Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift
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568 J. Pr¨uss and M. Wilke
(ii) Consider the term (λ
(ψ
0
) − λ
(u))Δv − (λ
(ψ
0
) − λ
(¯u))Δ¯v.
|(λ
(ψ
0
) − λ
(u))Δv − (λ
(ψ
0
) − λ
(¯u))Δ¯v|
X(T )
≤|(λ
(ψ
0
) − λ
(u))Δ(v − ¯v)|
X(T )
+ |(λ
(u) − λ
(¯u))Δ¯v|
X(T )
≤|ψ
0
−u|
∞,∞
|v − ¯v|
E
2
(T )
+ |u − ¯u|
∞,∞
|¯v|
E
2
(T )
≤ (|ψ
0
− u
∗
|
∞,∞
+ |u
∗
− u|
∞,∞
)|v − ¯v|
E
2
(T )
+ |u − ¯u|
E
1
(T )
(|¯v − v
∗
|
E
2
(T )
+ |v
∗
|
E
2
(T )
)
≤ C(μ(T )+R)|(u, v) − (¯u, ¯v)|
1
,
since λ ∈ C
4−
(R). Next, we consider the term ∇(λ
(ψ
0
) − λ
(u))∇v −∇(λ
(ψ
0
) −
λ
(¯u))∇¯v.Weobtain
|∇(λ
(ψ
0
) − λ
(u))∇v −∇(λ
(ψ
0
) − λ
(¯u))∇¯v|
X(T )
≤|∇(λ
(ψ
0
) − λ
(u))|
∞
|∇(v − ¯v)|
X(T )
+ |∇(λ
(u) − λ
(¯u))|
∞
|∇¯v|
X(T )
.
Since
∇(λ
(ψ
0
) − λ
(u)) = ∇ψ
0
(λ
(ψ
0
) − λ
(u)) + λ
(u)(∇ψ
0
−∇u),
and the same for ∇(λ
(u) −λ
(¯u)), we may argue as above, to conclude
|∇(λ
(ψ
0
) − λ
(u))|
∞,∞
|∇(v − ¯v)|
X(T )
+ |∇(λ
(u) − λ
(¯u))|
∞,∞
|∇¯v|
X(T )
≤ (μ(T )+R)|(u, v) − (¯u, ¯v)|
1
.
Finally, we estimate the remaining part with H¨older’s inequality to the result
|vΔ(λ
(ψ
0
) − λ
(u)) − ¯vΔ(λ
(ψ
0
) −λ
(¯u))|
X(T )
≤|v − ¯v|
∞,∞
|Δ(λ
(ψ
0
) − λ
(u))|
X(T )
+ |¯v|
r
p,r
p
|Δ(λ
(u) − λ
(¯u))|
rp,rp
, (6.2)
where 1/r +1/r
= 1. For the first part, we obtain
|Δ(λ
(ψ
0
) − λ
(u))|
X(T )
≤|Δψ
0
|
p
|λ
(ψ
0
) − λ
(u)|
∞,∞
+ |Δψ
0
− Δu|
p
|λ
(u)|
∞,∞
+ |∇ψ
0
|
2
∞,∞
|λ
(ψ
0
) − λ
(u)|
∞,∞
+ |λ
(u)|
∞,∞
|∇ψ
0
−∇u|
∞,∞
≤ C(|ψ
0
− u|
∞,∞
+ |∇ψ
0
−∇u|
∞,∞
+ |Δψ
0
− Δu|
p,p
)
≤ C(μ(T )+R),
since ψ
0
∈ H
2
p
(Ω)∩C
1
(Ω) and λ ∈ C
4−
(R). For the second term in (6.2) we obtain
|Δ(λ
(u) − λ
(¯u))|
rp,rp
≤|Δu|
rp,rp
|λ
(u) − λ
(¯u)|
∞,∞
+ |λ
(¯u)|
∞,∞
|Δu −Δ¯u|
rp,rp
+ |∇u|
2
∞,∞
|λ
(u) − λ
(¯u)|
∞,∞
+ |λ
(¯u)|
∞,∞
|∇u −∇¯u|
∞,∞
≤ C|u − ¯u|
E
1
(T )
,
since u, ¯u ∈ C(J; C
1
(Ω)) and r>1 can be chosen close enough to 1, due to the
fact that ¯v ∈ C(J; C(Ω)).
Conserved Penrose-Fife Type Models 569
Finally, we observe
|¯v|
r
p,r
p
≤|¯v − v
∗
|
r
p,r
p
+ |v
∗
|
r
p,r
p
≤ μ(T )+R.
(iii) For simplicity we set f(u, v)=a
0
λ
(ψ
0
) − a(v)λ
(u). Then we compute
|f(u, v)∂
t
u − f(¯u, ¯v)∂
t
¯u|
X(T )
≤|∂
t
u(f(u, v) −f(¯u, ¯v))|
X(T )
+ |f(¯u, ¯v)(∂
t
u − ∂
t
¯u)|
X(T )
(6.3)
≤ (|∂
t
u − ∂
t
u
∗
|
X(T )
+ |∂
t
u
∗
|
X(T )
)|f(u, v) − f(¯u, ¯v)|
∞,∞
+ |f(¯u, ¯v)|
∞,∞
|∂
t
u − ∂
t
¯u|
X(T )
≤ C(μ
3
(T )+R)|f (u, v) − f(¯u, ¯v)|
∞,∞
+ |f(¯u, ¯v)|
∞,∞
|∂
t
u − ∂
t
¯u|
X(T )
.
Next we estimate
|f(u, v) − f (¯u, ¯v)|
∞,∞
≤|a(v)(λ
(u) −λ
(¯u))|
∞,∞
+ |λ
(¯u)(a(v) − a(¯v))|
∞,∞
≤|a(v)|
∞,∞
|λ
(u) −λ
(¯u)|
∞,∞
+ |λ
(¯u)|
∞,∞
|a(v) − a(¯v)|
∞,∞
≤ C(|u − ¯u|
∞,∞
+ |v − ¯v|
∞,∞
) ≤ C|(u, v) − (¯u, ¯v)|
1
.
Furthermore, we have
|f(¯u, ¯v)|
∞,∞
≤|a
0
|
∞,∞
|λ
(ψ
0
) −λ
(¯u)|
∞,∞
+ |λ
(¯u)|
∞,∞
|a
0
−a(¯v)|
∞,∞
≤ C(|ψ
0
− ¯u|
∞,∞
+ |ϑ
0
− ¯v|
∞,∞
)
≤ C(|ψ
0
−u
∗
|
∞,∞
+ |u
∗
− ¯u|
∞,∞
+ |ϑ
0
− v
∗
|
∞,∞
+ |v
∗
− ¯v|
∞,∞
)
≤ C(μ(T )+R).
Theestimateof(a
0
−a(v))Δv −(a
0
−a(¯v))Δ¯v in L
p
(J; L
p
(Ω)) can be carried out
in a similar way.
(iv) We compute
|(a(v) −a(¯v)f
2
|
X(T )
≤|a(v) − a(¯v)|
∞,∞
|f
2
|
X(T )
≤|v − ¯v|
∞,∞
|f
2
|
X(T )
≤ μ(T )|v − ¯v|
E
2
(T )
≤ μ(T )|(u, v) − (¯u, ¯v)|
1
,
since f
2
∈ X(T ) is a fixed function, hence |f
2
|
X(T )
→ 0asT → 0.
(v) By trace theory, we obtain
|∂
ν
(Φ
(u) − Φ
(¯u))|
Y
1
(T )
≤ C|Φ
(u) − Φ
(¯u)|
H
1/2
p
(J;L
p
(Ω))
+ |Φ
(u) − Φ
(¯u)|
L
p
(J;H
2
p
(Ω))
.
The second norm has already been estimated in (i), so it remains to estimate
Φ
(u) − Φ
(¯u)inH
1/2
p
(J; L
p
(Ω)). Here we will use (6.1), to obtain
|Φ
(u) − Φ
(¯u)|
H
1/2
p
(L
p
)
≤ μ(T )(|u − ¯u|
H
s
0
p
(L
p
)
+ |u − ¯u|
∞,∞
)
≤ μ(T )C|u − ¯u|
E
1
(T )
≤ μ(T )C|(u, v) −(¯u, ¯v)|
1
,
since s
0
< 1.
570 J. Pr¨uss and M. Wilke
(vi) We may apply (ii) and trace theory, to conclude that it suffices to estimate
(λ
(ψ
0
) −λ
(u))v − (λ
(ψ
0
) − λ
(¯u))¯v
=(λ
(ψ
0
) − λ
(u))(v − ¯v) − (λ
(u) − λ
(¯u))¯v
in H
1/2
p
(J; L
p
(Ω)). This yields
|(λ
(ψ
0
) −λ
(u))(v − ¯v)|
H
1/2
p
(L
p
)
≤|λ
(ψ
0
) −λ
(u)|
H
1/2
p
(L
p
)
|v − ¯v|
∞,∞
+ |λ
(ψ
0
) − λ
(u)|
∞,∞
|v − ¯v|
H
1/2
p
(L
p
)
≤ (|λ
(ψ
0
) − λ
(u
∗
)|
H
1/2
p
(L
p
)
+ |λ
(u
∗
) − λ
(u)|
H
1/2
p
(L
p
)
)|v − ¯v|
E
2
(T )
+(|ψ
0
− u
∗
|
∞,∞
+ |u
∗
−u|
∞,∞
)|v − ¯v|
E
2
(T )
≤
|λ
(ψ
0
) −λ
(u
∗
)|
H
1/2
p
(L
p
)
+ μ(T )R +(μ(T )+R)
|v − ¯v|
E
2
(T )
.
Clearly λ
(ψ
0
) − λ
(u
∗
) ∈
0
H
1/2
p
(J; L
p
(Ω)), since ψ
0
does not depend on t and
since λ ∈ C
4−
(R). Therefore it holds that
|λ
(ψ
0
) −λ
(u
∗
)|
H
1/2
p
(L
p
)
→ 0
as T → 0. The second part (λ
(u) − λ
(¯u))¯v can be treated as follows.
|(λ
(u) − λ
(¯u))¯v|
H
1/2
p
(L
p
)
≤|λ
(u) − λ
(¯u)|
H
1/2
p
(L
p
)
|¯v|
∞,∞
+ |λ
(u) − λ
(¯u)|
∞,∞
|¯v|
H
1/2
p
(L
p
)
≤ C(μ(T )+R + μ(T ))|u − ¯u|
E
1
(T )
,
where we applied again (6.1). This completes the proof of the proposition.
Proof of Proposition 4.1 Let J
δ
max
:= [δ, T
max
] for some small δ>0. Setting
A
2
=Δ
2
N
with domain
D(A
2
)={u ∈ H
4
p
(Ω) : ∂
ν
u = ∂
ν
Δu =0on∂Ω},
the solution ψ(t) of equation (4.1)
1
may be represented by the variation of param-
eters formula
ψ(t)=e
−A
2
t
ψ
0
+
t
0
Ae
−A
2
(t−s)
λ
(ψ(s))ϑ(s) − Φ
(ψ(s))
ds, t ∈ J
max
, (6.4)
where e
−A
2
t
denotes the analytic semigroup, generated by −A
2
= −Δ
2
N
in L
p
(Ω).
By (H1), (H2) and (4.6) it holds that
Φ
(ψ) ∈ L
∞
(J
max
; L
q
0
(Ω)) and λ
(ψ) ∈ L
∞
(J
max
; L
6
(Ω)),
with q
0
=6/(γ + 2). We then apply A
r
,r∈ (0, 1), to (6.4) and make use of
semigroup theory to obtain
ψ ∈ L
∞
(J
δ
max
; H
2r
q
0
(Ω)), (6.5)
Conserved Penrose-Fife Type Models 571
valid for all r ∈ (0, 1), since q
0
< 6. It follows from (6.5) that ψ ∈ L
∞
(J
δ
max
; L
p
1
(Ω))
if 2r − 3/q
0
≥−3/p
1
,and
Φ
(ψ) ∈ L
∞
(J
δ
max
; L
q
1
(Ω)) as well as λ
(ψ) ∈ L
∞
(J
δ
max
; L
p
1
(Ω)),
with q
1
= p
1
/(γ + 2). Hence we have this time
ψ ∈ L
∞
(J
δ
max
; H
2r
q
1
(Ω)),r∈ (0, 1).
Iteratively we obtain a sequence (p
n
)
n∈N
0
such that
2r −
3
q
n
≥−
3
p
n+1
,n∈ N
0
with q
n
= p
n
/(γ +2)andp
0
= 6. Thus the sequence (p
n
)
n∈N
0
may be recursively
estimated by
1
p
n+1
≥
γ +2
p
n
−
2r
3
,
for all n ∈ N
0
and r ∈ (0, 1). From this definition it is not difficult to obtain the
following estimate for 1/p
n+1
.
1
p
n+1
≥
(γ +2)
n+1
p
0
−
2r
3
n
!
k=0
(γ +2)
k
=
(γ +2)
n+1
p
0
−
2r
3
(γ +2)
n+1
−1
γ
1
+1
=(γ +2)
n+1
1
p
0
−
2r
3γ +3
+
2r
3γ +3
,n∈ N
0
. (6.6)
By the assumption (H1) on γ we see that the term in brackets is negative if
r ∈ (0, 1) is sufficiently close to 1 and therefore, after finitely many steps the
entire right side of (6.6) is negative as well, whence we may choose p
n
arbitrarily
large or we may even set p
n
= ∞ for n ≥ N and a certain N ∈ N
0
.Inotherwords
this means that for those r ∈ (0, 1) we have
ψ ∈ L
∞
(J
δ
max
; H
2r
p
(Ω)), (6.7)
for all p ∈ [1, ∞]. It is important, that we can achieve this result in finitely many
steps!
Next we will derive an estimate for ∂
t
ψ. For all forthcoming calculations we
will use the abbreviation ψ = ψ(t)andϑ = ϑ(t). Since we only have estimates on
the interval J
δ
max
, we will use the following solution formula.
ψ(t)=e
−A
2
(t−δ)
ψ
δ
+
t−δ
0
Ae
−A
2
s
λ
(ψ)ϑ − Φ
(ψ)
(t −s) ds, t ∈ J
δ
max
where ψ
δ
:= ψ(δ). Differentiating with respect to t,weobtain
∂
t
ψ(t)=A
t−δ
0
e
−A
2
s
(λ
(ψ)ϑ∂
t
ψ + λ
(ψ)∂
t
ϑ −Φ
(ψ)∂
t
ψ)(t − s) ds
+ F (t, ψ
δ
,ϑ
δ
), (6.8)
572 J. Pr¨uss and M. Wilke
for all t ≥ δ and with
F (t, ψ
δ
,ϑ
δ
):=Ae
−A
2
(t−δ)
(λ
(ψ
δ
)ϑ
δ
− Φ
(ψ
δ
)) − A
2
e
−A
2
(t−δ)
ψ
δ
.
Let us discuss the function F in detail. By the trace theorem we have ψ
δ
∈
B
4−4/p
pp
(Ω) and ϑ
δ
∈ B
2−2/p
pp
(Ω). Since we assume p>(n +2)/2, it holds that
ψ
δ
,ϑ
δ
∈ L
∞
(Ω). Furthermore, the semigroup e
−A
2
t
is analytic. Therefore there
exist some constants C>0andω ∈ R such that
|F (t, ψ
δ
,ϑ
δ
)|
L
p
(Ω)
≤ C
1
(t − δ)
1/2
+
1
t −δ
e
ωt
,
for all t>δ. This in turn implies that
F (·,ψ
δ
,ϑ
δ
) ∈ L
p
(J
δ
max
× Ω)
for all p ∈ (1, ∞), where 0 <δ<δ
<T
max
. We will now use equations (5.1)
1,2
to
rewrite the integrand in (6.8) in the following way.
(λ
(ψ)ϑ − Φ
(ψ))∂
t
ψ + λ
(ψ)∂
t
ϑ
=(λ
(ψ)ϑ − Φ
(ψ))Δμ +
λ
(ψ)
b
(ϑ)
Δϑ −
λ
(ψ)
2
b
(ϑ)
Δμ
=div
.
λ
(ψ)ϑ −
λ
(ψ)
2
b
(ϑ)
− Φ
(ψ)
∇μ
/
+div
.
λ
(ψ)
b
(ϑ)
∇ϑ
/
(6.9)
−∇
λ
(ψ)ϑ −
λ
(ψ)
2
b
(ϑ)
− Φ
(ψ)
·∇μ −∇
λ
(ψ)
b
(ϑ)
·∇ϑ.
Thus we obtain a decomposition of the following form
(λ
(ψ)ϑ − Φ
(ψ))∂
t
ψ + λ
(ψ)∂
t
ϑ
=div(f
μ
∇μ + f
ϑ
∇ϑ)+g
μ
∇μ + g
ϑ
∇ϑ + h
μ
∇ϑ∇μ + h
ϑ
|∇ϑ|
2
,
with
f
μ
:= λ
(ψ)ϑ −
λ
(ψ)
2
b
(ϑ)
− Φ
(ψ),f
ϑ
:=
λ
(ψ)
b
(ϑ)
,
g
μ
:= −
λ
(ψ)ϑ − 2
λ
(ψ)λ
(ψ)
b
(ϑ)
− Φ
(ψ)
∇ψ, g
ϑ
:= −
λ
(ψ)
b
(ϑ)
∇ψ,
h
μ
:= λ
(ψ) −
b
(ϑ)λ
(ψ)
2
b
(ϑ)
2
,h
ϑ
:=
b
(ϑ)λ
(ψ)
b
(ϑ)
2
.
By Assumption (H3) and the first part of the proof it holds that f
j
,g
j
,h
j
∈
L
∞
(J
δ
max
× Ω) for each j ∈{μ, ϑ} and this in turn yields that
div(f
μ
∇μ + f
ϑ
∇ϑ) ∈ L
2
(J
δ
max
; H
1
2
(Ω)
∗
),
g
μ
·∇μ + g
ϑ
·∇ϑ ∈ L
2
(J
δ
max
×Ω),
h
μ
∇ϑ ·∇μ + h
ϑ
|∇ϑ|
2
∈ L
1
(J
δ
max
×Ω),
Conserved Penrose-Fife Type Models 573
wherewealsomadeuseof(4.6).Setting
T
1
= Ae
−A
2
t
∗ div(f
μ
∇μ + f
ϑ
∇ϑ),T
2
= Ae
−A
2
t
∗ (g
μ
·∇μ + g
ϑ
·∇ϑ)
and
T
3
= Ae
−A
2
t
∗ (h
μ
∇ϑ ·∇μ + h
ϑ
|∇ϑ|
2
),
we may rewrite (6.8) as
∂
t
ψ = T
1
+ T
2
+ T
3
+ F (t, ψ
0
,ϑ
0
).
Going back to (6.8) we obtain
T
1
∈ H
1/4
2
(J
δ
max
; L
2
(Ω)) ∩ L
2
(J
δ
max
; H
1
2
(Ω)) → L
2
(J
δ
max
×Ω),
T
2
∈ H
1/2
2
(J
δ
max
; L
2
(Ω)) ∩ L
2
(J
δ
max
; H
2
2
(Ω)) → L
2
(J
δ
max
×Ω), and
F (·,ψ
δ
,ϑ
δ
) ∈ L
2
(J
δ
max
× Ω).
Observe that we do not have full regularity for T
3
since A has no maximal regularity
in L
1
(Ω), but nevertheless we obtain
T
3
∈ H
1/2−
1
(J
δ
max
; L
1
(Ω)) ∩ L
1
(J
δ
max
; H
2−
1
(Ω)).
Here we used the notation H
s−
p
:= H
s−ε
p
and ε>0 is sufficiently small. An
application of the mixed derivative theorem then yields
H
1/2−
1
(J
δ
max
; L
1
(Ω)) ∩ L
1
(J
δ
max
; H
2−
1
(Ω)) → L
p
(J
δ
max
; L
2
(Ω)),
if p ∈ (1, 8/7), whence
∂
t
ψ ∈ L
2
(J
δ
max
× Ω) + L
p
(J
δ
max
; L
2
(Ω))
for some 1 <p<8/7. Now we go back to (6.9) where we replace this time only
∂
t
ϑ by the differential equation (5.1)
2
to obtain
(λ
(ψ)ϑ − Φ
(ψ))∂
t
ψ + λ
(ψ)∂
t
ϑ
=
λ
(ψ)ϑ − Φ
(ψ) −
λ
(ψ)
2
b
(ϑ)
∂
t
ψ
+div
.
λ
(ψ)
b
(ϑ)
∇ϑ
/
−
λ
(ψ)
b
(ϑ)
∇ψ ·∇ϑ +
λ
(ψ)b
(ϑ)
b
(ϑ)
2
|∇ϑ|
2
= f∂
t
ψ +div[g∇ϑ]+h ·∇ϑ + k|∇ϑ|
2
.
Rewrite (6.8) in the following way
∂
t
ψ = S
1
+ S
2
+ S
3
+ S
4
+ F (t, ψ
0
,ϑ
0
), (6.10)
574 J. Pr¨uss and M. Wilke
where the functions S
j
are defined in the same manner as T
j
.Sincef,g,h ∈
L
∞
(J
δ
max
× Ω) it follows again from regularity theory that
S
1
∈ H
1/2
2
(J
δ
max
; L
2
(Ω)) ∩ L
2
(J
δ
max
; H
2
2
(Ω))
+ H
1/2
p
(J
δ
max
; L
2
(Ω)) ∩ L
p
(J
δ
max
; H
2
2
(Ω)),
S
2
∈ H
1/4
2
(J
δ
max
; L
2
(Ω)) ∩ L
2
(J
δ
max
; H
1
2
(Ω)),
S
3
∈ H
1/2
2
(J
δ
max
; L
2
(Ω)) ∩ L
2
(J
δ
max
; H
2
2
(Ω)),
and it can be readily verified that
H
1/2
p
(J
δ
max
; L
2
(Ω)) ∩ L
p
(J
δ
max
; H
2
2
(Ω)) → L
2
(J
δ
max
× Ω),
whenever p ∈ [1, 2]. Now we turn our attention to the term S
4
= Ae
−A
2
t
∗k|∇ϑ|
2
.
First we observe that by the mixed derivative theorem the embedding
Z
q
:= H
1/2−
q
(J
δ
max
; L
1
(Ω)) ∩ L
q
(J
δ
max
; H
2−
1
(Ω)) → L
2
(J
δ
max
×Ω)
is valid, provided that q ∈ (8/5, 2]. Hence it holds that
|S
4
|
2,2
≤ C|S
4
|
Z
q
≤ C|k|∇ϑ|
2
|
q,1
≤ C|∇ϑ|
2
2q,2
,
with some constant C>0. Taking the norm of ∂
t
ψ in L
2
(J
δ
max
× Ω) we obtain
from (6.10)
|∂
t
ψ|
2,2
≤ C
⎛
⎝
3
!
j=1
|S
j
|
2,2
+ |∇ϑ|
2
2q,2
+ |F (·,ψ
δ
,ϑ
δ
)|
2,2
⎞
⎠
.
The Gagliardo-Nirenberg inequality in connection with (4.6) yields the estimate
|∇ϑ|
2
2q,2
≤ c|∇ϑ|
2a
2,2
|∇ϑ|
2(1−a)
∞,2
≤ c|∇ϑ|
2(1−a)
∞,2
,
provided that a =1/q. Multiply (4.1)
2
by ∂
t
ϑ and integrate by parts to the result
Ω
b
(ϑ(t, x))|∂
t
ϑ(t, x)|
2
dx +
1
2
d
dt
|∇ϑ(t)|
2
2
= −
Ω
λ
(ψ(t, x))∂
t
ψ(t, x)∂
t
ϑ(t, x) dx.
Making use of (H3) and Young’s inequality we obtain
C
1
|∂
t
ϑ|
2
2,2
+
1
2
|∇ϑ(t)|
2
2
≤ C
2
(|∂
t
ψ|
2
2,2
+ |∇ϑ
0
|
2
2
), (6.11)
after integrating w.r.t. t. This in turn yields the estimate
|∇ϑ|
2
2q,2
≤ c|∇ϑ|
2(1−a)
∞,2
≤ c(1 + |∂
t
ψ|
2(1−a)
2,2
).
In order to gain something from this inequality we require that 2(1 − a) < 1, i.e.,
q is restricted by 1 <q<2. Finally, if we choose q ∈ (8/5, 2) and use the uniform
boundedness of the L
2
norms of S
j
,j∈{1, 2, 3} we obtain
|∂
t
ψ|
2,2
≤ C(1 + |∂
t
ψ|
2(1−a)
2,2
).
Since by construction 2(1 −a) < 1, it follows that the L
2
-norm of ∂
t
ψ is bounded
on J
δ
max
× Ω. In particular, this yields the statement for ϑ by equation (6.11).
Conserved Penrose-Fife Type Models 575
Now we go back to (6.8) with δ replaced by δ
. By Assumption (H5), by the
bounds ∂
t
ϑ, ∂
t
ψ ∈ L
2
(J
δ
max
; L
2
(Ω)) and by the first part of the proof we obtain
λ
(ψ)ϑ∂
t
ψ + λ
(ψ)∂
t
ϑ −Φ
(ψ)∂
t
ψ ∈ L
2
(J
δ
max
; L
2
(Ω)).
Since the operator A
2
=Δ
2
with domain
D(A
2
)={u ∈ H
4
p
(Ω) : ∂
ν
u = ∂
ν
Δu =0}
has the property of maximal L
p
-regularity (cf. [6, Theorem 2.1]), we obtain from
(6.8)
∂
t
ψ − F (·,ψ
δ
,ϑ
δ
) ∈ H
1/2
2
(J
δ
max
; L
2
(Ω)) ∩ L
2
(J
δ
max
; H
2
2
(Ω)) → L
r
(J
δ
max
; L
r
(Ω)),
and the last embedding is valid for all r ≤ 2(n +4)/n. By the properties of the
function F it follows
∂
t
ψ ∈ L
r
(J
δ
max
; L
r
(Ω)),
for all r ≤ 2(n +4)/n and some 0 <δ
<T
max
. To obtain an estimate for the
whole interval J
max
, we use the fact that we already have a local strong solution,
i.e., ∂
t
ψ ∈ L
p
(0,δ
; L
p
(Ω)), p>(n +2)/2. The proof is complete.
Acknowledgement
The authors are grateful to the anonymous referee for valuable comments on the
literature.
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Jan Pr¨uss and Mathias Wilke
Institut f¨ur Mathematik
Martin-Luther-Universit¨at Halle-Wittenberg
Theodor-Lieser-Str. 5
D-06120 Halle, Germany
e-mail: jan.pruess@mathematik.uni-halle.de
mathias.wilke@mathematik.uni-halle.de (Corresponding author)
Progress in Nonlinear Differential Equations
and Their Applications, Vol. 80, 577–589
c
2011 Springer Basel AG
The Asymptotic Profile of Solutions of
a Class of Singular Parabolic Equations
F. Ragnedda, S. Vernier Piro and V. Vespri
Abstract. We consider a class of singular parabolic problems with Dirichlet
boundary conditions. We use the Rayleigh quotient and recent Harnack esti-
mates to derive estimates from above and from below for the solution. More-
over we study the asymptotic behaviour when the solution is approaching the
extinction time.
Mathematics Subject Classification (2000). Primary 35K55; Secondary 35B40.
Keywords. Singular parabolic equation, asymptotic behaviour.
1. Introduction
Let u(x, t) be the weak solution of the following initial boundary value problem:
u
t
=divA(x, t, ∇u), (x, t) ∈ Ω × (t>0), (1.1)
u(x, t)=0, (x, t) ∈ ∂Ω ×(t>0), (1.2)
u(x, 0) = u
0
(x) ≥ 0,x∈ Ω, (1.3)
where Ω is a bounded domain in R
N
with Lipschitz boundary, u
0
∈ L
1
(Ω) and
Ω
u
0
(x)dx > 0. The functions A := (A
1
,...,A
N
) are regular and are assumed to
satisfy the following structure conditions:
A(x, t, ∇u) ·∇u ≥ c
0
|∇u|
p
, (1.4)
|A(x, t, ∇u)|≤c
1
|∇u|
p−1
, (1.5)
with
2N
N+1
<p<2andc
0
,c
1
given positive constants.
A function u ∈ C
loc
(R
+
; L
2
loc
(Ω)) ∩ L
p
loc
(R
+
; W
1,p
loc
(Ω)) is a weak solution of
(1.1)–(1.3), if for any compact subset K of Ω and for every subinterval [t
1
,t
2
] ∈ R
+
,
K
uφdx|
t
2
t
1
+
t
2
t
1
K
(−uφ
t
+ A(x, t, ∇u) ·∇φ) dx dt =0, (1.6)