Stability of the steady state for the Falk model system of shape memory alloys

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2007; 30:2233–2245Published online 3 May 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/mma.889MOS subject classification: 35Q 72; 74N 30; 80A 22

Stability of the steady state for the Falk model systemof shape memory alloys

Takashi Suzuki1,‡ and Shuji Yoshikawa2,∗,†

1Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science,Osaka University, Toyonaka 560-8531, Japan

2Department of Business Administration, Ube National College of Technology, Ube 755-8555, Japan

Communicated by H.-D. Alber

SUMMARY

In this article a stability result for the Falk model system is proven. The Falk model system describes themartensitic phase transitions in shape memory alloys. In our setting, the steady state is a nonlocal ellipticproblem. We show the dynamical stability for the linearized stable critical point of the correspondingfunctional. Copyright q 2007 John Wiley & Sons, Ltd.

KEY WORDS: shape memory; elasticity; stability; phase transition

1. INTRODUCTION

In this article, we treat the initial boundary value problem of the following dispersive-parabolicsystem:

�utt + �uxxxx = (F ′1(ux )� + F ′

2(ux ))x (1)

CV �t − k�xx = �F ′1(ux )utx in (0,∞) × (0, l) (2)

u(0, ·) = u0, ut (0, ·) = u1, �(0, ·) = �0 (3)

ux (t, 0) = ux (t, l) = uxxx (t, 0)= uxxx (t, l) = �x (t, 0) = �x (t, l) = 0 (4)

where u ∈ R and � ∈ R denote the displacement and the temperature of shape memory alloys,respectively. This thermoelastic system is called the Falk model system describing the martensitic

∗Correspondence to: Shuji Yoshikawa, Department of Business Administration, Ube National College of Technology,Ube 755-8555, Japan.

†E-mail: [email protected]‡E-mail: [email protected]

Copyright q 2007 John Wiley & Sons, Ltd. Received 28 July 2006

2234 T. SUZUKI AND S. YOSHIKAWA

phase transitions occurring on a rod made by shape memory alloys of length l. Positive physicalconstants �, �, CV and k are the mass density, capillarity, specific caloric heat and heat conductivity,respectively. In this article we assume that the nonlinear term F1 ∈C4 and F2 ∈C4 satisfy that

F ′1(0)= 0 and F2�−C (5)

This nonlinearity includes the Falk nonlinearity

F1(ux ) = �1u2x and F2(ux ) = �3u

6x − �2u

4x − �1�cu

2x (6)

for positive physical constants �1, �2, �3 and the critical temperature �c. The nonlinear term (6)accounts quite well for the experimental observed behaviour (see [1]). For the physical backgroundof this system, we refer to the monograph by Brokate and Sprekels [2].

System (1)–(4) satisfies momentum conservation law

d

dtM(ut ) = 0 (7)

and energy conservation law

d

dtE(u, ut , �) = 0 (8)

where the momentum and the energy are defined as follows:

M(ut ) =∫ 1

0ut dx and E(u, ut , �) = �

2‖ut‖2L2 + �

2‖uxx‖2L2 +

∫ l

0(F2(ux ) + CV �) dx (9)

Throughout this article we put

M(u1) = a and E(u0, u1, �0) = b (10)

In this article, we prove the dynamical stability of certain steady state (u, �) of the Falk modelsystem. We consider the stationary problem satisfying ut = �t = 0. The stationary solution satisfiesthe following equation: ⎧⎨

⎩�uxxxx = (�F ′

1(ux ) + F ′2(ux ))x on (0, l)

ux (0) = ux (l) = uxxx (0)= uxxx (l) = 0(11)

for a positive constant �. In this case, from the above conservation laws (7) and (8) we also have

CV l� + �

2‖uxx‖2L2 +

∫ l

0F2(ux ) dx = b and a = 0 (12)

Eliminating the undetermined constant �, we derive the following nonlocal problem:⎧⎪⎪⎨⎪⎪⎩

�uxxxx ={

1

CV l

(b − �

2‖uxx‖2L2 −

∫ l

0F2(ux ) dx

)F ′1(ux ) + F ′

2(ux )

}x

in (0, l)

ux (0) = ux (l) = uxxx (0)= uxxx (l) = 0

(13)

Obviously, the arbitrary constant is the solution of Equation (11).

Copyright q 2007 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2007; 30:2233–2245DOI: 10.1002/mma

STABILITY OF THE STEADY STATE FOR THE FALK MODEL SYSTEM 2235

For simplicity, we write

J1(ux ) =∫ l

0F1(ux ) dx and J2(ux ) = �

2‖uxx‖2L2 +

∫ l

0F2(ux ) dx

If u satisfies Equation (11), it holds that

�J2(ux ) = ��J1(ux )

Then eliminating the undetermined constant � by (12), we can obtain

�J2(ux ) = − 1

CV l(b − J2(ux ))�J1(ux ) (14)

Here, we put

Jb(ux ) = 1

CV lJ1(ux ) − log(b − J2(ux ))

= 1

CV l

∫ l

0F1(ux ) dx − log

(b − �

2‖uxx‖2L2 −

∫ l

0F2(ux ) dx

)(15)

Using this functional, the variational equation (14) can be rewritten as follows:

�Jb(ux ) = 0 (16)

These calculations imply that the nonlocal problem (13) is the Euler–Lagrange equation for thevariational problem (16). Our main result of this article is as follows:

Theorem 1.1Let the initial data (u0, u1, �0) ∈ H4 × H2 × H2 satisfy minx ∈ [0,l] �0(x)>0 and

∫ l0 u1 dx = 0.

Assume that �>0 is a constant and that u ∈ H2(0, l) with ux (0)= ux (l) = 0 is a linearized stablecritical point of Jb defined by (15). Then (u, �) is a dynamically stable in the sense that for anyε>0 there exists �>0 such that

E(u0, u1, �0) = b, ‖(u0 − u)x‖H1(0,l)<�,

∣∣∣∣1l∫ l

0log �0(x) dx − log �

∣∣∣∣ <� (17)

implies

supt�0

‖(u(t) − u)x‖H1(0,l)<ε, supt�0

∣∣∣∣1l∫ l

0log �(t, x) dx − log �

∣∣∣∣ <ε

An essential ingredient of the proof is the following functional:

W (u, �) =∫ l

0(F1(ux ) − CV log �) dx

This functional acts as the Lyapnov function. If the Lyapnov function satisfies W (u, �)�W (u, �)

for the stationary solution (u, �), then the stability result follows immediately from the classicaltheory. In our case, the inequality does not hold. Instead, it holds that the following inequalitycalled semi-unfolding-minimality:

W (u, �)�C1 Jb(ux ) + C2 (18)



for some positive constants C1 and C2. A system is said to have the dual variational structureif a Lyapunov function of the system satisfies the above-mentioned semi-unfolding-minimality toa variational function. In [3], Ito and Suzuki gave such a result for the phase field equation byusing the dual variational principle. For more precise information of this theory, we refer to themonographs [4, 5].

We also show the global existence of the solution. Aiki [6] proved such a result for (u0, u1, �0) ∈H3 × H1 × H1 for more general nonlinearity, that is

F1, F2 ∈C3(R) and F2� − C

when the boundary condition is as follows:

u(t, 0)= u(t, l) = uxx (t, 0) = uxx (t, l) = �x (t, 0) = �x (t, l) = 0 (19)

In [7], the unique global existence of the energy class solution for (1)–(3) and (19) was proved.The energy class solution is the weak solution with (u0, u1, �0) ∈ H2 × L2 × L1 which is naturallydefined by the form of energy E(u, ut , �) given in (9). This is proved by using the Strichartzestimate for the Schrodinger equation and the maximal regularity estimate for the heat equation.The above boundary condition (19) is needed to decompose Equation (1) into the Schrodingerequations and to apply the Strichartz estimate. In [8], this result was proven without using theStrichartz estimate nor the above decomposition, which guarantees the unique global existence ofenergy class solution for the Falk model system with the boundary condition (4). This energy classsolution cannot assure the positivity of temperature � although we obtain its non-negativity.

Since we use the positivity of � to prove the dynamical stability, we introduce the strong solutionand prove its global existence in Section 2. Here, we apply the method of Sprekels and Zheng [9]employed for the other boundary condition, emphasizing the role of F ′

1(0)= 0 to treat (5). Finallyin Section 3, we prove Theorem 1.1.

Remarks

(i) Compactness of the orbit {(u(t), �(t))}t�0 and asymptotical stability of the linearized stablestationary solution of (13) are not expected, because of dispersive property for the firstequation of (1).

(ii) A similar result to the result of this article holds in the case of periodic boundary setting

⎧⎪⎪⎨⎪⎪⎩

�utt + �uxxxx = (F ′1(ux )(� − �c) + F ′

2(ux ))x

CV �t − k�xx = �F ′1(ux )utx in (0,∞) × T

u(0, ·) = u0, ut (0, ·) = u1, �(0, ·) = �0 on T

where T = R/Z.(iii) In the case of viscous materials, Equations (1)–(2) are modified as follows:

�utt + �uxxxx − �uxxt = (F ′1(ux )� + F ′

2(ux ))x (20)

CV �t − k�xx = ��t F1(ux ) + �|uxt |2 (21)



The viscosity term �uxxt in (20) simplifies the analysis because of the smoothing property. Infact, Hoffmann and Zochowski [10] established the unique global existence result by decomposing(20) into a system of two parabolic equations. Multi-dimensional models with viscosity were alsostudied by many authors ([11–13] and reference therein). Sprekels et al. [14] studied the asymptoticbehaviour of the solution for (20)–(21) as t → ∞. However, it seems that the asymptotic behaviourof the solution for (1)–(4) without viscosity has not been determined.

Another interesting property of shape memory alloys is hysteresis. There are a lot of modelsand results from this point of view. For related results to the hysteresis, we refer to e.g. [15, 16].

2. GLOBAL EXISTENCE OF SOLUTION

In this section we construct the strong solution. The solution satisfies that the temperature � isalways positive under the additional assumption that �0 is positive. Throughout this article frombelow we use the following notation:

‖ f ‖L pT X

:= ‖‖ f ‖X‖L p(0,T ) =(∫ T

0‖ f (t)‖p

X dt

)1/p

and the norm for L∞T X is defined in a similar fashion.

Lemma 2.1Assume that �0�0. For any (u0, u1, �0) ∈ H4 × H2 × H2 problem (1)–(4) has a unique globalsolution (u, �) satisfying

u ∈C([0,∞); H4), ut ∈C([0,∞); H2)

� ∈C([0,∞); H2), �t ∈ L2loc(0, ∞; H1)

Moreover, if we assume that �∗ = minx ∈ [0,l] �0(x)>0, then for any T>0 there exists �>0 deter-mined by �∗, ‖utx‖L∞

T L∞ and ‖ux‖L∞T L∞ such that

�(t)��∗ exp(−�t) for t ∈ [0, T ]

ProofThe time local solution can be constructed by using the contraction mapping principle without theassumption �0�0 similar to [7, 8]. The non-negativity of the temperature � is also obtained by thesame argument as in [7]. We give a priori estimates for the higher norm by several steps.

Step 1: From the energy conservation law (8) it follows that

�

2‖ut (t)‖2L2 + �

2‖uxx (t)‖2L2 + CV ‖�(t)‖L1

��

2‖u1‖2L2 + �

2‖u0‖2H2 + CV ‖�0‖L1 +

∫ l

0F2(�xu0) dx + Cl

Since ‖�xu0‖L∞�C‖u0‖H2 and F2 is continuous, we have

‖ut‖L∞T L2 + ‖uxx‖L∞

T L2 + ‖�‖L∞T L1�C(‖(u0, u1, �0)‖H2 × L2 × L1)



Since it holds that for t ∈ (0, T ]‖u(t)‖2L2 �

∣∣∣∣∫ t

0

d

dt

∫ l

0u2 dx ds

∣∣∣∣ + ‖u0‖2L2

� 2T 1/2‖u‖L2t L2‖ut‖L∞

T L2 + ‖u0‖2L2

�C‖u‖2L2t L2 + C + ‖u0‖2L2

by Gronwall’s lemma we obtain

‖ut‖L∞T L2 + ‖u‖L∞

T H2 + ‖�‖L∞T L1�C(T, ‖(u0, u1, �0)‖H2 × L2 × L1)

Step 2: It follows from Gagliardo–Nirenberg’s inequality that

‖�‖L∞�C‖�‖L1 + C‖�‖1/3L1 ‖�x‖2/3L2 (22)

Multiplying (2) by � and integrating over [0, l] yieldCV

2

d

dt‖�‖2L2 + k‖�x‖2L2 =

∫ l

0utx�

2F ′1(ux ) dx (23)

With the help of the estimate in Step 1 and (22), we have∫ l

0utx�

2F ′1(ux ) dx � [ut�2F ′

1(ux )]l0 + C‖ut��x‖L1 + C‖ut�2uxx‖L1

�C‖ut‖L2‖�‖L∞‖�x‖L2 + C‖ut‖L2‖uxx‖L2‖�‖2L∞

�C(1 + ‖�x‖L2 + ‖�x‖2/3+1L2 + ‖�x‖4/3L2 )

�C + k

2‖�x‖2L2

where we used the assumption F ′1(0)= 0. Then integrating (23) over [0, T ], we obtain

‖�‖L∞T L2 + ‖�x‖L2

T L2�C(T )

Step 3: By Sobolev’s embedding, it holds that for t ∈ (0, T ]‖ f ‖L2

t L∞�C‖(‖ f ‖L2 + ‖ fx‖L2)‖L2t�CT 1/2‖ f ‖L∞

T L2 + C‖ fx‖L2T L

2 (24)

Then, putting f = �, we have ‖�‖L2t L∞�C(T ).

Multiplying (2) by �t − �xx and integrating over [0, t] × [0, l] for t ∈ [0, T ] yieldCV ‖�t‖2L2

t L2 + k‖�xx‖2L2t L2 + CV + k

2‖�x (t)‖2L2

= CV + k

2‖�0‖2H1 +

∫ t

0

∫ l

0utx�t�F

′1(ux ) dx ds −

∫ t

0

∫ l

0utx�xx�F

′1(ux ) dx ds

�C‖�0‖2H1 + C‖utx‖L∞t L2‖�t‖L2

t L2‖�‖L2t L∞ + C‖utx‖L∞

t L2‖�xx‖L2t L2‖�‖L2

t L∞

�C‖�0‖2H1 + C‖utx‖2L∞t L2 + CV

2‖�t‖2L2

t L2 + k

2‖�xx‖2L2

t L2



Consequently, we have

‖�t‖2L2t L2 + ‖�xx‖2L2

t L2 + ‖�x (t)‖2L2�C‖�0‖2H1 + C‖utx‖2L∞t L2 (25)

From (24), the estimates in Steps 1 and 2 and Sobolev’s inequality it holds that

‖(F ′1(ux )� + F ′

2(ux ))xx‖L2t L2 �C(‖uxxx‖L2

t L2‖�‖L∞t L∞ + ‖uxx‖L∞

t L2‖�x‖L2t L∞

+‖uxx‖L∞t L2‖uxx‖L2

t L∞‖�‖L∞t L∞ + ‖�xx‖L2

t L2

+‖uxxx‖L2t L2 + ‖uxx‖L∞

t L2‖uxx‖L2t L∞)

�C(‖uxxx‖L2 + ‖�xx‖L2 + 1)

Multiplying (1) by −utxx and integrating over [0, t] × [0, l] with the help of (25) yields

�

2‖utx (t)‖2L2 + �

2‖uxxx (t)‖2L2

��

2‖u1‖2H1 + �

2‖u0‖2H3 +

∣∣∣∣∫ t

0

∫ l

0utxx (�F

′1(ux ) + F ′

2(ux ))x dx ds

∣∣∣∣

�C(‖u0‖H3, ‖u1‖H1) + C∫ t

0‖utx‖L2‖(F ′

1(ux )� + F ′2(ux ))xx‖L2 ds

�C(‖u0‖H3, ‖u1‖H1) + C∫ t

0‖utx‖L2(‖uxxx‖L2 + ‖�xx‖L2 + 1) ds

�C(‖u0‖H3, ‖u1‖H1) + C‖utx‖L2t ′ L

2‖uxxx‖L2t ′ L

2 + C‖utx‖L2t ′ L

2‖�xx‖L2t ′ L

2

�C(‖u0‖H3, ‖u1‖H1, ‖�0‖H1) + C‖utx‖2L2t ′ L

2 + C‖uxxx‖2L2t ′ L

2 + �

4‖utx‖2L∞

t ′ L2

for 0�t<t ′�T . Taking the supremum of the left-hand side in [0, t ′], we have

‖utx‖2L∞t ′ L

2+‖uxxx‖2L∞t ′ L

2�C(‖u0‖H3, ‖u1‖H1, ‖�0‖H1)+C∫ t ′

0(‖utx‖2L∞

s L2 + ‖uxxx‖2L∞s L2) ds

The application of Gronwall’s lemma implies that

‖utx‖L∞T L2 + ‖uxxx‖L∞

T L2 + ‖�t‖L2T L

2 + ‖�xx‖L2T L

2 + ‖�x‖L∞T L2

�C(T, ‖u0‖H3, ‖u1‖H1, ‖�0‖H1)



Step 4: It follows from the estimates in Steps 1–3, Sobolev’s inequality and (24) that

‖�‖L∞T L∞ �C(T )

‖uxx‖L∞T L∞ �C(T )

‖�x‖L2T L

∞ �C(T )

(26)

Then we have

‖(�F ′1(ux )utx )x‖L2

T L2 �C‖�x‖L2

T L∞‖utx‖L∞

T L2 + C‖�‖L∞T L∞‖utxx‖L2

T L2

+CT 1/2‖�‖L∞T L∞‖uxx‖L∞

T L∞‖utx‖L∞T L2

�C(T )(1 + ‖utxx‖L2T L

2)

Multiplying (2) by −�t xx and integrating over [0, t] × [0, l], sufficiently smooth solution (u, �)

satisfies that

k

2‖�xx (t)‖2L2 + CV

2‖�t x‖2L2

t L2

�k

2‖�0‖2H2 +

∣∣∣∣∫ t

0

∫ l

0�t x (�F

′1(ux )utx )x dx ds

∣∣∣∣�k

2‖�0‖2H2 + ‖�t x‖L2

t L2‖(�F ′1(ux )utx )x‖L2

t L2

�k

2‖�0‖2H2 + C(T )‖�t x‖L2

t L2(‖utxx‖L2t L2 + 1)

�C(T, ‖u0‖H3, ‖u1‖H1, ‖�0‖H2) + C‖utxx‖2L2t L2 + CV

4‖�t x‖2L2

t L2 (27)

By using (26), we have

‖(�F ′1(ux ) + F ′

2(ux ))xt‖L2t L2

�C(‖�xt‖L2t L2 + ‖�x‖L2

t L∞‖uxt‖L∞t L2 + ‖�t‖L2

t L2‖uxx‖L∞t L∞

+‖�‖L∞t L∞‖uxxt‖L2

t L2 + T 1/2‖uxx‖L∞t L∞‖uxt‖L∞

t L2 + ‖uxxt‖L2t L2)

�C(‖�xt‖L2t L2 + ‖uxxt‖L2

t L2 + 1) (28)

and

‖(�F ′1(ux ) + F ′

2(ux ))x (t)‖L2 � ‖(�F ′1(ux ) + F ′

2(ux ))x‖L∞T L2

�C(‖�x‖L∞T L2 + ‖�‖L∞

T L∞‖uxx‖L∞T L2 + ‖uxx‖L∞

T L2)

�C(T ) (29)



Then, multiplying (1) by utxxxx and integrating over [0, t] × [0, l] for t ∈ [0, T ], for sufficientlysmooth solution (u, �) we obtain

�

2‖utxx (t)‖2L2 + �

2‖uxxxx (t)‖2L2

=�

2‖�2xu1‖2L2 + �

2‖�4xu0‖2L2 +

∫ t

0

∫ l

0�t (uxxxx (�F ′

1(ux ) + F ′2(ux ))x ) dx ds

−∫ t

0

∫ l

0uxxxx (�F

′1(ux ) + F ′

2(ux ))xt dx ds

�C(‖u0‖H4, ‖u1‖H2) + [‖uxxxx (�F ′1(ux ) + F ′

2(ux ))x‖L1(s)]t0+C‖uxxxx‖L2

t L2‖(�F ′1(ux ) + F ′

2(ux ))xt‖L2t L2

�C(T, ‖u0‖H4, ‖u1‖H2, ‖�0‖H2) + C‖(�F ′1(ux ) + F ′

2(ux ))x (t)‖2L2

+�

4‖uxxxx (t)‖2L2 + C‖uxxxx‖L2

t L2(‖uxxt‖L2t L2 + ‖�xt‖L2

t L2)

�C(T, ‖u0‖H4, ‖u1‖H2, ‖�0‖H2) + C(‖utxx‖2L2t L2 + ‖uxxxx‖2L2

t L2) + �

4‖uxxxx (t)‖2L2

with the help of (27)–(29). By Gronwall’s lemma, we have

‖ut‖L∞T H2 + ‖u‖L∞

T H4 + ‖�‖L∞T H2 + ‖�t‖L2

T H2�C(T, ‖u0‖H4, ‖u1‖H2, ‖�0‖H2)

which is the desired estimate.These formal calculations can be justified by the same argument as [7, Lemma 5.1], which is

concerned with the regularized approximation of the solution.Next, we show the positivity of the temperature �. We note that from Sobolev’s inequality

‖utx‖L∞�C‖ut‖H2 and ‖ux‖L∞�C‖u‖H2 hold. Introducing the new function

� = � exp(�t)

for �>0 determined later, then we can rewrite Equation (2) as follows:

⎧⎪⎪⎪⎨⎪⎪⎪⎩

CV �t − k�xx = (� + utx F′1(ux )�)

�(0, ·) = �0

�x (t, 0) = �x (t, l) = 0

(30)

If we choose � such that

� + �∗‖utx‖L∞T L∞‖F ′

1(ux )‖L∞T L∞�0



then multiplying (30) by (� − �∗)− := min{(� − �∗), 0} and integrating over [0, l] yieldsCV

2

d

dt

∫ l

0(� − �∗)2− dx + k

∫ l

0|�x (� − �∗)−|2 dx

=∫ l

0(� + utx F

′1(ux )�∗)(� − �∗)− dx +

∫ l

0utx F

′1(ux )(� − �∗)2− dx

� �

�∗

∫ l

0(� − �∗)2− dx

Consequently, by Gronwall’s lemma we conclude that

∫ l

0(� − �∗)2− dx� exp

(�t

�∗

)∫ l

0(�0 − �∗)2− dx = 0

for t ∈ [0, T ]. Hence it follows that ��∗, which completes the proof. �

3. DYNAMICAL STABILITY

In this section we prove Theorem 1.1.

ProofRecall that

W (u(t), �(t)) :=∫ l

0(F1(ux ) − CV log �) dx

First, we verify that the free energy W (u(t), �(t)) acts as a Lyapnov function and that thesemi-unfolding-minimality (18) holds. Thus, the functional W (u(t), �(t)) is continuously non-decreasing with respect to time. Indeed, for the solution (u, �) obtained in the previous section forsystem (1)–(4) it holds that

d

dtW (u(t), �(t)) =

∫ l

0

(��t

F1(ux ) − CV�t�

)dx

= −k∫ l

0

�xx�

dx

= −k∫ l

0

∣∣∣∣�x�∣∣∣∣2

dx

� 0

thanks to �>0. Therefore, the functional W (u(t), �(t)) acts as a Lyapnov function.



We show the semi-unfolding-minimality between W (u, �) and Jb(ux ). First, from the energyconservation law, we can define the constant

b= �

2‖ut‖2L2 + �

2‖uxx‖2L2 + CV

∫ l

0� dx +

∫ l

0F2(ux ) dx (31)

Then from Jensen’s inequality

1

l

∫ l

0log � dx� log

(1

l

∫ l

0� dx

)

it follows that

W (u, �) �∫ l

0F1(ux ) dx − CV l log

(1

l

∫ l

0� dx

)

�∫ l

0F1(ux ) dx − CV l log

⎛⎜⎝b − �

2‖uxx‖2L2 − ∫ l

0 F2(ux ) dx

CV l

⎞⎟⎠

=CV l Jb(ux ) + CV l log(CV l)

Next, we prove the stability result using this. Recall that u ∈ H2(0, l) is a linearized stablecritical point of Jb(ux ). This means that there is ε0>0 such that any ε1 ∈ (0, ε0/2] admits �0>0such that if ‖(u − u)x‖H1<ε0 and Jb(ux ) − Jb(ux )<�0 then

‖(u − u)x‖H1<ε1 (32)

From the above properties it holds that

Jb(ux (t)) � 1

CV lW (u(t), �(t)) − log(CV l)

� 1

CV lW (u0, �0) − log(CV l) (33)

and for the constant �>0 defined by (12),

W (u, �) =CV l

[1

CV l

∫ l

0F1(ux ) dx − log

(b − �

2‖uxx‖2L2 −

∫ l

0F2(ux ) dx

)]+ CV l log(CV l)

=CV l Jb(ux ) + CV l log(CV l)

i.e.

Jb(ux ) = 1

CV lW (u, �) − log(CV l) (34)



Given ε>0, the choice of sufficiently small � ∈ (0, ε0/2] satisfying (17) yields

1

CV l|W (u0, �0) − W (u, �)| � 1

CV l(‖F ′

1(�xu0)‖L∞ + ‖F ′1(�x u)‖L∞)‖(u0 − u)x‖L1

+1

l

∣∣∣∣∫ l

0log �0 dx − log �

∣∣∣∣<min(�0, ε) (35)

Therefore from (33)–(35) we have

Jb(ux (t)) − Jb(ux )�1

CV l(W (u0, �0) − W (u, �))<�0

If ‖(u(t) − u)x‖H1 = � (�ε0/2<ε0), then we apply (32) for ε1 = �, and hence

‖(u(t) − u)x‖H1<�

This is a contradiction. Thus, we have

‖(u(t) − u)x‖H1 �= �

Here, from u ∈C([0,∞); H2) and ‖(u0 − u)x‖H1<� it follows that

‖(u(t) − u)x‖H1<� (36)

for any t�0.From the semi-unfolding-minimality (33)–(34) and the linearized stability of Jb it follows that

W (u, �)�CV l(Jb(ux ) + log(CV l))�CV l(Jb(ux ) + log(CV l))=W (u, �)

Then by (35) and (36) we have∣∣∣∣1l

∫ l

0log �(t, x) dx − log �

∣∣∣∣ � 1

CV l(W (u, �) − W (u, �)) + 1

CV l

∣∣∣∣∫ l

0(F1(ux ) − F1(ux )) dx

∣∣∣∣� 1

CV l(W (u0, �0) − W (u, �)) + C‖(u − u)x‖H1

� ε + C�

� 2ε

where we retake � which satisfies also �<ε/C . This completes the proof. �

ACKNOWLEDGEMENTS

The second author would like to thank Prof. Koichi Osaki for his valuable comments. The authors alsowishes to express their deep gratitude to the referee for valuable remarks.



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Stability of the steady state for the Falk model system of shape memory alloys