

Q∇Wλ(∇u+ ∇(vN+wN)): ∇(vN+wN)dxdt→0,DN=C_N⁽¹⁾+A⁽²⁾_N →0. Consequently, for sufficiently largeN, we have

2ε0

Q

|∇w_N|²dxdt+c₁ 2

E1

|∇v_N|²dxdtC₅√

μ. (5.8)

This implies that∇wN→0 strongly inL²(Q, M^N×n)asN → ∞and|∇vN|² is equi-integrable onΩ. Therefore

∇(v_N+w_N)→0 asN→ ∞. Sov_N+w_N converges strongly to 0 inL²(0, T;H¹(Ω)).

Therefore, we can take the limit in (5.1) to get a solution of (0.1).

Case II:We now haveλ=λ^∗. To simplify notation, we still useλin the text.

Since∇H_λ(X):X=2Hλ(X)and∇H_λ(X)is only quasimonotone, parallel to Case I, we get C_N⁽²⁾=

Q\E₁

∇H_λ(∇w_N): ∇w_Ndxdt

=

Q

2Hλ(∇w_N)dxdt−

E₁

2Hλ(∇w_N)dxdt

−2C0

E1

1+ |∇w_N|²

dxdt−4C0μ. (5.9)

This is because the constantε₀in Case I is now 0, whileC₀>0 is given by Proposition 2.1(6). Subsequently, (5.6) be- comes

A_ND_N−4C0μ (5.10)

whereD_N=A⁽²⁾_N +C_N⁽¹⁾→0. It is easy to see that estimate ofB_Nin Case I still holds, we obtain B_N=

E1

∇W_λ ∇u+ ∇(v_N+w_N)

: ∇(v_N+w_N)dxdt

=

E₁

∇Wλ ∇u+ ∇(vN+wN)

: ∇u+ ∇(vN+wN) dxdt−

E₁

∇Wλ ∇u+ ∇(vN+wN)

: ∇udxdt

c₁ 2

E1

|∇v_N|²dxdt−C₄√

μ, (5.11)

hence

Q

∇W_λ ∇u+ ∇(v_N+w_N)

: ∇(v_N+w_N)dxdtD_N−4C0μ+c1

2

E1

|∇v_N|²dxdt−C₄√

μ (5.12)

where

Q∇Wλ(∇u+ ∇(vN+wN)): ∇(vN+wN)dxdt→0,DN→0. Consequently, for sufficiently largej >0, we have

c1

2

E1

|∇v_N|²dxdtC₅√

μ, (5.13)

therefore|∇v_N|²is equi-integrable onΩwhich implies that∇v_N→0 inL²(Q)strongly. Therefore∇(v_N+w_N) 0 weakly inL²(Q)asN→ ∞and is equi-integrable onQ.

From now on, we writeW_λ∗(X)=G(P_A(X))+H (X)withG_PA(X)=G_λ∗(X)andH (X)=H_λ∗(X).

Sinceφ_N=v_N+w_Nas given above, we see that∇v_N→0 inL²,|∇w_N|²is equi-integrable inQ,∇w_N 0 and w_N=0 on∂Q. We also have

Q

∇G P_A(∇u+ ∇φ_N)

:P_A(∇φ_N)dxdt+

Q

∇H (∇u+ ∇φ_N): ∇φ_Ndxdt=t_N→0, (5.14)

asN→ ∞. We have in (5.14) that

Q

∇H (∇u+ ∇φ_N): ∇φ_Ndxdt

=

Q

∇H (∇u+ ∇v_N+ ∇w_N): ∇(v_N+w_N)dxdt

=2

Q

H (∇wN)dxdt+

Q

∇H (∇u+ ∇vN): ∇φNdxdt+

Q

∇H (∇u+ ∇φN): ∇vNdxdt

=2

Q

H (∇w_N)dxdt+r_Nr_N, (5.15)

where r_N=

Q

∇H (∇u+ ∇v_N): ∇φ_Ndxdt+

Q

∇H (∇u+ ∇φ_N): ∇v_Ndxdt→0

and 2

Q

H (∇w_N)dx0 (5.16)

because w_N ∈H₀¹(Q), for a.e.t ∈(0, T ),w_N(t,·)∈H₀¹(Ω). Also H is quasiconvex andH (0)=0 hence (5.16) holds. Similarly

Q

H (∇u+ ∇φ_N)dxdt=

Q

H (∇u)dxdt+

Q

H (∇w_N)dxdt+l_N (5.17)

wherelN→0. Substituting (5.15) into (5.14), we have

Q

∇G P_A(∇u+ ∇φ_N)

:P_A(∇φ_N)dx+2

Q

H (∇w_N)dxdt=t_N−r_N→0 (5.18)

asN→ ∞. SinceGis convex,

Q

∇G P_A(∇u+ ∇φ_N)

:P_A(∇φ_N)dxdt

Q

∇G P_A(∇u)

:P_A(∇φ_N)dxdt→0 (5.19)

asN→ ∞. Therefore (5.16), (5.18) and (5.19) imply

Nlim→∞

Q

∇G P_A(∇u+ ∇φ_N):P_A(∇φ_N)

dxdt=0, lim

N→∞

Q

H (∇w_N)dxdt=0. (5.20) Thus from (5.17) and (5.20),

Nlim→∞

Q

H (∇u+ ∇φ_N)dxdt=

Q

H (∇u)dxdt. (5.21)

SinceGis convex,

Q

G P_A(∇u+ ∇φ_N) dxdt−

Q

G P_A(∇u) dxdt

Q

∇G P_A(∇u)

:P_A(∇φ_N)dxdt→0,

Q

G P_A(∇u) dxdt−

Q

G P_A(∇u_N) dxdt

Q

∇G P_A(∇u+ ∇φ_N)

:P_A(−∇φ_N)dx→0,

hence

Nlim→∞

Q

G P_A(∇u+ ∇φ_N) dxdt=

Q

G P_A(∇u)

dxdt. (5.22)

From (5.21) and (5.22), we obtain, asN→ ∞, that

Q

G P_A(∇u+ ∇φ_N) dxdt+

Q

H (∇u_N)dxdt+

Q

f ·(u+φ_N)dxdt

→

Q

G P_A(∇u) dxdt+

Q

H (∇u)dxdt+

Q

f ·udxdt.

Now we prove thatuis a solution of (0.1).

SinceHis a quadratic form, for any test functionψ, we have

Q

∇H (∇u+ ∇φ_N): ∇ψdxdt→

Q

∇H (∇u): ∇ψdxdt.

We only need to prove that

Q

∇G P_A(∇u+ ∇φ_N)

:P_A(∇ψ )dxdt→

Q

∇G P_A(∇u)

:P_A(∇ψ )dxdt. (5.23)

We also have, from Proposition 2.1 that

Q

∇G P_A(∇u+ ∇φ_N)

− ∇G P_A(∇u)²dxdt

C

Q

∇G P_A(∇u+ ∇φ_N)

− ∇G P_A(∇u)

·P_A(∇φ_N)dxdt→0

asN→ ∞and (5.23) follows. Therefore,uis a solution of (0.1).

Furthermore, asT is arbitrary, we have proved in fact

Theorem 5.1.The problem(0.1)admits at least one weak solution on(0,∞)in the following sense:for anyT >0, anyφ∈L²(0, T;H¹(Ω)),

⎧⎪

⎪⎨

⎪⎪

⎩ T 0

Ω

u_t·φ+ ∇W_λ(∇u): ∇φ+f·φ

dxdt=0, u(x,0)=u0(x).

(5.24)

Moreover, the energy functional

Ω

Wλ(∇u)+f (x)·u

dx (5.25)

is a decreasing function of time.

Proof. The existence proof is obvious. The fact that the energy is decreasing in time can be proved by standard differentiation argument. For the proof under a more general setting, we refer to [8] for details. 2

Remark 5.1.The use of Proposition 2.1 is not essential for establishing the weak continuity of the integrals. In fact, it is possible to establish same kind of results in a much more general setting forW =G+H whereGis convex andH a quadratic form under various boundedness and coercivity conditions. However, without the special form of G(X)=f (A·X), the proof is more complicated and involves the use of Young measures. We will examine more general cases later.

6. Long time behaviour of solutions

6.1. The case whereλ < λ^∗

First, we establish a general results on the existence ofω-limit by using similar ideas explored in [4] and [20]. First, we concentrate our discussion to the case whenλ < λ^∗.

Definition 6.1.A generalized semi-flowGonXis a family of mappingsφ:[0,∞)→X(called solutions) satisfying (H1) (Existence) For eachz∈X, there exists at least oneφ∈Gwithφ(0)=z;

(H2) (Translation) Ifφ∈Gandτ 0, thenφ^τ(t )=φ(t+τ )∈G;

(H3) (Concatenation) Ifφ, ψ∈G,t0 withψ (0)=φ(t)thenθ∈Gwhere θ (s)=

φ(s), s∈ [0, t], ψ (s−t ), t < s.

(H4) (Upper-semi-continuity) Ifφj ∈Gwithφj(0)→z, then there exists a subsequenceφμofφj andφ∈Gwith φ(0)=zsuch that

φ_μ(t )→φ(t) for a.e.t0.

The definition of upper-semi continuity is weaker than that found in [4] but it is enough for most application problems.

Proposition 6.1.The solution set established in Theorem5.1for all possible initial data inH¹(Ω)defines a general- ized semi-flow.