Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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548 J. Pr¨uss and M. Wilke

by the mixed derivative theorem. It follows readily that

∂

1/2

(η

u) ∈

1/4

(J; H

(Ω))

and

(∂

+ A)

−1

(I + A)

1/2

∂

1/2

(η

u) ∈

5/4

(J; L

(Ω)) ∩

1/4

(J; H

(Ω)).

Since the operator (I + A)

1/2

with domain D((I +A)

1/2

)=H

(Ω) commutes with

the operator (∂

+ A)

−1

, this yields

(∂

+ A)

−1

∂

1/2

(η

u) ∈

5/4

(J; H

(Ω)) ∩

1/4

(J; H

(Ω))

for each ﬁxed u ∈

(T ). By the mixed derivative theorem we obtain furthermore

5/4

(J; H

(Ω)) ∩

1/4

(J; H

(Ω)) →

3/4

(J; H

(Ω)).

Therefore

F (u)=−∂

1/2

(∂

+ A)

−1

∂

1/2

(η

u) ∈

1/4

(J; H

(Ω)),

and there exists a constant C>0 being independent of T>0andu ∈

(T )

such that

F (u)|

1/4

(J;H

(Ω))

≤ C|u|

(T )

for each u ∈

(T ). In particular this yields the estimate

F (u)|

(J;H

(Ω))

≤ T

1/2p

F (u)|

(J;H

(Ω))

≤ T

1/2p

F (u)|

1/4

(J;H

(Ω))

≤ T

1/2p

C|u|

(T )

by H¨older’s inequality and C>0 does not depend on the length T of the interval

J.Wehavethusshownthat

|Δ

G(u)|

(J;L

(Ω))

≤ μ

(T )C|u|

(T )

wherewehavesetμ

(T ):=T

1/2p

(1+T

1/2p

). Observe that μ

(T ) → 0

as T → 0

The next step consists of estimating the term ∂

G(u)in

1/4−1/4p

(J; L

(∂Ω)) ∩

(J; W

1−1/p

(∂Ω)). To this end, we recall the trace map

1/2

(J; L

(Ω)) ∩ L

(J; H

(Ω)) →

1/4−1/4p

(J; L

(∂Ω)) ∩ L

(J; W

1−1/p

(∂Ω))

for the Neumann derivative on ∂Ω. Therefore, by the results above, it remains to

estimate

G(u)in

1/2

(J; L

(Ω)). By the complex interpolation method we have

|w|

1/2

(J;L

(Ω))

≤ C|w|

1/2

(J;L

(Ω))

|w|

1/2

(J;L

(Ω))

for each w ∈

(J; L

(Ω)), and C>0 does not depend on T>0. Using H¨older’s

inequality, this yields

|w|

1/2

(J;L

(Ω))

≤ T

1/4p

C|w|

1/2

(J;L

(Ω))

|w|

1/2

(J;L

(Ω))

≤ T

1/4p

C|w|

(J;L

(Ω))

Conserved Penrose-Fife Type Models 549

Finally we obtain the estimate

G(u)|

1/2

(J;L

(Ω))

≤ T

1/2p

|η

∞

(Ω)

C|u|

(T )

which in turn implies

|∂

G(u)|

(J)

≤|

G(u)|

1/2

(J;L

(Ω))

+ |

G(u)|

(J;H

(Ω))

≤ μ

(T )C|u|

(T )

where μ

(T ):=T

1/4p

(1+T

1/4p

)andμ

(T ) → 0

as T → 0

. Deﬁne two operators

L, B :

(T ) →

(T )bymeansof

Lu :=

⎡

⎣

∂

u +Δ

∂

Δu

∂

⎤

⎦

and Bu :=

⎡

⎣

G(u)

∂

G(u)

⎤

⎦

With these deﬁnitions, we may rewrite (2.7) in the abstract form

Lu = Bu + f, f := (

, ˜g

) ∈

(T ).

By [6, Theorem 2.1], the operator L is bijective with bounded inverse L

−1

, hence

u ∈

(T ) is a solution of (2.7) if and only if (I −L

−1

B)u = L

−1

f.Observethat

−1

B is a bounded linear operator from

(T )to

(T )and

−1

Bu|

(T )

≤|L

−1

B(E

(T ),E

(T ))

|Bu|

(T )

≤ (μ

(T )+μ

(T ))C|u|

(T )

Here the constant C>0 as well as the bound of L

−1

are independent of T>0.

This shows that choosing T>0 suﬃciently small, we may apply a Neumann series

argument to conclude that (2.7) has a unique solution u ∈

(T ) on a possibly

small time interval J =[0,T]. Since the linear system (2.7) is invariant with respect

to time shifts, we may set J = J

. 

3. Local well-posedness

In this section we will use the following setting. For T

> 0, to be ﬁxed later, and

agivenT ∈ (0,T

] we deﬁne

(T ):=E

(T ) × E

(T ),

(T ):={(u, v) ∈ E

(T ):(u, v)|

t=0

=0}

and

(T ):=X(T ) × X(T ) ×Y

(T ) × Y

(T ),

as well as

(T ):={(f

) ∈ E

(T ):g

t=0

= g

t=0

= g

t=0

=0},

with canonical norms |·|

and |·|

, respectively. The aim of this section is to ﬁnd

a local solution (ψ, ϑ) ∈ E

(T ) of the quasilinear system

∂

ψ − Δμ = f

,μ= −Δψ +Φ



(ψ) − λ



(ψ)ϑ, t ∈ J, x ∈ Ω,

∂

(b(ϑ)+λ(ψ)) − Δϑ = f

,t∈ J, x ∈ Ω,

∂

μ = g

,∂

ψ = g

,∂

ϑ = g

,t∈ J, x ∈ ∂Ω,

ψ(0) = ψ

,ϑ(0) = ϑ

,t=0,x∈ Ω.

(3.1)

550 J. Pr¨uss and M. Wilke

To this end, we will apply Banach’s ﬁxed point theorem. For this purpose let

p>(n +2)/2, p ≥ 2, f

∈ X(T

), g

∈ Y

(0,T

),j=1, 2, 3, ψ

∈ X

and

∈ X

be given such that the compatibility conditions

∂

Δψ

−∂



(ψ

)+∂

(λ



(ψ

)ϑ

)=−g

t=0

,∂

= g

t=0

and ∂

= g

t=0

are satisﬁed, whenever p>5, p>5/3andp>3, respectively. In the sequel

we will assume that λ, φ ∈ C

4−

(R), b ∈ C

3−

(0, ∞)andb



(s) > 0 for all s>0.

Note that by the Sobolev embedding theorem we have ϑ

∈ C(

Ω) as well as



(ϑ

) ∈ C(

Ω). Since ϑ represents the inverse of the absolute temperature of the

system, it is reasonable to assume ϑ

(x) > 0 for all x ∈

Ω. Therefore, there exists

a constant σ>0 such that ϑ

(x),b



(ϑ

(x)) ≥ σ>0 for all x ∈

Ω. We deﬁne

(x):=1/b



(ϑ

(x)), η

(x)=λ



(ψ

(x)) and η

(x)=a

(x)η

(x). By assumption,

it holds that a

∈ B

2−2/p

(Ω), η

∈ B

4−4/p

(Ω) and η

∈ B

2−2/p

(Ω), cf. [15, Section

4.6 & Section 5.3.4].

Thanks to Theorem 2.1 we may deﬁne a pair of functions (u

∗

) ∈ E

)

as the solution of the problem

∂

∗

+Δ

∗

+Δ(η

∗

)=f

,t∈ [0,T

],x∈ Ω,

∂

∗

− a

Δv

∗

+ η

∂

∗

= a

,t∈ [0,T

],x∈ Ω,

∂

Δu

∗

+ ∂

(η

∗

)=−g

−e

−B

,t∈ [0,T

],x∈ ∂Ω,

∂

∗

= g

,t∈ [0,T

],x∈ ∂Ω,

∂

∗

= g

,t∈ [0,T

],x∈ ∂Ω,

∗

(0) = ψ

∗

(0) = ϑ

,t=0,x∈ Ω,

(3.2)

where B = −Δ

∂Ω

is the Laplace-Beltrami operator on ∂Ωande

−B

is the analytic

semigroup which is generated by −B

.Furthermoreg

=0ifp<5andg

−g

t=0

− (∂

Δψ

+ ∂

(η

)) if p>5.

Deﬁne a linear operator L :

) →

)by

L(u, v)=

⎡

⎢

⎣

∂

u +Δ

u + η

Δv

∂

v − a

Δv + η

∂

Δu + ∂

(η

∂

⎤

⎥

⎦

Then, by Theorem 2.1, the operator L :

) →

) is bounded and bi-

jective, hence an isomorphism with bounded inverse L

−1

. For all (u, v) ∈

(T )

we set

(u, v)=(λ



(ψ

) − λ



(u))v +Φ



(u),

(u, v)=(a



(ψ

) − a(v)λ



(u))∂

u − (a

− a(v))Δv − (a

− a(v))f

Conserved Penrose-Fife Type Models 551

where a(v(t, x)) = 1/b



(v(t, x)) and a

= a(ϑ

). Lastly we deﬁne a nonlinear

mapping G : E

(T ) ×

(T ) →

(T )by

G((u

∗

); (u, v)) =

⎡

⎢

⎣

ΔG

(u + u

∗

,v+ v

∗

)

(u + u

∗

,v+ v

∗

)

∂

(u + u

∗

,v+ v

∗

) − ˜g

⎤

⎥

⎦

where ˜g

=0ifp<5and˜g

= e

−B

∂

(ψ

,ϑ

)ifp>5. Then it is easy to see

that ψ = u + u

∗

∈ E

(T )andϑ = v + v

∗

∈ E

(T ) is a solution of (1.2) if and

only if

L(u, v)=G((u

∗

); (u, v))

or equivalently

(u, v)=L

−1

G((u

∗

); (u, v)).

In order to apply the contraction mapping principle we consider a ball B

×B

⊂

(T ), where R ∈ (0, 1]. Furthermore we deﬁne a mapping T : B

→

(T )byT (u, v)=L

−1

G((u

∗

); (u, v)). We shall prove that T B

⊂ B

and

that T deﬁnes a strict contraction on B

. To this end we deﬁne the shifted ball

∗

)=B

∗

) ×B

∗

) ⊂ E

(T )by

∗

)={(u, v) ∈ E

(T ):(u, v)=(˜u, ˜v)+(u

∗

), (˜u, ˜v) ∈ B

To ensure that the mapping G

is well deﬁned, we choose T

> 0andR>0

suﬃciently small. This yields that all functions v ∈ B

∗

)haveonlyasmall

deviation from the initial value ϑ

.Toseethis,write

|ϑ

(x) − v(t, x)|≤|ϑ

(x) − v

∗

(t, x)| + |v

∗

(t, x) − v(t, x)|≤μ(T )+R,

for all functions v ∈ B

∗

), where μ = μ(T ) is deﬁned by

μ(T )= max

(t,x)∈[0,T ]×Ω

∗

(t, x) − ϑ

(x)|.

Observe that μ(T ) → 0asT → 0, by the continuity of v

∗

and ϑ

. This in turn

implies that v(t, x) ≥ σ/2 > 0andb



(v(t, x)) ≥ σ/2 > 0for(t, x) ∈ [0,T] ×

and all v ∈ B

∗

), with T

> 0, R>0 being suﬃciently small. Moreover, for all

v, ¯v ∈ B

∗

) we obtain the estimates

|a(ϑ

(x)) − a(v(t, x))|≤C|ϑ

(x) − v(t, x)| (3.3)

and

|a(¯v(t, x)) − a(v(t, x))|≤C|¯v(t, x) − v(t, x)|, (3.4)

valid for all (t, x) ∈ [0,T] ×

Ω, with some constant C>0, since b



is locally

Lipschitz continuous.

The next proposition provides all the facts to show the desired properties of

the operator T .

552 J. Pr¨uss and M. Wilke

Proposition 3.1. Let n ∈ N and p>(n +2)/2, p ≥ 2, b ∈ C

2−

(0, ∞), b



(s) > 0

for all s>0, λ, Φ ∈ C

4−

(R) and ϑ

(x) > 0 for all x ∈

Ω. Then there exists a

constant C>0, independent of T , and functions μ

= μ

(T ) with μ

(T ) → 0 as

T → 0, such that for all (u, v), (¯u, ¯v) ∈ B

∗

) the following statements hold.

1. |ΔG

(u, v) − ΔG

(¯u, ¯v)|

X(T )

≤ (μ

(T )+R)|(u, v) − (¯u, ¯v)|

(T )

2. |G

(u, v) − G

(¯u, ¯v)|

X(T )

≤ C(μ

(T )+R)|(u, v) − (¯u, ¯v)|

(T )

3. |∂

(u, v) − ∂

(¯u, ¯v)|

(T )

≤ C(μ

(T )+R)|(u, v) − (¯u, ¯v)|

(T )

The proof is given in the Appendix.

It is now easy to verify the self-mapping property of T .Let(u, v) ∈ B

.By

Proposition 3.1 there exists a function μ = μ(T )withμ(T ) → 0asT → 0such

that

|T (u, v)|

= |L

−1

G((u

∗

), (u, v))|

≤|L

−1

||G((u

∗

), (u, v))|

≤ C(|G((u

∗

), (u, v)) − G((u

∗

), (0, 0))|

+ |G((u

∗

), (0, 0))|

)

≤ C(|ΔG

(u + u

∗

,v+ v

∗

) −ΔG

∗

X(T )

+ |G

(u + u

∗

,v+ v

∗

) −G

∗

X(T )

+ |∂

(u + u

∗

,v+ v

∗

) −∂

∗

(T )

+ |G((u

∗

), (0, 0))|

)

≤ C(μ(T )+R)|(u, v)|

+ |G((u

∗

), (0, 0))|

≤ C(μ(T )+R)R + |G((u

∗

), (0, 0))|

HenceweseethatT B

⊂ B

if T and R are suﬃciently small, since

G((u

∗

), (0, 0)) is a ﬁxed function. Furthermore for all (u, v), (¯u, ¯v) ∈ B

have

|T (u, v) −T(¯u, ¯v)|

= |L

−1

(G((u

∗

), (u, v)) − G((u

∗

), (¯u, ¯v)))|

≤|L

−1

||G((u

∗

), (u, v)) − G((u

∗

), (¯u, ¯v))|

≤ C(|ΔG

(u + u

∗

,v+ v

∗

) −ΔG

(¯u + u

∗

, ¯v + v

∗

X(T )

+ |∂

(u + u

∗

,v+ v

∗

) − ∂

(¯u + u

∗

, ¯v + v

∗

(T )

+ |G

(u + u

∗

,v+ v

∗

) − G

(¯u + u

∗

, ¯v + v

∗

X(T )

)

≤ C(μ(T )+R)|(u, v) − (¯u, ¯v)|

Thus T is a strict contraction on B

,ifT and R are again small enough. Therefore

we may apply the contraction mapping principle to obtain a unique ﬁxed point

(˜u, ˜v) ∈ B

of T . In other words the pair (ψ, ϑ)=(˜u + u

∗

, ˜v + v

∗

) ∈ E

(T )isthe

unique local solution of (1.2). We summarize the preceding calculations in

Theorem 3.2. Let n ∈ N, p>(n +2)/2, p ≥ 2, p =3, 5, b ∈ C

3−

(0, ∞), b



(s) > 0

for all s>0 and let λ, Φ ∈ C

4−

(R). Then there exists an interval J =[0,T] ⊂

[0,T

]=J

and a unique solution (ψ, ϑ) of (1.2) on J,with

ψ ∈ H

(J; L

(Ω)) ∩ L

(J; H

(Ω))

Conserved Penrose-Fife Type Models 553

and

ϑ ∈ H

(J; L

(Ω)) ∩ L

(J; H

(Ω)),ϑ(t, x) > 0 for all (t, x) ∈ J ×

Ω,

provided the data are subject to the following conditions.

1. f

∈ L

×Ω),

2. g

∈ W

1/4−1/4p

; L

(∂Ω)) ∩ L

; W

1−1/p

(∂Ω)),

3. g

∈ W

3/4−1/4p

; L

(∂Ω)) ∩ L

; W

3−1/p

(∂Ω)),

4. g

∈ W

1/2−1/2p

; L

(∂Ω)) ∩ L

; W

1−1/p

(∂Ω)),

5. ψ

∈ B

4−4/p

(Ω), ϑ

∈ B

2−2/p

(Ω),

6. ∂

Δψ

−∂



(ψ

)+∂

(λ



(ψ

)ϑ

)=−g

t=0

, if p>5,

7. ∂

= g

t=0

, ∂

= g

t=0

, if p>3,

8. ϑ

(x) > 0 for all x ∈

Ω.

The solution depends continuously on the given data and if the data are inde-

pendent of t,themap(ψ

,ϑ

) → (ψ, ϑ) deﬁnes a local semiﬂow on the natural

(nonlinear) phase manifold

:= {(ψ

,ϑ

) ∈ B

4−4/p

(Ω) × B

2−2/p

(Ω) : ψ

and ϑ

satisfy 6. − 8.}.

4. Global well-posedness

In this section we will investigate the global existence of the solution to the con-

served Penrose-Fife type system

∂

ψ − Δμ =0,μ= −Δψ +Φ



(ψ) − λ



(ψ)ϑ, t > 0,x∈ Ω,

∂

(b(ϑ)+λ(ψ)) − Δϑ =0,t>0,x∈ Ω,

∂

μ =0,∂

ψ =0,∂

ϑ =0,t>0,x∈ ∂Ω,

ψ(0) = ψ

,ϑ(0) = ϑ

,t=0,x∈ Ω,

(4.1)

with respect to time if the spatial dimension n is less or equal to 3. Note that

the boundary conditions are equivalent to ∂

ϑ = ∂

ψ = ∂

Δψ = 0. A successive

application of Theorem 3.2 yields a maximal interval of existence J

max

=[0,T

max

)

for the solution (ψ, ϑ) ∈ E

(T )×E

(T ) of (4.1), where T ∈ (0,T

max

). In the sequel

we will make use of the following assumptions.

(H1) Φ ∈ C

4−

(R) and there exist some constants c

> 0, γ ≥ 0 such that

Φ(s) ≥−

−c

, |Φ



(s)|≤c

(1 + |s|

for all s ∈ R,whereη<λ

with λ

being the smallest nontrivial eigenvalue of

the negative Laplacian on Ω with Neumann boundary conditions and γ<3

if n =3.

(H2) λ ∈ C

4−

(R)andλ



,λ



∈ L

∞

(R). In particular, there is a constant c>0

such that |λ



(s)|≤c(1 + |s|) for all s ∈ R.

554 J. Pr¨uss and M. Wilke

(H3) b ∈ C

3−

((0, ∞)), b



(s) > 0on(0, ∞) and there is a constant κ>1 such that

≤ ϑ(t, x) ≤ κ

on J

max

× Ω. In particular, there exists σ>1 such that

≤ b



(ϑ(t, x)) ≤ σ,

on J

max

× Ω.

Remark: Condition (H1) is certainly fulﬁlled, if Φ is a polynomial of degree 2m,

m<3.

We prove global well-posedness with respect to time by contradiction. For this

purpose, assume that T

max

< ∞. Multiply ∂

ψ =Δμ by μ and integrate by parts

to the result



|∇ψ|



Φ(ψ) dx



+ |∇μ|

−





(ψ)ϑ∂

ψdx=0. (4.2)

Next we multiply (4.1)

by ϑ and integrate by parts. This yields



ϑb



(ϑ)∂

ϑdx+ |∇ϑ|





(ψ)ϑ∂

ψdx=0. (4.3)

Set β



(s)=sb



(s) and add (4.2) to (4.3) to obtain the equation



|∇ψ|



Φ(ψ) dx +



β(ϑ) dx



+ |∇μ|

+ |∇ϑ|

=0.

(4.4)

Integrating (4.4) with respect to t,weobtain

E(ψ(t),ϑ(t)) +





|∇μ(s)|

+ |∇ϑ(s)|



dt = E(ψ

,ϑ

), (4.5)

for all t ∈ J

max

,where

E(u, v):=

|∇u|



Φ(u) dx +



β(v) dx.

It follows from (H1) and the Poincar´e-Wirtinger inequality that



|∇ψ(t)|

dx +

1 − ε



|∇ψ(t)|

dx +



Φ(ψ(t)) dx

≥



|∇ψ(t)|

dx +

(1 − ε)λ

− η

|ψ(t)|

− c

|Ω|−

2|Ω|







since by equation ∂

ψ =Δμ and the boundary condition ∂

μ = 0, it holds that



ψ(t, x) dx ≡



(x) dx, t ∈ J

max

Hence for a suﬃciently small ε>0 we obtain the a priori estimates

ψ ∈ L

∞

max

; H

(Ω)) and |∇μ|, |∇ϑ|∈L

max

; L

(Ω)), (4.6)

Conserved Penrose-Fife Type Models 555

since β(ϑ(t, x)) is uniformly bounded on J

max

× Ω, by (H3). However, things are

more involved for higher-order estimates. Here we have the following result.

Proposition 4.1. Let n ≤ 3, p>(n +2)/2, p ≥ 2 and let (ψ, ϑ) be the maximal

solution of (4.1) with initial value ψ

∈ B

4−4/p

(Ω) and ϑ

∈ B

2−2/p

(Ω).Suppose

furthermore b ∈ C

3−

(0, ∞), b



(s) > 0 for all s>0, λ, Φ ∈ C

4−

(R) and let (H1)–

(H3) hold.

Then ψ ∈ L

∞

max

× Ω) and ϑ ∈ H

max

; L

(Ω)) ∩ L

∞

max

; H

(Ω)).

Moreover, it holds that ∂

ψ ∈ L

max

× Ω),wherer := min{p, 2(n +4)/n}.

Proof. The proof is given in the Appendix. 

Deﬁne the new function u = b(ϑ). Then u satisﬁes the nonautonomous linear

diﬀerential equation in divergence form

∂

u − div(a(t, x)∇u)=f, (4.7)

subject to the boundary and initial conditions ∂

u =0andu(0) = b(ϑ

)=:u

where a(t, x):=1/b



(ϑ(t, x)) and f := −λ



(ψ)∂

ψ. With (H3), the regularity of ϑ

from Proposition 4.1 carries over to the function u;inparticularu

∈ B

2−2/p

(Ω).

This yields, that u is a weak solution of (4.7) in the sense of Lieberman [12] &

DiBenedetto [7], and u is bounded by (H3).

Furthermore, by (H3)

0 <

≤ a(t, x) ≤ σ<∞,

for all (t, x) ∈ J

max

×Ω. Note that by Proposition 4.1 it holds that f = −λ



(ψ)∂

ψ ∈

max

×Ω), r := min{p, 2(n +4)/n}. Consider the case r =2(n +4)/n.Thenit

can be readily checked that

n +2

2(n +4)

= r

provided n ≤ 5. It follows from Lieberman [12] & DiBenedetto [7] that there

exists a real number α ∈ (0, 1/2) such that u ∈ C

α,2α

(Ω

max

), provided f ∈

max

×Ω) and p>(n +2)/2. Here C

α,2α

(Ω

max

) is deﬁned as

α,2α

(Ω

max

):={v ∈ C(Ω

max

): sup

(t,x),(s,y)∈Ω

max

|v(t, x) − v(s, y)|

|t −s|

+ |x − y|

2α

< ∞}.

and we have set Ω

max

=(0,T

max

) ×Ω. The properties of the function b then yield

that ϑ = b

−1

(u) ∈ C

α,2α

(Ω

max

). In a next step we solve the initial-boundary

value problem

∂

ϑ −a(t, x)Δϑ = g, t ∈ J

max

,x∈ Ω,

∂

ϑ =0,t∈ J

max

,x∈ ∂Ω,

ϑ(0) = ϑ

,t=0,x∈ Ω,

(4.8)

556 J. Pr¨uss and M. Wilke

with g := −a(t, x)λ



(ψ)∂

ψ ∈ L

max

× Ω) and r =2(n +4)/n > (n +2)/2. By

[6, Theorem 2.1] we obtain

ϑ ∈ H

max

; L

(Ω)) ∩ L

max

; H

(Ω)),

of (4.8), since

∈ B

2−2/p

(Ω) → B

2−2/r

(Ω),p≥ r.

At this point we use equation (6.8) from the proof of Proposition 4.1 to conclude

∂

ψ ∈ L

max

×Ω), with s =min{p, q} where q is restricted by

≥

−

n +4

For the case r =2(n +4)/n, this yields

≥

n −4

2(n +4)

i.e., q may be arbitrarily large in case n ≤ 3andwemaysets = p.Nowwesolve

(4.8) again, this time with g ∈ L

max

× Ω), to obtain

ϑ ∈ H

max

; L

(Ω)) ∩ L

max

; H

(Ω))

and therefore ϑ(T

max

) ∈ B

2−2/p

(Ω) is well deﬁned. Next, consider the equation

∂

ψ +Δ

ψ =ΔΦ



(ψ) − Δ(λ



(ψ)ϑ),

subject to the initial and boundary conditions ψ(0) = ψ

and ∂

ψ = ∂

Δψ =0.

By maximal L

-regularity there exists a constant M = M(J

max

) > 0 such that

|ψ|

(T )

≤ M (1 + |ΔΦ



(ψ)|

X(T )

+ |Δ(λ



(ψ)ϑ)|

X(T )

). (4.9)

for each T ∈ J

max

.Sinceϑ ∈ E

max

) we may apply [13, Lemma 4.1] to the result

|ΔΦ



(ψ)|

X(T )

+ |Δ(λ



(ψ)ϑ)|

X(T )

≤ C(1 + |ψ|

(T )

), (4.10)

with some δ ∈ (0, 1) and C>0 being independent of T ∈ J

max

. Combining (4.9)

with (4.10), we obtain the estimate

|ψ|

(T )

≤ C(1 + |ψ|

(T )

which in turn yields that |ψ|

(T )

is bounded as T + T

max

,sinceδ ∈ (0, 1).

Therefore the value ψ(T

max

) ∈ B

4−4/p

(Ω) is well deﬁned and we may continue the

solution (ψ, ϑ)beyondthepointT

max

, contradicting the assumption that J

max

[0,T

max

) is the maximal interval of existence.

We summarize these considerations in

Theorem 4.2. Let n ≤ 3, p>(n+2)/2, p ≥ 2 and p =3, 5. Assume that (H1)–(H3)

hold. Then for each T

> 0 there exists a unique solution

ψ ∈ H

; L

(Ω)) ∩ L

; H

(Ω)) = E

)

and

ϑ ∈ H

; L

(Ω)) ∩ L

; H

(Ω)) = E

Conserved Penrose-Fife Type Models 557

of (1.2), provided the data are subject to the following conditions:

1. ψ

∈ B

4−4/p

(Ω), ϑ

∈ B

2−2/p

(Ω);

2. ∂

Δψ

=0, if p>5, ∂

=0;

3. ∂

=0, if p>3, ϑ

(x) > 0 for all x ∈

Ω.

The solution depends continuously on the given data and the map (ψ

,ϑ

) → (ψ,ϑ)

deﬁnes a semiﬂow on the natural phase manifold

:= {(ψ

,ϑ

) ∈ B

4−4/p

(Ω) × B

2−2/p

(Ω) : ψ

and ϑ

satisfy 2. &3.}.

5. Asymptotic behavior

Let n ≤ 3. In the following we will investigate the asymptotic behavior of global

solutions of the homogeneous system

∂

ψ − Δμ =0,μ= −Δψ +Φ



(ψ) − λ



(ψ)ϑ, t > 0,x∈ Ω,

∂

(b(ϑ)+λ(ψ)) − Δϑ =0,t>0,x∈ Ω,

∂

μ =0,t>0,x∈ ∂Ω,

∂

ψ =0,t>0,x∈ ∂Ω,

∂

ϑ =0,t>0,x∈ ∂Ω,

ψ(0) = ψ

,ϑ(0) = ϑ

,t=0,x∈ Ω,

(5.1)

as t →∞. To this end let (ψ

,ϑ

) ∈M

, p>(n +2)/2, p ≥ 2 and denote by

(ψ(t),ϑ(t)) the unique global solution of (5.1). In the sequel we will make use of

the following assumptions.

(H4) b ∈ C

3−

((0, ∞)), b



(s) > 0on(0, ∞) and there is a constant κ>1 such that

≤ ϑ(t, x) ≤ κ

on J

max

× Ω. In particular, there exists σ>1 such that

≤ b



(ϑ(t, x)) ≤ σ,

on J

max

× Ω.

(H5) The functions Φ, λ and b arerealanalyticonR.

We remark that assumption (H4) is identical to (H3) for a global solution. We

stated it here for the sake of readability.

Note that the boundary conditions (5.1)

3,5

yield



ψ(t, x) dx ≡



(x) dx,

and



(b(ϑ(t, x)) + λ(ψ(t, x))) dx ≡



(b(ϑ

(x)) + λ(ψ

(x))) dx.