Some results on the well-posedness 121
so that, in the abstract formulation, we can introduce the corresponding functions F ∈ L
2
0, T ; H ,
V
′
hF, vi
V
= Z
f v, v ∈ V 23
H ∈ H
1
0, T ; V
′
,
V
′
hH, vi
V
= Z
Ŵ
hv
|
Ŵ
, v ∈ V ,
24
G ∈ H
1
0, T ; W
′
,
W
′
hG, vi
W
= Z
Ŵ
1
g · v
|
Ŵ1
, v ∈ W,
25
S ∈ H
1
0, T ; H
3
,
W
′
hS, vi
W
= Z
s · v, v ∈ W. 26
Moreover, we have to precise the assumptions on the kernel k in 19 and the function α
. Precisely, we require that 27
k ∈ W
1,1
0, T , and α fulfils
28 α ∈
C
2
R and c
α
:= α
′′ L
∞
R
is sufficiently small. Hence, as at high temperatures shape memory alloys present mostly an elastic behavior,
αθ = 0 for θ ≥ θ
c
and in addition we assume 29
{γ ∈ R : α
′
γ 6= 0} ⊂ [0, θ
c
]. Note that, as a consequence, the functions of the variable θ in the nonlinear terms of
3 turn out continuous and uniformly bounded. Indeed, we observe that 28 and 29 imply
30 |α
′
γ | ≤ θ
c
c
α
, |γ α
′
γ | ≤ θ
2 c
c
α
, ∀γ ∈ R.
R
EMARK
1. As to concerns the constant c
α
and the second of 28, it turns out nec- essary to assume some compatibility conditions satisfied by physically realistic data
between the quantities involved in the model and the heat capacity of the system. In- deed, the coefficient of the temperature time derivative in the energy balance represents
the specific heat of the solid-solid phase transition and it seems physically consistent to require it is positive everywhere. To this aim, later we will specify 28 by letting a
suitable bound for c
α
. Now, we are in the position of stating the existence and uniqueness result referring
to 3-5 and 8-12 combined with 19. T
HEOREM
1. Assume that 21-22, 23-26 and 27-29 hold. Then, there exists a unique quadruplet θ, χ
1
, χ
2
, u, with
θ ∈ H
1
0, T ; H ∩ L
∞
0, T ; V , 31
χ
j
∈ W
1,∞
0, T ; H ∩ L
∞
Q, j = 1, 2
32
u ∈ H
1
0, T ; W, div u ∈ L
∞
Q, 33
122 E. Bonetti
fulfilling θ
0 = 2 ,
34 χ
1
0, χ
2
0 = χ
1
, χ
2
, 35
and satisfying, almost everywhere in 0, T , c
− θ α
′′
θ χ
2
div u∂
t
θ + k
Aθ + k ∗ Aθ = F + H + L∂
t
χ
1
+θ α
′
θ − αθ div u∂
t
χ
2
+ θ α
′
θ χ
2
∂
t
div u in
V
′
36 ζ ∂
t
χ
1
χ
2
+ ∂ I
K
χ
1
, χ
2
∋ −lθ − θ
∗
−αθ div u
in H
2
37
Hu + Bαθ χ
2
= S + G
in W
′
. 38
In particular, the boundedness result in 33 for div u follows from the next lemma,
which can be proved by use of the Lax-Milgram theorem and exploiting standard esti- mates and regularity results on elliptic equations cf. [9].
L
EMMA
1. Let θ , χ
2
belong to L
2
Q such that |χ
2
| ≤ c
K
a.e. in Q. Then, under
assumptions 25, 26, 28, and 29, there exists a unique solution u ∈ L
∞
0, T ; W
