THERMODYNAMIC PROPERTIES AND STABILITY

QUARTERLY OF APPLIED MATHEMATICS VOLUME LI, NUMBER 2 JUNE 1993, PAGES 343-362

THERMODYNAMIC PROPERTIES AND STABILITY

FOR THE HEAT FLUX EQUATION WITH LINEAR MEMORY

By C. GIORGI (Dip. di Maternatica. Univ. della Calabria, Arcavacata di Rende (CS). Italy) AND

G. GENTILI (Dip. di Maternatica, Universitil di Bologna. Italy)

Abstract. Within the linearized theory of heat conduction with fading memory, some restrictions on the constitutive equations are found as a direct consequence of thermodynamic principles. Such restrictions allow us to obtain existence, uniqueness, and stability results for the solution to the heat flux equation. Both problems, which respectively occur when the instantaneous conductivity k o is positive or vanishes, are considered. 1. Introduction. This paper deals with the asymptotic behaviour of the solution of the heat flux equation with memory. We restrict our attention to a homogeneous and isotropic rigid heat conductor with linear memory occupying a fixed bounded domain n C]R3 with smooth boundary an. Moreover, if we consider only small variations of the temperature O(x, t) from a reference value 00 and small temper ature gradients 'VO(x, t) we may assume that the internal energy e(x, t) and heat flux q(x, t) are described by the following linearized constitutive equations: 00

e(x, t) q(x, t)

= eo + aou(x, t) + 10 a' (s)u l (x, s) ds,

=

-ko'Vu(x, t)-

10

( 1.l)

00

k'(s)'Vul(x,s)ds,

where u(x, t) = O(x, t) - 00 is the temperature variation field, and u (x, s) = u(x, t -s). As usual in material with memory, a fading memory principle for the relaxation functions a' and k' is required. Following Day's argument [8] we may state the principle in a weak form, which is equivalent to l

a', k'

E L'(]R+).

( 1.2)

Hence, the heat capacity and thermal conductivity are defined respectively by

a(t)

------

= a o + 10

1

a' (s) ds,

k(t)

= k o + 10

1

k' (s) ds,

( 1.3)

Received August 5, 1991. 1991 Mathematics Subject Classification. Primary 80A20, 45K05. @1993 Brown University 343

344

C. GIORGI

AND

G. GENTILI

and the values a co = lim/_co a(t) and k co = lim/_co k(t) represent respectively the equilibrium heat capacity and the equilibrium thermal conductivity. Moreover, we impose the a priori condition ( 1.4) which follows from the physical evidence that, if the temperature of a body is constant up to a certain time t = 0 and increases instantaneously at t = 0, then the internal energy of the body increases too. The ensuing evolution problem with mixed boundary conditions and initial history data is given by i(x, t) + V'. q(x, t) = r(x, t) q(x, t) . n(x) = ¢(x, t)

on on

O(x, t) = 00 O(x, t) = 00 + uo(x, t)

on

nx

IR+,

anq x IR+ , on anu x IR+,

(1. 5)

n x IR- ,

where r, ¢, and U o represent respectively the heat source, the heat flux on the boundary q , and the temperature variation field up to the time t = 0, while n is a bounded domain with boundary an = u n q q = 0) and n(x) is the outward normal on . q Problems such as (1.5) have been studied by several authors. For instance, Nun ziato [7] and Grabmueller [19] proved existence and uniqueness for generalized so lutions; Miller [11] has studied stability and continuous dependence on parameter. This paper differs from the previous ones in its investigation of the connection between the complete set of thermodynamic restrictions on the relaxation functions, a and k, and the solvability and stability of the problem (1.1), (1.5). In essence we prove that the fading-memory principle, stated by (1.2), condition (1.4), and the Second Law of Thermodynamics, in the form of the Clausius property, are sufficient conditions to obtain existence, uniqueness, and stability for the solution. Finally, we stress that a Neumann boundary condition involving the heat flux, instead of the temperature gradient flux, is considered. Such conditions, in fact, are not completely equivalent in materials that satisfy (1.1)2' The complete set of restrictions imposed on the heat capacity and thermal con ducivity by the Second Law of Thermodynamics is derived in Sec. 2. It is interesting to compare such results (especially Corollaries 2.1 and 2.3) with the set of assump tions required by Miller to prove existence, uniqueness, and stability in [II]. In fact all these sufficient conditions turn out to be thermodynamic consequences. In order to study the evolution problem (1.5), we discuss separately the cases k o > 0 and ko = O. In effect, these cases are quite different because the positiveness of k o leads to a parabolic problem, whereas when ko vanishes the problem might become hyperbolic. The parabolic problem, when ko > 0, is considered in Sec. 3. Existence, unique ness, and asymptotic behaviour of the solution are proved, and a stability result is given. Moreover, under suitable regularity hypotheses on the data, the solution turns out to be asymptotically stable. Similar results are obtained when ko = 0 in Sec. 4,

an

an

anu an (an u an

HEAT FLUX EQUATION WITH LINEAR MEMORY

345

whereas in Sec. 5 we discuss the meaning of some functional spaces and restrictions on the initial-history data introduced in the previous section. As a concluding remark we emphasize that weaker conditions than those of Miller are needed here, and nevertheless more accurate stability results are obtained when k o is positive or vanishes. In particular, our assumptions allow k' (t) to have an integrable singularity at t = 0 whereas Miller implicitly assumes that k' (0) is finite. More importantly, in the case k o = 0 Miller's assumption (S') (see [11, p. 325]) fails to be valid, as proved by Corollary 2.3, so that no stability property follows from his argument. Thus, in that case, our stability results are new. Next we give some notation for Laplace and Fourier transforms and convolution. For later convenience we denote by ] the (formal) Fourier transform of any scalar or vector-valued function f defined on Q x JR, namely

=

](x, w)

I:

f(x, s)exp(-iws)ds,

(x, w)

E Q

x JR.

Similarly, letting the subscripts s (c) denote the half-range Fourier sine (cosine) trans form, for any function g defined on Q x JR+ we have (formally)

1

00

gc(x, w) =

1

00

g(x, s)coswsds,

gs(x, w) =

If g(x,') belongs to V, where V transform, defined by

1

= L'(JR+)

or V

g(x, s)sinwsds.

= L 2 (JR+) ,

(1.6)

then its Laplace

00

g(x, p) =

g(x, s) exp(-ps) ds,

(x, p)

E Q X

c+

H

(c+ = {p E c; Re(p) 2: O}), is analytic on C = {p E C; Re(p) > O}. Un less otherwise stated, we identify a function g with its causal extension to JR, i.e., g(x, t) = 0 when t < O. In this way the Laplace transform g(x, p) , when p = iw, can be considered as the Fourier transform, i.e., g(x, iw) = g(x, w). Moreover, we have g(x, w) =gc(x, w) - ig/x, w). (1.7) Finally, given any pair (f, g) of scalar- or vector-valued functions defined on Q x JR, let "*" and "*F" denote respectively the Laplace and Fourier time-convolution products, namely

f * g(t)

=

l'

f(s)g(t - s) ds

and

2. Thermodynamic restrictions. Let the l l (u , g') , where gl = '\luI and u (s) = u(t thermal histories be denoted by 9'. Then rigid heat conductors in the Clausius form

f *F g(t)

lo+d [

1 0

- s) ds.

thermal history AI be defined as AI = s), s E JR+ , and let the set of all periodic the Second Law of Thermodynamics for is stated as follows (see [! 5]).

SECOND LAW. For each periodic thermal history AI and for each to E JR, the following inequality holds:

j

= l~ f(s)g(t

=

(u

l ,

l)

E 9' with period d

e(x, t) + q(x, t) og(x, t)] dt < O. eo + u(x, t) (eo + u(x, t))2

346

C. GIORGI

AND

G. GENTILI

Under linear approximation (i.e., small variations U(X, t) with respect to a given reference temperature 00 and small g(x, t)), the quantity O-I(x, t) = (Oo+u(x, t))-I can be substituted by its linear Taylor polynomial, thus yielding the approximate

inequality -2 jlo+d

00

[e(x, t)(Oo - u(x, t)) +q(x, t) ·g(x, t)]dt S; 0,

(2.1 )

10

where e and q are given by the linearized constitutive equations (1.1). Moreover, because of periodicity, (2.1) is equivalent to

0~2 hd[e(x, t)u(x, t) +q(x, t)· g(x, t)]dt S; O.

(2.2)

The complete statement of the Second Law, however, specifies that the inequality refers to an irreversible process, whereas the equality occurs in reversible processes only (see for instance [18]). Actually, for a heat conductor satisfying (1.1) a process is reversible if and only if it holds the temperature constant in time (O(t) = 00) and l uniform in space (g(t) = 0) , so that the ensuing thermal history AI = (u , gl) E .9' vanishes identically. Thereby, the inequality (2.2) is verified as an equality only if u = 0 and g = O. Summarizing and putting (1.1) into (2.2), the Second Law for hereditary rigid heat conductors under linear approximation can be reformulated as follows: SECOND LAW. Relative to rigid heat conductors satisfying (1.1)-(1.2), for each d l periodic thermal history AI = (u , gl) E.9' not vanishing identicallY, the following strict inequality holds:

d h

[h

OO

l

a' (s)u (x, s)u(x, t) ds - kog(x, t) . g(x, t)

-loo

k'(S)gl(X, s) ·g(x, t)dS] dt

< O. (2.3)

In order to exploit the consequences of this statement we consider a thermal history AI generated by the function A(x, t)

= AI (x) cos wt +

A 2 (x) sin wt

(2.4)

with A;(x) = (u;(x), g;(x)) depending only on the spatial variable. Such a history is periodic with period d = 2n/w. THEOREM 2.1. For a rigid heat conductor satisfying (1.1 )-( 1.2), the Second Law of Thermodynamics holds if and only if the relaxation functions k' and a' satisfy:

k o + k~(w) > 0 'v'w

E

IR,

(2.5)

w&:(w) > 0

i= O.

(2.6)

'v'w

Proof. Substituting A(x, t) as expressed by (2.4) into (2.3) and taking into ac

count that I

A (x, s) == A(x, t- s) A(x, t)

' = A(x, t) cosws - w- IA(x, t) sin ws,

= wA 2 (x) cos ws -

wA) (x) sin ws ,

347


it follows that

rd[o:~(w)U(x, t)u(x, t) - ~o::(w)u(x, t)2 10 w "

- (ko + kc(w))!g(x, t)1

=-

2

1, ,

+ -ks(w)g(x, t)· g(x, t)]dt w

~d [~O::(W)U(x, t)2 + (ko + k~(w))lg(x, t)1

2 ]

(2.7)

dt < O.

Now if Al = (0, g) and A 2 = (u, 0) from (2.7) we have

1r

d

2

"

2

[wO:s(w)u cos wt

0

d"

= l[wo: s(w)u

2

"2 2 + (ko + kc(w))lgl cos wt] dt

" 2 + (ko + kc(w))lgl ] > 0 Vw E lR

and by the arbitrariness of U and g relations (2.5)-(2.6) follow. Paralleling the argument explained in [15], it is easily shown that (2.5)-(2.6) are also sufficient conditions for the validity of the Second Law for rigid heat conductors in the form (2.3). 0 As a consequence of Theorem 2.1 we have the following corollaries. COROLLARY 2.1. For a rigid heat conductor satisfying (I.l )-( 1.2), if 0: 0 > 0 the Second Law of Thermodynamics yields

Re{ko+k'(p)} >0 Re{p[o:o

(2.8)

VpEC+,

+ 0:'(P)]} > 0 Vp

E

(2.9)

C+\{O}.

Proof. We first observe that the left-hand sides of (2.8) and (2.9) are real parts of analytic functions on C++. Then, by using the Hilbert integral representation for Laplace transforms, that is, -

f(P)

= -21= n

0

2

w 2!s(w)dw, A

p +w

f(P) -

= -21"? n

0

2

P

p +w

2!c(w)dw, A

we have Re{pa' (p)}

-,

=~

Re{ko + k (P)} = .

r=

wo::(w)(P~ + p~ + ( 2)rp(P1 ,P2' w) dw,

n 10 21= " 2 2 2 (ko + kc(W))(PI + P2 + W )({J(PI' P2' w)dw, n 0 2

2

2 2

(2.10) (2.11)

2 2 -I

where P = PI + IP2 and rp(P I , P2' w) dw = PI ((PI - P2 + W ) + 4PIP2) (PI> 0). Because of (2.5)-(2.6) both of the integrals (2.10) and (2.11) are positive. Then, since 0: 0 > 0 and Re P > 0, (2.8) and (2.9) follow. 0 COROLLARY

2.2. Under previous assumptions, the Second Law leads to the following

properties: (a) 0: 0 < 0:(/) < 20:= - 0: 0

(2.12 )

and moreover, if 0:'(0) exists, it must be nonnegative; (b) ko 2 0 and k= > O.

(2.13)

348

C. GIORGI

AND

G. GENTILI

Proof. From the inversion formula 0

1(t) = ~ rOO o:(w) sin wt dw 7r

and (2.6) it foJlows that o(t) -

10

21

°0 =

-

7r

00

w -1 (I - coswt)o 1 (w)dw > 0

0

2

00 -

roo w

°0 = n10

Combining (2.14) and (2. I 5) we obtain o(t) -

°

00

21

=-

7r

0

(2.14)

s

and, moreover,

°

a.e. on R+

_I

1

21

00

(2.15 )

0s(w)dw> O.

1 -w -1 cos ws 0s(w) dw < -

7r

00

0

1 w -I 0s(w) dw =

°

00

-

°0

1

so that (2.12) follows. FinaJly, if 0 (0) exists by (2.14) it must be nonnegative. Furthermore, by taking the limit as w ---> 00 in (2.5) we obtain (2.13)1' whereas taking w = 0 there we obtain ko + ie~(O)

= k oo > o.

0

COROLLARY 2.3. Under the previous assumptions, if k o = 0 and k ' (0) does exist, then, as a thermodynamic consequence, we have

Ik' (t)1 < k ' (0)

k ' (0) > 0 and

a.e. on R+.

(2.16)

Moreover, if k" (t) has Laplace transform on C++ , we have k~+ie"(p)::f:O V'PEC+\{O}.

(2.17)

Proof. When ko = 0 and k'(O) exists, in the sense that limt->o+ k'(t) = k'(O) E R, then the inversion formula

k'(t)=~ (ooie'(w)~oswtdw 7r

10

and inequality (2.5) lead to

a.e.onR+

c

21

k 1 (0) = -

7r

0

(2.18)

00 "I kc(w)dw > O.

(2.19)

Inequalities (2.16) foJlow from (2.18) and (2.19). On the other hand, well-known properties of the Laplace transform lead to l P = PI + ipz E C+ . (2.20) k ' (0) + ie"(p) = pie (P), l oo First observing that ie (0) = k' (s) ds = k oo < 00 > from (2.20) it easily foJlows that k'(O) + ie"(p) = 0 when p = O. (2.21)

Io

Moreover, if p yield

= ipz

then piel(p) =Pz(ie:(Pz)+iie~(pz)), so (2.5), (2.20), and k o = 0

k'(O)+ie"(p)::f:O

when PI =Oandpz::f:O.

(2.22)

349


Applying once again the Hilbert integral representation for Laplace transforms we obtain -, 4 2 A' Im{pk (p)} = 1 p 2 w kc(w)rp(P1 'P 2 ' w) dw.

n roo 0

Thus, by (2.5) with k o = 0 and (2.20), it follows that k'(O) Finally, if P

-,

= PI > 0

pk (p)

1

+ k"(P) =I-

when PI> 0 and P2 =I- O.

0

then (2.18) and (2.5), with k o = 0, lead to the inequality

21

00

= PI

o

(2.23)

,

exp( -p1t)k (t) dt

=-

n

00

2

2

PI (PI

2 _I A' + w) kc(w) dw

> 0

0

so that from (2.20) it follows that k'(O)

+ k"(P)

=I- 0 when PI> 0 and P2 = O.

(2.24)

The collection (2.21)-(2.24) proves (2.17). 0 Note that the conditions required by Miller (see [II]) to ensure the existence, uniqueness, and stability of solutions to the heat flux equation when k o i- 0 are exactly (2.8)-(2.13). Thus Theorem 2.1 and the ensuing corollaries allow us to give a physical interpretation of these conditions; that is, the constitutive equations of the model need to be compatible with thermodynamics. In the case ko = O. however, in order to obtain stability results Miller needs a stronger condition for k(t) which fails to be valid. In essence (see assumption (S'), p. 325 in [II]), he requires that k~ + k"(P) =I- 0 'tip E C+ . Unfortunately, it is apparent from (2.21) that this cannot occur if P = 0 and k' ELI (jR+) ; so no stability property follows from his argument when k o vanishes. 3. Existence, uniqueness, and stability results when k o is positive. Substituting the constitutive equations (1.1) into problem (I. 5) one obtains the following initial history boundary value problem:

a

a t + Joroo' cr. (s) atU (x, s)ds - k ot!1u(x, t) - Iooo k' (s)t!1u t (x, s) ds = r(x, t) oo t) + I o k' (s)Vu' (x, s) ds] . n(x) = -¢(x,

cr.oatu(x, t) [ko Vu(x, u(x, t) u(x, t)

t)

=0

= uo(x,

on n x jR+ , on anq x jR+ , on an u x jR+ , on n x jR-,

t)

where an is smooth and constituted by the union of anq and an u ' We exclude the case an = an q , when no temperature is assigned at the boundary. Then, by setting uo(x)

= uo(x, 0),

.

Qo(x, t)

1

00

Uo(x,t)= , h(x, t)

= r(x,

,

= JOO k' (s)Vu~(x, s) ds,

a,

cr.(S)atUo(x,s)ds,

t) - Uo(x, t)

+ v· Qo(x,

lfI(X, l) = -¢(x, l) - Qo(x, l) . n(x) ,

t),

C. GIORGI

350

AND

G. GENTILI

the previous problem is replaced by the following equivalent system:

a

, a

O:oatu(x, t)

+ 0: * atu(x,

[koV'u(x, t)

+ k' * V'u(x,

t)

- kotl.u(x, t) - k' * tl.u(x, t) = h(x, t) t)]· n(x) = 'II (x , t)

on n x jR+ , on anq x jR+ ,

u(x,t)=O u(x, 0) = uo(x)

(3.1)

onanuxjR+, in n.

For later convenience we shall now give some notation for functional spaces, n ) norms, and scalar products. Set n ' = n uI an q and define J.Q = H' o(H) (see [13] . 2

Let Ilull) denote the norm in Vo , lIull the norm in L (n) , and III u L 2 (jR+ x n). Moreover, from now on we use the following notation:

(a, b) =

In

a(x)b(x) dx,

(a, b)q

= (

10 n

III

the norm in

a(x)b(x) da, q

In order to achieve a stability result relative to problem (3.1) we impose the fol lowing assumptions: (HI) Uo E

J.Q; 2

2

2

(H2) r E L (jR+, L (n)) , ¢ E L (jR+, L 2 (an q ));

(H3) uo(x,t) on nxjR- is such that UOE L 2 (]R+, L 2 (n)) , and V'. Qo E L 2 (]R+, H-1(n'));

QOE L

2

2

(jR+, L (n))

2

(H4) uo(x, t) on an x jR- is such that Qo' n E L (jR+, H- 1/ 2 (an q )).

REMARK

3.1. (H2)-(H4) ensure that

hE L 2 (jR+, H-1(n'))

and

2

'II E L (]R+ , H-

12 / (an )). q

First consider the Laplace transformed"problem of (3.1) which is defined for each

p E C+ as p[o:o + a'(p)]u(x, p) - [ko + k'(p)]tl.u(x, p) [ko + k'(p)]V'u(x, p). n(x) = ljJ(x, p) u(x,p)=O

= F(x, p)

on n, on an q

(3.2)

,

on an u '

where F(x, p) = h(x, p) DEFINITION OF WEAK SOLUTION.

+ uo(x)[o:o + a'(p)].

A function

uE

J.Q

(3.3)

is called a weak solution of

(3.2) if I { {p[o:o

1n

+ a'(p)]ui/ + [ko + k'(p)]V'u. V'i/} dx = { Fi/ dx + {

holds for every iJ E I

v'

1n

J.Q.

denotes the complex conjugate of

v.

10 n

ljJi/ dx q


351

3.1. If a and k verify inequalities (2.8)-(2.9) and uo ' ¢, and r satisfy (Hl)-(H4), then problem (3.2) has one and only one weak solution u(', p) E Vo for every p E C+ . Proof. First, note that h(.,p) and F(·,p) are well defined in H-1(n), and 1 2 r[J(., p) is well defined in H- / (an q ) for every p E C+. Then, by virtue of well known theorems on elliptic problems, (3.2) has one and only one weak solution if and only if the operator L(p) defined by

LEMMA

= -p[a o + &'(p)] + [k o + k'(p)],1

L(p)

is coercive for every p E C+ , i.e., if and only if the associated bilinear form a(u, v; p) =

1

{p[ao + &' (p)]uv· + [ko + k' (p)]V'u . V'v·} dx

satisfies la(u, u; p)1 ~ C(p)lIulI;

Vu E VO ' Vp E C+, with C(p) > O.

(3.4)

If p f- 0 then y(p)

,

~,

= Re(p[ao + & (p)]) Re([ko + k

,

(p)])[Re(p[a o + & (pm

-'-I

+ Re([ko + k (pm]

turns out to be positive because of (2.8) and (2.9); thus, 2 2 2 + la(u, u; p)1 ~ I Re(a(u, u; p))1 ~ y(p)(llull + IIV'uli ) = y(P)llulll Vp E C \{O}. On the other hand, when p = 0 we recall that (see [13, Lemma 1.46]) lIV'ull ~ C(n)lluli l

Vu E Vo '

(3.5)

and from (2.13)2 it follows that la(u, u; 0)1 = k oo llV'u ,2 dx

~ Collull;

. with Co

= kC

2 (ny> 0

and the proof of (3.4) is complete. 0 We can represent the solution U of (3.2) as follows: u(x,p)= {r(x,x';p)F(x',p)dx'+ {r(x,x';p)r[J(x',p)da', Jn Jan

where the Green function

r

(3.6)

for almost all x E nand p E C+ solves the problem

p[a o + &'(p)]r(x, x'; p) - [k o + k'(p)],1'r(x, x'; p) V"r(x, x'; p). n(x') = 0

r(x,x';p)=O

= l5(x -

x')

x' E n, x' E an q , X'Ean u '

(3.7)

LEMMA 3.2. Under the hypotheses of Lemma 3.1, for almost all x E nand p E C+ problem (3.7) has one and only one solution r(x, x' ; p) with the following properties: (a) r(x, x' ; .) is continuous on C+; (b) lim 1pl _ oo fnpaor(x, x'; p)u(x')dx' = u(x) VJ1 E C;;'(r);

(c) V'xr(x, x' ; .) is continuous on C+;

352

C. GIORGI

AND

G. GENTILI

(d) limlpl-->DO In pao \7 xr(x, x' ;p )u(x') dx' = \7u(x) \:Ill E c~ (r) Proof. (a) follows from the continuity of a(u, v,·) with respect to p (see [la, Lemma 44.1]). Now, multiplying (3.7), by tp(x') E C~(n) and integrating over n we have

L

pr(x, x'; p)[(a o + (j' (p))tp(x') ,- p -, (k o + i(' (p))~tp(x')] dx'

so that the limit as p

tp(x)=

=

-+ 00

= tp(x)

yields

lim f pr(x,x';p)[(ao+(j'(p))tp(x')-p-I(ko+i('(p))~tp(x')]dx' Ipl-->DO In lim f paor(x, x' ; p)tp(x') dx' Ipl-->DO In

for every tp E C~(n) and property (b) follows. Finally, by applying the operator \7 x to (3.7)1 we can easily obtain

p[a o + (j'(p)]\7xr(x, x'; p) - [k o + i('(p)]~'\7xr(x, x'; p)

= o'(x

(3.7)',

- x'),

so item (c) follows directly from (a), while the proof of (d) parallels that of (b), replacing (3.7), by (3.7)', and remarking that 0' E [C~(n)]'. 0 THEOREM 3.1. Under the hypotheses of Lemma 3.1 problem (3.1) has one and only 2 one solution u E L (]R+, Vo)' Proof. Since Lemma 3.1 states that the transformed problem (3.2) has one and only one solution U E Vo for every p E C+ , our aim is to prove that, by virtue of Lemma 3.2, this implies the existence and uniqueness of the solution to (3.1). In order to do this, we have to investigate the properties of u(x, p) when p = iw. First observe that lim F(x, p) Ipl-->DO

= aouo(x) a.e.

in

n

and

lim Iji(x, p) Ipl-->DO

=a

a.e. in

an.

Then, from property (b) of Lemma 3.2 it follows that

[f

lim pu(x,p)= lim pF(x',p)r(x,x';p)dx' Ipl-DO Ipl-oo In

+ fan pr(x, x'; p)lji(x' , p) dO']

=

lim f pr(x, x' ; p)aouo(x') dx' Ipl-DOln

= uo(x)

a.e. in

n

and property (d) of Lemma 3.2 leads to lim p\7x u(x,p) Ipl-DO

=

[f

lim pF(y,p)\7 x r(x,y;p)dy Ipi-DO In

+ fan p \7xr(x, y;P)Iji(Y'P)dOy ]

=

lim f p\7 J'(x, y; p )aouo(x) dy Ipl-->DO In

= \7uo(x)

a.e. in n.

353


Setting p = iw and observing that that u(x, iw) = u(x, w) and u E L~oc(lR, Vo), 2 by (3.6) and previous lemmas, it follows that u E L (lR, Vo)' Finally, by setting

t < 0,

0, v(x,t)= { u(x, t),

i:

Parseval's Theorem yields 2;rr-1

2 2 (Iu(x, w)1 + l\7u(x, w)1 )dw

=

~

t

i: (JO

=J

o

0,

2 2 (Iv(x, t)1 + l\7v(x, t)1 )dt 2

2

(Iu(x, t)1 + l\7u(x, t)1 )dt.

Hence it follows that

[00 (1Iu(x, t)1/2 + l/\7u(x, t)11 2) dt

h

1 = -2 /+00 (I/u(x, W)1/2 + II\7u(x, W)1/2) dt ;rr

-00

< 00

and the theorem is so proved. 0 We now complete the treatment of the parabolic case by showing how the solution is controlled by data. In particular, we are going to get a theorem that may be considered both a control theorem, when sources rand cP vary but Uo = "It < and a stability theorem in the usual sense, when rand cP vanish. Let (Hl)-(H4) be valid, so that we choose only initial histories belonging to

° °,

%

= {u o: 0

x lR-

-+

lR;

U

o satisfies (HI), (H3), and (H4)} .

Consider problem (3.1) again and set

f(x, t) = r(x, t) - Uo(x, t) . Thus (3.1)1 can be replaced by

a

a

I

O:oatu(x, t)+o: * atu(x,

I

t)-ko~u(x, t)-k *~u(x,

t) = f(x, t)+\7·Qo(x, t). (3.8)

2

3.2. (H2)-(H3) yield f E L (lR+, L 2 (0)). In order to obtain a priori estimates we take the product of (3.8) by u and integrate 2 over 0 x [0, T], yielding REMARK

T

Info

[o:o~~u+ (O:/*~~)U-(ko~U)U-(k'*~U)U]dxdt

= InfoTUU + \7. Qou]dxdt, 1

2

_

2

[T /

I

au)

[T

20: o(lI u (T)11 -Iluol/ ) + J \0: * at' u dt + J o

o

2 I [kol/\7ull + (k * \7u, \7u)]dt

= foT[(f, u) - (Qo' \7u) - (cP, U)q]dt.

(3.9)

2 ln the sequel, for ease of writing, the dependence on x and I within the integral expressions is iJnderstood

but not written except for the dependence on w in the Fourier transformed functions.

C. GIORGI

354

AND

G. GENTILI

If we extend the solution U(X, t) as follows, t E [0, T], t (j. [0, T],

U(t),

v (t)- { T 0,

a at T() -

-v t {

f,u(t),

t E]O, T[,

()(t)uo' -()(t - T)u(T),

t

= 0, t = T,

t (j. [0, T], (3.10) by Parseval's theorem the third term on the left-hand side of (3.9) becomes 0,

r [kollV'ull T

J

2'

o

+ (k

/+00 -00 [kollV'vTII 2 + (k' *F V'V T , V'V T )] dt 1 /+00" = 27r -00 (ko + k (w))IIV'v T (w)1I dw =

* V'u, V'u)] dt

2

=

n1 Jr+o

oo

(3.11 )

2

"

(k o + kc(w))IIV'vT(w)11 dw.

Note that the relation (3.11) defines a norm for V'u(x, t) because of (2.5), so

Jr [kollV'ull 2 + (k' * V'u, V'u)]dt T

2

III V'u IlIk,T =

o and with the aid of (3.5) we can define 2

Lk(O, T; Vo)

= {u:

(0, T)

Now, by assumption limjwl->ooko + k~(w)

III V'u IIl k2 , T

1 = 7r

~ -Y 7r

=

VO ,

1+00 1+00 0

0

2

III V'u

Illk,T < oo}.

(3.12)

k o > 0 and by (2.5) we obtain

= wER inf(ko + k' (w)) c

y

which implies

-+

>0

"

,

(k o + -k (w))IIV'v T(w)1I 2 dw c

1 T

IIV'v T(w)1I 2 dw

=y

0

lIV'ull 2 dt.

Since (ko + k~(w)) is bounded and (3.5) holds we can conclude: REMARK 3.3. If k > 0, the norm of L~(O, T; Vo) is equivalent to the usual o 2 norm of L (0, T; Vo)' Applying (3.10) again, the second term of (3.9) becomes

iT \a' * ~~ , u) dt =

I:\

T) dt

a' *F a;r, V

-iT (a'uo ' u) dt + IT (a' (t - T)u(T), u) dt. But a' (t - T) = 0 a.e. in (0, T) , so Parseval's theorem yields

IT \a' * ~~ , u) dt =

*i

oo

T 2

w&:(w)llv (w)11 dw -

iT a' (uo ' u) dt.

(3.13)

355


Finally, substituting (3.11) and (3.13) into (3.9) we obtain 1 2 -2 O:ollu(T)11 +

~

2 roo w6::(w)lI v T (w)1I dw + ~ roo (k o + k~(w»IIVvT(w)1I2 dw nJo nJo

= ~0:01luo1l2 + faT[((I + O:'Uo ), u) -

(Qo' Vu) - (¢, u)q] dt. (3.14)

On the basis of (3.14) we can state the following theorem. THEOREM 3.2. If the thermal conductivity k and the heat capacity 0: are such that (a) ko > 0 and 0: 0 > 0, (b) (1.2), (2.5), and (2.6) are satisfied; then, under assumptions (H I )-(H4), the problem (3.1) has one and only one solution 2 2 u E Loo(]R+, L (Q» nL (]R+ , Vo) and, moreover, the corresponding norms of u are bounded by the norms of the data in their respective spaces. Proof. We have just observed that

-1

1

00

(k o + k"Ic (w»II Vv• T (w)1I 2dw = III Vu Ill k2 T > CJ

n o '

I

0

T

lIull,2 dt

and moreover, by virtue of (2.6), we have

so (3.14) leads to I 2 {T 2 1 _ 2

20:01Iu(T)11 + C, J lIull, dt < 20:01luoII

o

+ faT [((I + O:' Uo ),

u) -

~Qo' Vu) -

(¢, u) q] d t .

Because of (H I )-(H4) for any positive constant e we can set

e 2 12 I(Qo,Vu)I:s 2 " vU Ii +2e IlQolI ,

e 2 12 1(I,u)I:S 2 1I u ll +2ell/ll, e 2 I 2 I(¢, u) q I :s 211ullq + 2e 11¢ll q ,

and by Theorem 3.1 there exists a constant C2 > 0 such that (see [13, Lemma 1.48]) 2

2

Ilullq :s C211Vuli .

Thus, if we choose e such that v that

= (C,

- e(l

+ C2 )/2) > 0, from

(3.15) (3.14) it follows

~0IlU(T)112+vfaTllull7dt:s ~0IluoIl2+ 2IefaT(lIQoIl2+11/112+1I¢11~)dt + faT (o:' UO '

u) dt

356

C. GIORGI

AND

G. GENTILI

so that T T u Ilu(T)11 2:::; lI uoll 2+ _I r (IIQ oIl 2+ 11/11 2+ II¢II~) dt + 211 oll la'llluli dt, aoe J o a o J o

rT 2 a 2 I rT 2 2 2 rT la'lliulidt. VJ lI ull l dt :::; 2°lluoll +2eJo (IIQ oll +11/11 + 1I¢ll q )dt + lI u oli

r

Jo o Applying Gronwall's generalized Lemma (see [17]) the following inequalities hold for any T E IR+ :

1 aoe

lIu(T)II:::; ( lIuoll2 + -c(Qo' /, ¢)

)

1/2

+ Iluoll

r lIulI; dt :::; ;0 lIuoll2 + fc(Qo' /, ¢) + Iluoll J T

v where

o

e

I 'I r ~ dt, Jo a o Ilu(t)11 r la'i dt, J (3.17) T

(3.16 )

T

sup tE[O,Tj

o

c(Qo' /, ¢) = Jo (IIQ oI1 2+ 11/11 2+ II¢II~) dt < 00 by assumption. Uo E Vo and la' (t)1 ELI (IR+) , taking the limit in (3.16) and (3.17) as OO

Finally,

T ---> 00 since the conclusion follows. 0 Observe that, if the data are suitably regular, (3.1) has a solution belonging to HI with respect to time; therefore Theorem 3.2 yields asymptotic stability.

4. Existence, uniqueness, and stability results when ko vanishes. Assuming ko = 0 , problem (3.1) becomes

8 , 8, + a 08t u(x,t)+a *8tu(x,t)-k *~u(x,t)=h(x,t) onnxlR, k' * V'u(x, t)· n(x) u(x, t) = 0 u(x, 0) = uo(x)

= 'II (x , t)

on an q x IR+, on anux IR+, in n.

(4.1)

REMARK 4.1. In the case ko = 0 a compatibility condition between the data arises, i.e., limf-;O- q(x, t) . n(x) = ¢(x, 0+) , and yields the following restriction on the initial histories:

loo

k' (s)V'uo(x, -s) ds . n(x) = -¢(x, 0)

on

an q .

Nevertheless, by using the method of the previous section, with slight modifications we can still obtain existence and uniqueness results. In order to avoid repetitions, we give only the statement of the theorem. THEOREM 4.1. If a and k, when k o = 0, satisfy inequalities (2.8)-(2.9), and data u O ' ¢, and r satisfy (H I)-(H4) then (4.1) has one and only one solution

u E L 2 (1R+, Vo).

On the other hand, investigating how the norm of the solution is controlled by the norms of the data, in the case k o = 0 we are no longer able to obtain results close to the previous ones. In essence, when k o > 0 Theorem 3.2 states that the 2 2 norm of u in L (IR+ , Vo) is controlled by some L -type norms of the data, whereas if k o

=0

we can hardly control the norm of u relative to L~(IR+, Vo) and, what

357


is more important, by stronger conditions on the data. To clarify this assertion we make some remarks. Let L~/k(IR+, L 2(Q)) denote the Banach space defined by the norm

III flll~/k=

i:

I 2n

(k o +

k~(w))-1Ilj(w)112 dw.

2

2

2

2

REMARK 4.2. The spaces L;/k(JR+, L (Q)), Li(JR+, L (Q)), and L (1R+, L (Q)) are equivalent when k o > O. On the contrary, when ko vanishes they are not equivalent anymore, but their norms are ordered as follows:

III . 1I1~ :::; III . 111

2 :::;

III . III~/k .

In the following section we shall scrutinize L~/k in detail. Some conditions on the data involving norms of the L~/k-type have to be assumed in order to achieve a stability result in the case k o = O. In particular we impose (H5) uo(x, t) on Q x IR- is such that

°

0,

Q o E L~/k(lR+ , L 2(Q)), 2

(H6) ¢ E L;/k(lR+ , L \aQq)) ' r E L;/k(lR+ , L (Q)) so that

f

= r ~

° 0

E

L~/k(JR+ , L (Q)). Choose

= {uo:

2

Q x IR-

o varying in the set

o satisfies (HI), (H4), and (H5)}. THEOREM 4.2. If the thermal conductivity k and the heat capacity 0: are such that (a ) ko = 0, 0: 0 > 0 , -+

IR;

U

U

(b) (1.2), (2.5), and (2.6) are satisfied, then, under the assumptions (H 1)-(H6), problem (4.1) has one and only one solution 2 2 u E L 00 (IR+ , L (Q)) n L (IR+ , Vo); nevertheless only the norms of u in the spaces OO

2

L (IR+, L (Q)), and Li(IR+, Vo) are bounded by. suitable norms of the data in their respective spaces. Proof. By working out (4.1) 1 we are led to the following equation: I 2 "2O: o(lI u (T)11 -

= faT[(f,

r

T

_

2

Iluoll ) + i

o

[/

\a

I

au u ) + (k * Vu, * 7ii'

u} - (Qo' vu) - (¢, u}q]dt

and after some manipulations

I

Vu)

]

dt

)58

C. GIORGI

AND

G. GENTILI

Note that (H5)-(H6), (3.15), and Poincare's inequality yield the following:)

10rT (Qo' \lu)dt I = 2nI

I

1

/+00 ~ • I 1 2 e 2 -00 (Qo(w), \lvT(w))dt :::; 2e IIIQollil/k +2: III \lulll k ,T'

10rT (¢, u)qdt I = 2n1 1/+ -00

00

I

10rT (f, u) dt I = 2n1 1/+ -00

00

I

I I

2

'. 1 1I1¢IIII/k,q 2 e (¢(w) , vT(w))qdt :::; 2e +C22:III\lulllk,T'

2

•• 1 III flil 2/ +C)2:e III \lu Ill , T . (f(w) , vT(w)) dt :::; 2e k l k (4.4)

Thus we obtain

r

T a 2 2 a 2 1 , , 2°llu(T)1I +,uIII\lulllk,T:::; 2°lluoll +2e C (Qo,f'¢)+10 Ilauolillulidt

I - e(l + C2 + C))/2 > 0 and C' (Qo, f, ¢) = (III Qo III~/k + III ¢ III~/k ,q + III flll~/k)' Applying again Gronwall's generalized lemma, for any T E lR+ it follows that

with ,u

=

1 ) 2 Ilu(T)II:::; ( lI uoll + - C ' (Qo' f, ¢) aoe

2

ao

2

1

1/2

+ lIuoll

,

10r

T

,u III \lu IIl k T:::; -2 Iluoll + -2 C (Qo' f, ¢) + sup Ilu(t)11

,

e

IE(O, T)

I 'I !:- dt, ao

iT Iia ,uoll dt. °

Thus taking the limit as T - t 00 and taking account of Poincare's inequality, the theorem is proved. 0 REMARK 4.3. If a' (0) > 0, we can control the norm of the solution in L 2(lR+, L 2(0)). In fact, by substituting Poincare's inequality into (4.2), IIvT(w)1I :::; 2c(w)lI\lv T (w)1I

and by setting P(w) = w&:(w) we see that

with C(w) > 0 Vw E lR

+ C(w)k.~(w) > 0 (and limw->oo P(w)

I /+00 2 P /+00 2 2n -00 P(w)llvT(w)11 dw ~ 2n -00 II v T (w)11 dw

where P

f

=

.

r

= a'(O)

T

= P 10 Ilu(t)11

2

> 0),

dt,

= infwER P(w) > O. Moreover, in this case we do not need the requirement 2

+

2

r - V o E L1/k(lR ,L (0)).

5. Restrictions on the data. In the sequel (H.5)-(H.6), imposed on the data when k o = 0, will be considered in detail. As an introduction we remark that, as k' ELI, k.~ is bounded, continuous on lR+ , and goes to 0 as w - t 00. Then, by (2.5) k' is a positive definite kernel, that is (see [20]), foT folk'(t-S)U(S)U(t)dsdt>o

utO

VTElR+,


and, moreover, k' (0)

*

= ftX> k~(w) dw;

359

so k' (0) is finite and positive if and only

if k~ ELI. Finally we recall (see (20]) that k' is a strongly positive definite kernel if k' (s) - rTe -s is a positive definite kernel for some rT > O. It has been shown that if k' ELI then k' is strongly positive definite if and only if

A'

kc(W) ~

rT

rT > 0, Vw E [0, 00).

--2'

1+ w

(5.1 )

In order to examine some properties of the functional spaces L~/k(lR+ , V), where 2

2

V is L (O) or L (aO q ) , we distinguish three cases: 1. k' is singular at the origin (k~

3a ~ 1, y > 0, w

f/.

L I) and

such that

k~(w) ~ ~ Vw > w.

(5.2)

w

By the continuity of k~ and (2.5), there exists a positive constant e(ij such that k~(w) ~ e(ij > 0 Vw E [0, W], and letting rT = min{y, e(ij} > 0 we get

A'

kc(W) ~ 1

rT

Vw E [0,00).

a

+w

(5.3)

THEOREM 5.1. When (5.2) holds there exists a real index s E (0, 1/2] such that if 2 Q o E HSCR+ ,L (O)) and ¢ E HS(R+ , L 2 (aO q )) then (H5)-(H6) are satisfied. 2 2 Proof. Let I belong to HS(R+, V), 0 < s ~ 1/2, where V is L (O) or L (aO q ) and 11·11 denotes the norm on V, and let Then, by (5.3)

III . 111~s

S

denote the norm on H (lR+ , V).

i:

1I1/111~/k = 2n (k~(w)flllj(w)112 dw 2 /+00 2s 2 2 ~ n -00 (1 + w )rT II/(w)11 'dw = rT 111/1I1Hs. 1

_I

-I

A

For instance let us consider a singularity of "log" type, i.e., k' (s) (1 - Xt(s))k'(s) where Xt(s)

={

In this case (5.2) holds with a

I

0

=1

s e,

-100

= - P logs Xt(s) +

.

loge

and

k~(w) = 1°O[-p 10gsXt(s) + (1 =

P = _ k'(e)

and

0

xt(s))k'(s)]coswsds

[PX;(S) _ (I _ Xt(S))k"(S)] Si:WS ds.

Taking into account that k~(w) "-' O(w-') as w such that

k~(w) < y1 - -+w

--+

00, there exist y ~ 0 and w

Vw> w.

360

C. GIORGI

AND

G. GENTILI

On the other hand, there is a posItIve constant M such that k~(w) :::; M and in particular k~(w) :::; M(I + w)/(I + w), 'r/w E [0, w]. So letting ( = max{y, M( I + w)} it follows that (5.4) COROLLARY 5.1. If k' has a singularity of "log" type at t = 0, then (H5)-(H6) hold 2 2 if and only if Qo E HSC'R.+ ,L (O)) and ¢ E HSCIT{+, L (aO q )) , with s = 1/2. In fact, by (5.3)-(5.4) we obtain

1

( III . IIi H I/2 2.

k' (0)
0, so that Q o may be represented as a convolution product Q o = k' *F 'Vv o for every t > O. Let k' be the even-extension on IT{ of the kernel, namely k' (s) = k' (-s) , and set (5.5)


Thus inequality (4.4) is valid even if

III Q o III~/k

361

is replaced by

Added in proof. Although Theorem 3.1 and Theorem 4.1 state existence and 2 uniqueness in L (lR+, Vo)' this kind of result is not crucial to prove stability. In deed, one can build Theorem 3.2 and Theorem 4.2 starting from the weaker condition 2 u E L (0, T; ~) that easily follows from Lemma 3.1.

Acknowledgment. This work has been performed under the auspices of G.N.F.M. C.N.R. and partially supported by Italian M.U.R.S.T. through the 40% project "Prob lemi di evoluzione nei fluidi e e nei solidi". REFERENCES

[I) B. D. Coleman and V. J. Mizel, Thermodynamics and departures from Fourier's law of heal con duclion, Arch. Rational Mech. Anal. I3 (1963), 245-261 [2] B. D. Coleman and V. J. Mizel, Norms and semi-groups in lhe lheory offading memory, Arch. Rational Mech. Anal. 23 (1966), 87-123 [3] B. D. Coleman and M. E. Gurtin, Equipresence and conslilUlive equalionsfor rigid heal conduClors, Z. Angew. Math. Phys. 18 (1967), 199-208 [4) M. E. Gurtin, On the thermodynamics of materials with memory, Arch. Rational Mech. Anal. 28 (1968), 40-50 [5) M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), 113-126 [6) w. A. Day and M. E. Gurtin, On the symmetry of conductivity tensor and other restrictions in the nonlinear theory ofheat conduction, Arch. Rational Mech. Anal. 33 (1968), 26-32 [7) J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187-204 [8] W. A. Day, The thermodynamics of simple materials with fading memory, Springer, Berlin, 1972 [9) B. D. Coleman and D. R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. S4 (1974),1-104 [10) F. Treves, Basic Linear Partial DijJerential Equations, Academic Press, New York, 1975 [11) R. K. Miller, An integrodijJerential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), 313-332 [12) V. L. Kolpashchikov and A. I. Shnip, Linear thermodynamic theory of heat conduction Wilh memo ory, J. Engrg. Phys. 46 (1984), 732-739 [13) G. M. Troianiello, Elliptic Differential Equation and Obstacle Problems, Plenum Press, New York and London, 1987 [14] M. Fabrizio and B. Lazzari, On the stabilily of linear viscoelastic fluids, Differential Integral Equa tions, to appear [15] G. Gentili, Dissipativity conditions and variational principles for the heat flux equation with mem ory, Differential Integral Equations 4 (1991), 977-989 [16] C. Giorgi, Alcune conseguenze delle reslrizioni termodinamiche per mezzi viscoelastici lineari, Quad. Sem. Mat. Univ. Brescia, 6/91 [17) C. Baiocchi, Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni dijJerenziali astralle lineari del II ordine in spazi di Hilbert, Ric. Mat. 16 (1967), 27-95

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C. GIORGI

AND

G. GENTILI

[18] M. Fabrizio and A. Morro, On uniqueness in linear viscoelasticity: A family of counterexamples, Quart. Appl. Math. 4S (1987), 321-325 [19] H. Grabmueller, On linear theory of heat conduction in materials with memory, Proc. Roy. Soc. Edinburgh, A-76 (1976-1977), 119-137 [20] M. Renardy, W. J. Hrusa, and J. A. Nohe1, Mathematical problems in viscoelasticity, Pitman monographs and surveys in pure and applied mathematics, Longman, Essex, )987