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Asymptotic and Other Properties of a Nonlinear Diffusion Model
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
228, 119]140 Ž1998.
AY986121
Asymptotic and Other Properties of a Nonlinear Diffusion Model J. N. FlavinU Department of Mathematical Physics, National Uni¨ ersity of Ireland, Galway, Ireland
and S. Rionero† Department of Mathematics and Applications ‘‘R. Caccioppoli,’’ Uni¨ ersity ‘‘Federico II,’’ Naples, Italy Submitted by Jack K. Hale Received December 19, 1997
1. INTRODUCTION This paper derives a number of properties}asymptotic, stability, and other properties}for a certain Žparabolic. nonlinear diffusion equation. The essential underlying assumption is that the diffusivity is bounded below by a positive constant. An initial boundary value problem Ži.b.v.p.. for such an equation is considered where the assigned boundary values are independent of time together with the associated steady state solution. The purpose of this paper is to demonstrate how related, no¨ el Liapuno¨ functionals, of two general types, may be used to elucidate characteristics of the i.v.b.p., and its relationship to those of the corresponding b.v.p. Section 2 discusses the general context, particularly the unsteady and steady states associated with the nonlinear diffusion equation. Section 3 introduces some Liapunov functionals Žof type 1, say. and proves relevant properties Žlemmas. thereof. It establishes that the steady * E-mail: [email protected]. † E-mail: [email protected]. 119 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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FLAVIN AND RIONERO
state solution is exponentially asymptotically stable in these, and in related, measures, and discusses some related properties. Section 4 considers the i.b.v.p. both for the forward in time and for the backward in time p.d.e. One of the Liapunov functionals Žof type 1., under appropriate assumptions, is proved to be a logarithmically convex function of time. This leads to two conclusions for the zero solution of the appropriate i.b.v.p.: Ži. its uniqueness; Žii. Holder continuous dependence ¨ thereof for the backward in time problem, under appropriate assumptions. Section 5 introduces another type of Liapunov functional or measure Žof type 2, say.. A simple proof is given which establishes that the steady state solution is exponentially asymptotically stable in this measure. As an important consequence of this, exponential asymptotic stability in the pointwise norm follows in the one-dimensional case. In Section 6 we consider an i.b.v.p., in one spatial dimension, with moving boundaries and zero boundary values Žof temperature, for example.. Some stability and decay properties are established for functionals or measures of type 2, under various assumptions. Flavin and Rionero w5x have used Liapunov functionals of type 1 to derive some estimates of the general type occurring in this paper for a problem involving a thermal diffusivity depending quadratically on temperature}a constitutive property appropriate to cold ice. A very close analogue of such functionals or measures had earlier been used w12x in a mathematically analogous context. The context was that of nonlinear stability studies of Couette]Poiseuille flow in anisotropic magnetohydrodynamics when the perturbations are laminar. Such a Liapunov functional has also been considered in various other contexts Že.g., w4x and w1x.. Concerning other approaches to issues similar to that dealt with in this paper we mention the following: Kawanago w8, 9x considers the asymptotic behaviour as t ª ` of i.b.v.p.s for nonlinear diffusion equations of the general type considered in this paper but with zero Žor homogeneous. boundary conditions. In w8x he considers the nondegenerate case Ži.e., essentially when thermal diffusivity is strictly positive., while in w9x, in addition to this, he considers the degenerate case Ži.e., when the diffusivity is nonnegative, vanishing when the temperature or other relevant field variable vanishes.. It should be noted, however, that Kawanago also discusses Neumann boundary conditions and existence questions. We emphasize that what distinguishes this work from the two works just cited, and from other approaches to issues of the type addressed here, is its use of novel, conceptually simple, versatile Liapunov functionals. Further evidence of the versatility of such functionals is provided by the fact that one has recently been used as an essential ingredient in the construction of a very effective Liapunov functional appropriate to the Benard problem ´ for a thermofluid w6x.
PROPERTIES OF A DIFFUSION MODEL
121
2. PRELIMINARIES Let V g R 3 be a sufficiently smooth bounded fixed domain ensuring the validity of divergence-like theorems. We will consider the initial boundary value problem Ži.b.v.p..
Ž x, t . g V = Rq,
Ž 1.
u Ž x, 0 . s u 0 Ž x . ,
x g V,
Ž 2.
u Ž x, t . s u1 Ž x . ,
Ž x, t . g V = Rq,
Ž 3.
ut s D F Ž u. ,
where F g C 2 ŽR. and u 0 g C 2 Ž V ., u i g C Ž V ., i s 0, 1, are assigned functions. Typically Žthough not necessarily . u represents temperature, and F Ž u. is connected to the temperature dependent diffusivity k Ž u. by F Ž u. s
u
H0 k Ž u . du.
Our principal goal is to study matters relating to the asymptotic behaviour of solutions of Ž1. ] Ž3.. In particular, we are interested in obtaining conditions on F under which each solution tends toward a steady state when t ª `. Smooth solutions are assumed throughout, but most of the arguments are easily adapted to cater for weak solutions. Having this in mind, let us consider the steady boundary value problem Žb.v.p.. D F Ž U . s 0, U s u1 Ž x . ,
x g V,
Ž 4.
x g V,
Ž 5.
and let us put u s U q¨.
Ž 6.
Then from Ž1. ] Ž3. and Ž4. ] Ž6. it turns out that ¨ t s D L,
Ž x, t . g V = Rq,
¨ Ž x, 0 . s u 0 y u1 s ¨ 0 , ¨ s 0,
x g V, x g V,
Ž 7. Ž 8.
where L s L Ž U, ¨ . s F Ž U q ¨ . y F Ž U . .
Ž 9.
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FLAVIN AND RIONERO
Throughout this paper we shall assume that FX Ž u. s
dF du
) 0,
; u g R.
Ž 10 .
Remark 1. Let us remark that ¨ s0
«
L s 0.
Ž 11 .
¨ s0
m
L s 0.
Ž 12 .
Moreover, Ž10. implies Remark 2. Concerning the solvability of the b.v.p. Ž4. ] Ž5., we emphasize that, under the assumption Ž10., on setting F1 s F Ž u1 .
Ž 13 .
this problem depends on the solvability of the b.v.p. of the Dirichlet problem for the Laplace equation D F s 0, F s F1 ,
x g V,
Ž 14 .
x g V.
Ž 15 .
In fact, let F U be the solution of Ž14. ] Ž15., and let w be the Žunique. inverse function of F Žbearing Ž10. in mind.. From F Ž U . s FU
Ž 16 .
U s w Ž FU .
Ž 17 .
it follows that is the solution of Ž4. ] Ž5.. Of course, uniqueness for the b.v.p. Ž14. ] Ž15. implies Žin view of assumption Ž10.. uniqueness for Ž4. ] Ž5.. Let us note that conditions ensuring the solvability of the b.v.p. Ž14. ] Ž15., in the strong or in the weak sense, are well known and can be found, for instance, in Gilbarg and Trudinger w7x. Further, for many domains V, and for many examples of boundary data, explicit analytic solutions are known, many of which can be found, for instance, in Carslaw and Jaeger w2x.
3. CONVERGENCE TO ASYMPTOTIC STATES For the remainder of this paper we shall make the following assumption Žunless otherwise stated.: Assumption. For u g R let F X Ž u . G m,
Ž 18 .
PROPERTIES OF A DIFFUSION MODEL
123
where m is a positive constant. ŽIn the context of standard heat conduction this means that the thermal diffusivity is bounded below by m.. For an easy proof of theorems related to the asymptotic states, we need the following lemmas, which are to a large extent a generalization of those contained in w5x. LEMMA 1.
Supposing that ¨ g R and setting GŽ ¨ , U . s
¨
H0 L Ž ¨ , U . d¨ ,
Ž 19 .
one has
G
G Ž 0, U . s
Ž i.
¨
2G
Ž ii .
¨2
s 0,
Ž 20 .
¨ s0
) m,
Ž 21 .
G Ž ¨ , U . G 12 m¨ 2 ,
Ž iii . G
2
s L2 Ž ¨ , U . G 2 mG Ž ¨ , U . .
ž /
Ž iv.
Ž 22 .
¨
Ž 23 .
Proof. Note that Ž11., Ž18., and Ž19. immediately imply Ž20. ] Ž21., while Ž22. comes from the Taylor expansion and Ž20. ] Ž21.. Finally, from
L
L Ž 0, U . s 0,
¨
L2
) m,
Ž 24 .
L G 2 Lm, s 2L ¨ ¨ F 2 Lm,
¨ ) 0, ¨ - 0,
½
one obtains Ž23.. LEMMA 2.
Let ¨ g R and let Ž19. hold. On setting Gn s
¨
2 nq1
H0 L
Ž ¨ , U . d¨ ,
G0 s G
Ž 25 .
Ž n being a positi¨ e integer or zero., it follows that 0 F Gn - ¨ L2 nq1 , Gn G
m2 nq1 2 Ž n q 1.
¨ 2Ž nq1. ,
L2 nq2 G mGn .
Ž 26 .
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FLAVIN AND RIONERO
Proof. Because Ž19. ] Ž21. imply LŽ ¨ , U . ¨
) m,
; ¨ / 0,
Ž 27 .
it follows that L2 nq1 Ž ¨ , ? . G m2 nq1 ¨ 2 nq1 , 2 nq1
L
Ž ¨ , ?. F m
2 nq1
¨
2 nq1
¨ G 0,
Ž 28 .
¨ F 0.
,
Integration of Ž28. yields Ž26 2 .. Turning to Ž26 1 ., let us notice that
¨
L2 nq1 ) Ž 2 n q 1 . mL2 n ) 0,
; ¨ g R, ¨ / 0.
Therefore, because of Ž18., it follows that ¨ )0 ¨ -0
« «
¨
2 nq1
H0 L
0
0 2 nq1
HL ¨
¨
Ž ¨ , ? . d¨ - L2 nq1 Ž ¨ , ? . H d¨ s ¨ L2 nq1 Ž ¨ , ? . , 0
Ž ¨ , ? . d¨ ) L2 nq1 Ž ¨ , ? . H d¨ s y¨ L2 nq1 Ž ¨ , ? . , ¨
which together imply Ž26 1 .. The proof of Ž26 3 . is now trivial. In fact, for ¨ / 0, Ž26 1 . and Ž27. imply L2 nq1 Ž ¨ , ? . ¨
2 nq1
H0 L
Ž ¨ , ? . d¨
)
L2 nq2 Ž ¨ , ? . ¨L
2 nq1
Ž ¨ , ?.
s
L Ž ¨ , ?. ¨
) m.
We are now in a position to prove some theorems concerning asymptotic states, the first of these being: THEOREM 1. Let Ž18. hold and let the b.¨ . p. Ž4. ] Ž5. be sol¨ able. Then U is exponentially asymptotically stable in the L2-norm and is the asymptotic state of any solution of the i.b.¨ . p. Ž1. ] Ž3. in this norm. Proof. Let us introduce the Liapunov functional VŽ t. s
HVG
¨ Ž x, t . , U Ž x . dV
Ž 29 .
PROPERTIES OF A DIFFUSION MODEL
125
and notice that, by Ž20. and Ž22., it is positive definite with respect to ¨ . Along the relevant solutions, it turns out that V˙ s s
HVL¨
t
dV s
HVL D L dV s HVL= ? Ž =L . dV
H V L=L ? n dS y HV Ž =L .
2
dV ,
Ž 30 .
where n is the outward unit normal vector to V. On taking into account Ž8 2 ., Ž11., and the Poincare ´ inequality, it follows that V˙ F yg
2
HVL
dV
Ž 31 .
where g ) 0 is the positive constant in the Poincare ´ inequality Žsee Appendix 1.. Using Ž23., Ž29., and Ž31., it turns out that V˙ F y2 mg V ,
Ž 32 .
V F V Ž 0 . ey2 mg t .
Ž 33 .
which implies
Whereas this latter result means exponential asymptotic stability in the measure V, we proceed to the weaker, but more con¨ entional, result embodied in Theorem 1: this follows from Ž33. on noting that Ž22. implies
HV¨
2
dV F
2 m
V Ž 0 . ey2 mg t .
Ž 34 .
Remark 3. The decay constants occurring in the foregoing results Ž33. ] Ž34. are not quite as good as in their counterparts in an earlier paper w5x on cold ice Žwhich has a particular constitutive equation.. The reason is that it was possible to use stronger assumptions in the circumstances of the latter paper. We now proceed to a generalization of Theorem 1. THEOREM 2. Let Ž18. hold and let the b.¨ . p. Ž4. ] Ž5. be sol¨ able. Then U is asymptotically exponentially stable in the L2 n-norm, n g Rq, and is the asymptotic state of each solution of Ž1. ] Ž3. in the L2 n-norm as t ª `. Proof. Let us introduce the Liapunov functional Vn s
HVG
n
¨ Ž x, t . , U Ž x . dV
Ž 35 .
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FLAVIN AND RIONERO
and notice that, by Ž25. and Ž26 2 ., it is positive definite in ¨ . Along the relevant solutions, it turns out that V˙n s s
2 nq1
HVL
¨ t dV s
2 nq1
H V L
2 nq1
HVL
D L dV s
=L ? n dS y Ž 2 n q 1 .
2 nq1
HVL 2n
HV L
= ? Ž =L . dV
2 Ž =L . dV ,
Ž 36 .
and hence by Ž8 2 . and Ž11. V˙n s y
Ž 2 n q 1. Ž n q 1.
nq1 2
HV Ž =L
2
. dV .
Ž 37 .
From Ž37. and the relevant Poincare ´ inequality Žsee Appendix 1., one obtains V˙n F y
Ž 2 n q 1. Ž n q 1.
2
g
2Ž nq1.
HVL
dV ,
and in view of Ž26 3 . it follows that V˙n F y
Ž 2 n q 1. Ž n q 1.
2
g mVn
and hence Vn F Vn Ž 0 . exp y Ž 2 n q 1 . Ž n q 1 .
y2
g mt ,
Ž 38 .
where VnŽ0. s VnŽ ¨ 0 , UŽ x ... Finally Ž26 2 . gives
HV¨
2Ž nq1.
dV F
2 Ž n q 1. m2 nq1
Vn Ž 0 . exp y Ž 2 n q 1 . Ž n q 1 .
y2
g mt . Ž 39 .
On raising both sides of Ž39. to the power of 2Ž n q 1.4 y1 and, on taking the limit as n ª `, one finds that sup ¨ Ž x, t . F my1 sup L Ž ¨ 0 , U . xgV
xgV
F sup v0 Ž x . ,
Ž x, t . g V = Rq,
Ž 40 .
xgV
where use has been made of Ž26 3 .. Thus one has THEOREM 3a. Let Ž18. hold and let Ž4. ] Ž5. be sol¨ able: Then the steady state U is stable in the L`-norm.
127
PROPERTIES OF A DIFFUSION MODEL
Greater effort is required to establish convergence in the pointwise norm. To this end, let us note: if r, s, t 1 ) 0, are constants such that r ) s ) t 1 ) 0, it follows from Ž37. that my1 L2 nq2 Ž x, s . G Vn Ž s . G Ž 2 n q 1 . Ž n q 1 .
y2
r
nq1 2
Hs HV Ž =L
. dV dt, Ž 411 .
where Ž26 3 . has also been used. Proceeding as in Evans w3, pp. 600, 601x, this together with a Sobolev inequality implies the following inequality connecting two norms at different times: L Ž ?, r .
6 Ž nq1 .
F C Ž2 nq2.
y1
1 q Ž r y s.
y1 y1 Ž2 nq2 .
L Ž ?, s .
2 nq2 ,
Ž 41 2 .
where C is a constant depending on V. Using a Moser iteration of this, one may prove, as in Evans w3, pp. 600, 601x, that for t ) t 1 ) 0 L Ž x, t .
`
F cty3 r4 L Ž x, tr2 .
2,
Ž 41 3 .
where c is a constant depending on V. Suppose now that, in addition to assumption Ž18., one assumes F X Ž u . - M,
Ž 42 1 .
where M is a positive constant; it is obvious in this case that one can derive upper and lower bounds mirroring the lower and upper ones occurring in Ž26 2 . and Ž26 3 ., respectively. Now Ž18. and Ž42 1 . together with Ž33. imply that L Ž x, t .
2
F Ž Mrm .
1r2
L Ž x, 0 . 2exp Ž ymg t . .
Ž 42 2 .
Using Ž41 3 ., Ž42 2 ., and Ž26 3 . one finds that for any t ) t 0 , t 0 being a constant, one has Žcf. w8x. m ¨ Ž x, t .
`
F L Ž x, t .
`
F N exp yg m Ž t y t 0 . r2 4 ,
Ž 43 .
where N is a constant depending inter alia on V. Consequently we have THEOREM 3b. Let Ž18. and Ž42 1 . hold and let Ž4. ] Ž5. be sol¨ able. Then the steady state U is exponentially asymptotically stable in the L`-norm, and it is the asymptotic state in this norm. Remark 4. Kawanago w8x also requires an assumption additional to Ž18. when deriving some of his results on exponential asymptotic stability.
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FLAVIN AND RIONERO
Remark 5. Let Ž18. hold and let Ž4. ] Ž5. be solvable. We will consider here the stability of U, when perturbations in the boundary data are also allowed. Denoting these by f Žx, t ., Ž8 2 . becomes
Ž x, t . g V = Rq.
¨ Ž x, t . s f Ž x, t . ,
Ž 44 .
Then from Ž30., by using the Friedrichs inequality and Ž18., we obtain V˙ F y2 mg 1V q C Ž t . ,
Ž 45 .
where CŽ t. s
H V
ž
L
L
q g 2 L2 dS,
/
n
Ž 46 .
in which g 1 , g 2 are the positive constants in the Friedrichs inequality Žsee Appendix 1., the second term in the integrand being the outward normal derivative. On integrating, it turns out that V Ž t . F V Ž 0 . ey2 mg t q ey2 mg t
t
H0 c Ž t . e
2 mgt
dt .
Ž 47 .
On setting S Ž t . s sup C Ž t . w0, t x
Ž 48 .
it turns out that V Ž t . F V Ž 0 . ey2 mg t q
SŽ t . 2 mg
Ž 1 y ey2 mg t . .
Ž 49 .
Therefore, because of Ž22., V Ž 0. -
m 2
«,
Ž 50 .
S Ž t . - m g« 2
imply
HV¨
2
dV F 2 « ,
Ž 51 .
i.e., the continuous dependence in the L2-norm on w0, t x. Moreover, lim S Ž t . s 0
tª`
Ž 52 1 .
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PROPERTIES OF A DIFFUSION MODEL
implies that U is the asymptotic state in the L2-norm of any solution of Ž1. ] Ž2. and on V .
u Ž x, t . s u1 Ž x . q f Ž x, t .
Ž 52 2 .
By using the methodology used in proving Theorem 2, conditions ensuring the continuous dependence and the asymptotic behaviour in the L2Ž nq1. norm can be obtained. Analogously, conditions ensuring continuous dependence in the pointwise norm can be found. Remark 6. We notice that the Poincare ´ inequality holds also on noncompact domains, bounded at least in one direction. Therefore Theorems 1]3 and Remark 2 continue to hold also for these domains, at least with respect to perturbations spatially periodic in the direction in which the domains are unbounded.
4. LOGARITHMIC CONVEXITY CONSIDERATIONS In this section we establish, under appropriate assumptions, the logarithmic convexity of the Liapunov functional V Ž t . defined by Ž29. in the context previously described, both for the forward and backward in time equation. Recalling that ŽŽ19., Ž30.. V˙ s
HVG ¨
¨ t
dV s y
HV Ž =G . ¨
2
dV ,
Ž 53 .
where G¨ s Gr ¨ etc. Differentiating Ž532 . we obtain V¨Ž t . s y2
HV =G
¨
? =G¨ t d¨ s 2
HV DG G ¨
¨t
dV
s 2 G¨ ¨ ¨ t2 dV ,
Ž 54 .
H
where Ž7., Ž8 2 ., Ž20., etc. have been used together with the divergence theorem. Now using Ž53. ] Ž54. together with Schwarz’s inequality, we obtain 2
VV¨ y V˙ 2 s 2
G dV
HV
G¨ ¨ ¨ t2 dV y
HV
G 2 G¨ ¨¨ t2 dV
H
HV Ž G
¨¨
.
½H
y1
G¨ ¨ t dV
V
5
Ž GG¨ ¨ y 12 G¨2 . dV
,
Ž 55 .
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FLAVIN AND RIONERO
recalling that G¨ ¨ ) 0 in view of Ž18. and Ž21.. From this it follows that VV¨ y V˙ 2 G 0
Ž 56 .
GG¨ ¨ y 12 G¨2 G 0
Ž 57 .
provided that
Ži.e., that 'G is convex w.r.t. ¨ }assuming G / 0.. It is easily shown that the foregoing result continues to hold in the context of the backward in time equation, i.e., when y¨ t replaces ¨ t in Ž7.. Thus we have: THEOREM 4. In the context of the initial boundary ¨ alue problem defined by Ž7. ] Ž8., subject to Ž18., or for its counterpart for the backward in time equation Ž wherein y¨ t replaces ¨ t in Ž7.., suppose that there is no ¨ alue of t for which ¨ ' 0, ;x g V, and that G1r2 Ž ¨ , ? . s G Ž ¨ , ? .
'
Ž ) 0.
is a con¨ ex function of ¨ g Rq. Then VŽ t. s
HVG Ž ¨ , ?. dV
is a logarithmically con¨ ex function1 of t. Remark 7. Condition Ž57., concerning G, arising in the foregoing, is satisfied, for example, if the diffusivity k Ž u. s k 0 Ž 1 q « u2 n . , where k 0 , « are positive constants and n is a positive integer. Two implications of the foregoing result are now discussed. If the conditions of the foregoing theorem hold, logarithmic convexity implies Žcf. w4x. that V Ž t . F V Ž t1 .
Ž t 2 yt .r Ž t 2 yt 1 .
V Ž t2 .
Ž tyt 1 .r Ž t 2 yt 1 .
Ž 58 .
for 0 F t 1 F t F t 2 . Suppose that, for some fixed T, ¨ Ž x, t . ' 0, ¨ Ž x, T . ' 0. 1
Its logarithm is convex.
;x g V , for each t , 0 F t F T ,
Ž 59 .
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PROPERTIES OF A DIFFUSION MODEL
In Ž58. write t 1 s 0, t 2 s T y « , where « is a constant. T ) « ) 0 to obtain V Ž t . F V Ž 0.
Ž Ty « yt .Ž Ty « .y1
VŽT y « .
t Ž Ty « .y 1
.
Ž 60 .
Letting « ª 0, one obtains, in view of Ž59. and the positi¨ e definiteness of G, that for 0 F t F T VŽ t. ' 0 and that ¨ ' 0.
Thus supposition Ž59. cannot hold. Similarly, the following cannot hold: For some fixed T, T1 , ¨ Ž x, t . ' 0,
;x g V for each t , T F t F T1 ,
¨ Ž x, T . ' 0.
Thus one has the following THEOREM 5. In the context of the initial boundary ¨ alue problem gi¨ en by Ž7. ] Ž8. etc., or that of its backward in time counterpart, and subject to the restrictions Ž18. and Ž57., one has ¨ Ž x, T . ' 0,
;x g V ,
for some Ž finite. time T ŽG 0., implies that ¨ ' 0,
;x g V , ; t ,
i.e., one has uniqueness of the zero solution. Remark 8. This provides, inter alia, a complement to our earlier results which imply asymptotic decay to zero, in various measures, of the solution Žfor the forward in time problem.: the foregoing proposition establishes that the solution cannot decay identically to zero in finite time. A continuous dependence result also follows for the backward in time equation, using Ž58. etc. THEOREM 6. For the backward in time counterpart of the i.b.¨ . p. Ž7. ] Ž8. etc., subject to the restrictions Ž18. and Ž57., one has the following: pro¨ ided that, for some fixed T, V Ž T . F M,
Ž 61 .
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FLAVIN AND RIONERO
where M is a gi¨ en positi¨ e constant, then V Ž t . F M t r T V Ž 0.
1y trT
Ž 62 .
for 0 F t F T, establishing Holder continuous dependence, on the initial ¨ data, for the Ž zero. solution of Ž7. ] Ž8.. In the foregoing, the constant M may be related to the relevant melting temperature Žcf. w10x.. It is, of course, known that, in the absence of a restriction of type Ž61., the relevant i.b.v.p. is ill-posed w10x.
5. ANOTHER APPROACH TO ASYMPTOTIC STATES In this section we introduce a different Liapunov functional Žof type 2., yet one which is related to the first one used in Section 3, and exponential asymptotic stability etc. is established in this measure. The proof is especially simple and it is this simplicity which makes it attractive. Moreover, in one dimension, exponential asymptotic stability in the pointwise norm is implied. In the original context ŽŽ1. ] Ž9. etc.. we introduce the Liapunov functional EŽ t . s
1 2
HV Ž =L .
2
dV .
Ž 63 .
In view of Ž8 2 ., Ž9., and the Poincare ´ inequality Žcf. Appendix 1. there is a positive constant g such that E Ž t . G 12 g
2
HVL
dV .
Ž 64 .
In view of this and Ž11., E is positive definite with respect to ¨ , and thus serves as a Liapunov stability measure for the context in question. Along the solutions of Ž7. ] Ž8. we obtain, using the divergence theorem etc., E˙Ž t . s
HV =L ? =L
t
dV s
L
H V L =L ? n dS y HV n ¨ t
t
D L dV . Ž 65 .
But L s 0 on V « L t s 0 on V , L s FX Ž U q ¨ . ) m n
Ž 66 .
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PROPERTIES OF A DIFFUSION MODEL
on recalling Ž9. and Ž18.. Using Ž7., Ž65., and Ž66. one obtains E˙ s ym
HV Ž D L .
2
dV .
Ž 67 .
But L s 0 on V implies
g
HV Ž =L .
2
dV F
HV Ž D L .
2
dV ,
Ž 68 .
where g is the constant in the Poincare ´ inequality arising previously Žsee . Ž . Ž . Ž . Appendix 1 . Thus 63 , 67 , and 68 yield E˙Ž t . F y2 mg E Ž t . .
Ž 69 .
E Ž t . F E Ž 0 . ey2 mg t ,
Ž 70 .
Integration of this yields
wherein EŽ0. is given in terms of data by E Ž 0. s
HV
F X Ž ¨ 0 q U . = Ž U q ¨ 0 . y F X Ž U . =U
s
HV
F X Ž ¨ 0 q U . y F X Ž U . 4 =U q F X Ž ¨ 0 q U . =¨ 0
2
dV 2
dV . Ž 71 .
The foregoing result may be summarized thus: THEOREM 7. Let Ž18. hold and let Ž4. ] Ž5. be sol¨ able. Then U is asymptotically exponentially stable in the measure Ž65. Ž as con¨ eyed by Ž70. ] Ž71.. and is the asymptotic state of any solution of Ž1. ] Ž3. in this measure. It is readily verified that Ž70. implies that
HV
L2 q Ž =L .
2
dV F KE Ž 0 . ey2 mg t ,
Ž 72 .
where K is a computable constant. This implies a slight variant of the previous theorem, under the same underlying assumptions: THEOREM 7a. The steady state U is asymptotically exponentially stable in the H 1-norm of L and is the asymptotic state of any solution of Ž1. ] Ž3. in this norm.
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FLAVIN AND RIONERO
We conclude this section with two remarks: Remark 9. Consider a fixed, one-dimensional domain 0 F x F l, to which the previous analysis is applicable mutatis mutandis. The steady state is exponentially asymptotically stable in the pointwise norm. That this is so is a consequence of Ž70. together with the readily verified inequalities m2 ¨ 2 Ž x, t . F L2 Ž x, t . F
žH
2
x
L x dx
0
/
F x2
l 2
H0 L
x
F 2 l 2EŽ t . .
Remark 10. Using the considerations of this section, it is also possible to deduce pointwise decay to zero as t ª ` of ¨ , in R 3 : Note that Žin standard norm notation. Ži. it follows from Ž67. that t
H0
D L Ž x, t .
2 2
dt F my1 E Ž 0 . ;
Žii. one has the Sobolev inequality sup L Ž x, ? . F C D L Ž x, ? . V
2,
where C is a constant depending on V Že.g., w13x.. Using these and Ž26 2 . ] Ž26 3 . one finds that t
H0 ½ sup V
¨ Ž ?, t .
2
5
dt F c 2 my3 E Ž 0 . .
It follows from this, bearing in mind the assumed continuity of ¨ , that sup ¨ Ž ?, t . ª 0 V
as t ª `.
6. CASE OF MOVING BOUNDARIES In this section we consider an i.b.v.p. in one spatial dimension with moving boundaries and zero boundary values Že.g., of temperature.. Some stability and decay results are established for certain measures Žor norms. of type 2. These depend on a lemma given in Appendix 2 and also in w4x. Suppose ut s F Ž u.
xx,
xy Ž t . - x - xq Ž t . , t g Rq,
Ž 73 .
u Ž x, 0 . s u 0 Ž x . ,
Ž 74 .
u Ž xq Ž t . , t . s u Ž xy Ž t . , t . s 0.
Ž 75 .
PROPERTIES OF A DIFFUSION MODEL
135
It is supposed that xq Ž t ., xy Ž t . are single-valued, continuously differentiable functions such that xq Ž t . ) xy Ž t ., and that xq? Ž t . G 0,
xy? Ž t . F 0
Ž 76 .
Ž‘‘expanding’’ region.. Moreover, assumption Ž18. again obtains. Suppose that f Ž w . }arising in the Lemma in Appendix 2}is a twice continuously differentiable con¨ ex function Ži.e., f Y G 0. such that f Ž0. s 0; for such a function wf X Ž w . y f Ž w . G 0.
Ž 77 .
In what follows it is natural}though not necessary}to think of f Ž w . as a positive definite function of its argument. Defining JŽ t. s
HG f Ž F . dx
Ž 78 .
x
t
it follows from the Lemma in Appendix 2, Ž77., and Ž18. that J ?Ž t . s
HG f
Y
Ž Fx .
t
HG f
F ym
Y
F u
Fx2x dx
Ž Fx . Fx2x dx.
Ž 79 .
t
Thus we have THEOREM 8. For the i.b.¨ . p. with mo¨ ing boundaries, defined by Ž73. ] Ž75., subject to the restrictions on the boundary motion stated in the second paragraph of this section, the measure defined by Ž78. Ž wherein f Ž w . is a con¨ ex function of its argument such that f Ž0. s 0. is a nonincreasing function of the time t. If f is positive definite in its argument the foregoing proposition may be viewed as a stability estimate. As an example, take
f Ž w. s w2n, n being a positive integer. The foregoing implies 1r2 n
HG
t
Fx2 n
dx
1r2 n
F
HG
0
Fx2 n
dx
,
t G 0.
136
FLAVIN AND RIONERO
Letting n ª `, we obtain THEOREM 9. restrictions
For the i.b.¨ . p. defined by Ž73. ] Ž76. subject to the stated sup Fx Ž x, t . F sup Fx Ž x, 0 . , x
t G 0.
Ž 80 .
x
As Fx is interpretable as heat flux in the context of standard heat flow, this means that the absolute value of the heat flux is a nonincreasing function of time. As another example, take
f Ž w. s w2 and Ž79. takes the form J ? Ž t . F y2 m
2 xx
HG F
dx.
Ž 81 .
t
Hence J ? Ž t . F y2 mp 2 ly2 Ž t .
2 x
HG F
dx,
Ž 82 .
t
where l Ž t . denotes the width of the region at time t; this follows on noting that HGt Fx dx s 0 and on using the well-known Poincare ´ inequality L
H0 F
2 xx
dx dx G p 2 Ly2
L
H0 F
2 x
dx
for smooth functions F Ž x . such that L
H0 F
x
dx s 0.
Integration of Ž82. yields a global decay Žasymptotic stability . result, whence pointwise decay Žasymptotic stability . also follows: THEOREM 10. restrictions
For the i.b.¨ . p. defined by Ž73. ] Ž76. subject to the stated JŽ t. s
2 x
HG F
dx
Ž 83 .
t
satisfies J Ž t . F J Ž 0 . exp y2p 2 m
t y2
H0 l
Ž tX . dtX .
Ž 84 .
137
PROPERTIES OF A DIFFUSION MODEL
Further, ' a positi¨ e constant C. u Ž x, t . F C J 1r2 Ž 0 . exp yp 2 m
t y2
H0 l
Ž tX . dtX .
Ž 85 .
That Ž85. follows from Ž84. is a simple consequence of Schwarz’s inequality and the zero boundary conditions satisfied by u. It is possible to enhance the decay rate in Ž84. if one retains the boundary terms on applying the Lemma of Appendix 2 to Ž83.. Specifically one obtains xsxq
J ? Ž t . F y2 m
HG
Fx2x dx y Fx2 x ? xsxy
t
F y2 m l Ž t . J Ž t . ,
Ž 86 .
where lŽ t . is the greatest value for which the inequality 2 xx
HG F
dx q Ž 2 m .
y1
xsxq
Ž x ? . Fx2
xsxy
t
G lŽ t .
2 x
HG F
dx
Ž 87 .
t
holds in the class of smooth F Ž x .. F Ž xq . s F Ž xy . s 0. This value may be obtained, by standard variational means Že.g., w11x., as the lowest Žpositive. eigenvalue of a problem consisting of a fourth order differential equation subject to the two zero boundary conditions and two natural ones. Integration of Ž86. yields an enhancement of Ž84. ] Ž85. with lŽ t . ) p 2 ly2 Ž t .. It may also be easily proven that lŽ t . cannot exceed 4p 2 ly2 Ž t ..
´ AND FRIEDRICHS INEQUALITIES APPENDIX 1: POINCARE For three-dimensional regions V with sufficiently smooth boundary V one has the following inequalities: Ži. For arbitrary functions F regular in the region V, vanishing on the boundary V, and such that =F is square integrable, one has
HV Ž =F .
2
dV G g
HVF
2
dV ,
where g is the lowest Žpositive. eigenvalue of DC q lC s 0
in V ,
Cs0
on V .
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FLAVIN AND RIONERO
Žii. One also has
HV Ž D x .
2
dV G g
HV Ž =x .
2
dV
for arbitrary regular functions such that x s 0 on V. This follows from the previous inequality, Schwarz’s inequality, etc. as follows: 1r2
H Ž =x . 2 dV s yH x D x dV F V
HV
V
H Ž D x . 2 dV
x 2 dV
V
1r2
F gy1
HV Ž =x .
2
HV Ž D x .
dV
2
dV
.
Analogues of the foregoing inequalities hold in all dimensions. Žiii. Let V be a domain with Lipschitz boundary. Then for every function u g C 1 Ž V . there exist positive constants g i Ž V . Ž i s 1, 2. such that w11x
HVu
2
dV F gy1 1
HV Ž =u .
2
dV q g 2 gy1 1
H Vu
2
dV .
APPENDIX 2 LEMMA. Consider a region R in the t]x plane bounded Ži. Žii.
by a straight line segment with abscissa t s 0, by two cur¨ es Cq, Cy defined respecti¨ ely by x s xq Ž t . ,
x s xy Ž t . ,
which are Ž a . single-¨ alued continuously differentiable functions, Ž b . such that xq Ž t . ) xy Ž t .. Let Gt denote the straight line segment contained in the region ha¨ ing abscissa t. Let c Ž t, x . be a twice continuously differentiable function defined in the region which takes zero ¨ alues on Cq, Cy. Let f Ž w . be a twice continuously differentiable function of its argument w.
139
PROPERTIES OF A DIFFUSION MODEL
Then in the context and notation of the last two paragraphs the quantity JŽ t. s
HG f Ž c . dx x
t
satisfies J ?Ž t . s
HG
f Y Ž c x . c t c x x dx y c x f X Ž c x . y f Ž c x . 4 x ? Ž t .
xsxq Ž t .
. xsxy Ž t .
t
The result is a simple consequence of the Leibnitz theorem and differentiation of the boundary conditions. A more general version is given in w4x.
ACKNOWLEDGMENTS This research has largely been performed during visits of S. Rionero to the Department of Mathematical Physics, National University of Ireland, Galway, in May 1996 and June 1997. This author ŽS.R.. thanks Ži. the warm hospitality extended to him by N.U.I Galway and Žii. the Italian M.U.R.S.T. Ž40% and 60% contracts.. Both authors acknowledge the G.N.F.M. of the Italian C.N.R. for the help given toward their scientific cooperation. The authors thank the referee for his remarks which have led them to obtain additional results.
REFERENCES 1. F. Capone, S. Rionero, and I. Torcicollo, On the stability of solutions of the remarkable equation u t s D F Žx, u. y g Žx, u., Rend. Circ. Mat. Palermo Ž 2 . Suppl. 45 Ž1996., 83]90. 2. H. S. Carslaw and S. C. Jaeger, ‘‘Conduction of Heat in Solids,’’ 2nd ed., Clarendon, Oxford, 1959. 3. L. C. Evans, Regularity properties for the heat equation subject to nonlinear boundary conditions, Nonlinear Anal. 1 Ž1977., 593]602. 4. J. N. Flavin and S. Rionero, ‘‘Qualitative Estimates for Partial Differential Equations, An Introduction,’’ CRC Press, Boca Raton, FL, 1995. 5. J. N. Flavin and S. Rionero, On the temperature distribution in cold ice, Rend. Mat. Acad. Lincei, 99Ž8. Ž1997., 299]312. 6. J. N. Flavin and S. Rionero, The Benard problem for nonlinear heat conduction, to ´ appear in Quart. J. Mech. Appl. Math. 7. D. Gilbarg and N. S. Trudinger, ‘‘Elliptic Partial Differential Equations of Second Order,’’ 2nd ed., Springer-Verlag, BerlinrNew York, 1983. 8. T. Kawanago, The behaviour of solutions of quasilinear heat equations, Osaka J. Math. 27 Ž1990., 769]796. 9. T. Kawanago, Asymptotic behaviour as t ª ` of solutions of quasilinear heat equations, evolution equations and nonlinear problems, Surikaisekikenkyusho Koyuroku 785 Ž1992., 152]165.
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10. L. E. Payne, ‘‘Improperly Posed Problems in Partial Differential Equations,’’ SIAM Regional Conference Series in Applied Mathematics, Vol. 22, 1975. 11. K. Rektorys, ‘‘Variational Methods in Mathematics, Science and Engineering,’’ Reidel, Dordrecht, 1980. 12. S. Rionero and M. Maiellaro, On the stability of Couette]Poiseuille flows in the anisotropic MHD via the Liapunov direct method, Rend. Acad. Sci. Fis. Mat. Napoli 62 Ž1995., 315]332. 13. B. Straughan, ‘‘The Energy Method, Stability and Natural Convection,’’ Springer-Verlag, BerlinrNew York, 1993.
228, 119]140 Ž1998.
AY986121
Asymptotic and Other Properties of a Nonlinear Diffusion Model J. N. FlavinU Department of Mathematical Physics, National Uni¨ ersity of Ireland, Galway, Ireland
and S. Rionero† Department of Mathematics and Applications ‘‘R. Caccioppoli,’’ Uni¨ ersity ‘‘Federico II,’’ Naples, Italy Submitted by Jack K. Hale Received December 19, 1997
1. INTRODUCTION This paper derives a number of properties}asymptotic, stability, and other properties}for a certain Žparabolic. nonlinear diffusion equation. The essential underlying assumption is that the diffusivity is bounded below by a positive constant. An initial boundary value problem Ži.b.v.p.. for such an equation is considered where the assigned boundary values are independent of time together with the associated steady state solution. The purpose of this paper is to demonstrate how related, no¨ el Liapuno¨ functionals, of two general types, may be used to elucidate characteristics of the i.v.b.p., and its relationship to those of the corresponding b.v.p. Section 2 discusses the general context, particularly the unsteady and steady states associated with the nonlinear diffusion equation. Section 3 introduces some Liapunov functionals Žof type 1, say. and proves relevant properties Žlemmas. thereof. It establishes that the steady * E-mail: [email protected]. † E-mail: [email protected]. 119 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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FLAVIN AND RIONERO
state solution is exponentially asymptotically stable in these, and in related, measures, and discusses some related properties. Section 4 considers the i.b.v.p. both for the forward in time and for the backward in time p.d.e. One of the Liapunov functionals Žof type 1., under appropriate assumptions, is proved to be a logarithmically convex function of time. This leads to two conclusions for the zero solution of the appropriate i.b.v.p.: Ži. its uniqueness; Žii. Holder continuous dependence ¨ thereof for the backward in time problem, under appropriate assumptions. Section 5 introduces another type of Liapunov functional or measure Žof type 2, say.. A simple proof is given which establishes that the steady state solution is exponentially asymptotically stable in this measure. As an important consequence of this, exponential asymptotic stability in the pointwise norm follows in the one-dimensional case. In Section 6 we consider an i.b.v.p., in one spatial dimension, with moving boundaries and zero boundary values Žof temperature, for example.. Some stability and decay properties are established for functionals or measures of type 2, under various assumptions. Flavin and Rionero w5x have used Liapunov functionals of type 1 to derive some estimates of the general type occurring in this paper for a problem involving a thermal diffusivity depending quadratically on temperature}a constitutive property appropriate to cold ice. A very close analogue of such functionals or measures had earlier been used w12x in a mathematically analogous context. The context was that of nonlinear stability studies of Couette]Poiseuille flow in anisotropic magnetohydrodynamics when the perturbations are laminar. Such a Liapunov functional has also been considered in various other contexts Že.g., w4x and w1x.. Concerning other approaches to issues similar to that dealt with in this paper we mention the following: Kawanago w8, 9x considers the asymptotic behaviour as t ª ` of i.b.v.p.s for nonlinear diffusion equations of the general type considered in this paper but with zero Žor homogeneous. boundary conditions. In w8x he considers the nondegenerate case Ži.e., essentially when thermal diffusivity is strictly positive., while in w9x, in addition to this, he considers the degenerate case Ži.e., when the diffusivity is nonnegative, vanishing when the temperature or other relevant field variable vanishes.. It should be noted, however, that Kawanago also discusses Neumann boundary conditions and existence questions. We emphasize that what distinguishes this work from the two works just cited, and from other approaches to issues of the type addressed here, is its use of novel, conceptually simple, versatile Liapunov functionals. Further evidence of the versatility of such functionals is provided by the fact that one has recently been used as an essential ingredient in the construction of a very effective Liapunov functional appropriate to the Benard problem ´ for a thermofluid w6x.
PROPERTIES OF A DIFFUSION MODEL
121
2. PRELIMINARIES Let V g R 3 be a sufficiently smooth bounded fixed domain ensuring the validity of divergence-like theorems. We will consider the initial boundary value problem Ži.b.v.p..
Ž x, t . g V = Rq,
Ž 1.
u Ž x, 0 . s u 0 Ž x . ,
x g V,
Ž 2.
u Ž x, t . s u1 Ž x . ,
Ž x, t . g V = Rq,
Ž 3.
ut s D F Ž u. ,
where F g C 2 ŽR. and u 0 g C 2 Ž V ., u i g C Ž V ., i s 0, 1, are assigned functions. Typically Žthough not necessarily . u represents temperature, and F Ž u. is connected to the temperature dependent diffusivity k Ž u. by F Ž u. s
u
H0 k Ž u . du.
Our principal goal is to study matters relating to the asymptotic behaviour of solutions of Ž1. ] Ž3.. In particular, we are interested in obtaining conditions on F under which each solution tends toward a steady state when t ª `. Smooth solutions are assumed throughout, but most of the arguments are easily adapted to cater for weak solutions. Having this in mind, let us consider the steady boundary value problem Žb.v.p.. D F Ž U . s 0, U s u1 Ž x . ,
x g V,
Ž 4.
x g V,
Ž 5.
and let us put u s U q¨.
Ž 6.
Then from Ž1. ] Ž3. and Ž4. ] Ž6. it turns out that ¨ t s D L,
Ž x, t . g V = Rq,
¨ Ž x, 0 . s u 0 y u1 s ¨ 0 , ¨ s 0,
x g V, x g V,
Ž 7. Ž 8.
where L s L Ž U, ¨ . s F Ž U q ¨ . y F Ž U . .
Ž 9.
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FLAVIN AND RIONERO
Throughout this paper we shall assume that FX Ž u. s
dF du
) 0,
; u g R.
Ž 10 .
Remark 1. Let us remark that ¨ s0
«
L s 0.
Ž 11 .
¨ s0
m
L s 0.
Ž 12 .
Moreover, Ž10. implies Remark 2. Concerning the solvability of the b.v.p. Ž4. ] Ž5., we emphasize that, under the assumption Ž10., on setting F1 s F Ž u1 .
Ž 13 .
this problem depends on the solvability of the b.v.p. of the Dirichlet problem for the Laplace equation D F s 0, F s F1 ,
x g V,
Ž 14 .
x g V.
Ž 15 .
In fact, let F U be the solution of Ž14. ] Ž15., and let w be the Žunique. inverse function of F Žbearing Ž10. in mind.. From F Ž U . s FU
Ž 16 .
U s w Ž FU .
Ž 17 .
it follows that is the solution of Ž4. ] Ž5.. Of course, uniqueness for the b.v.p. Ž14. ] Ž15. implies Žin view of assumption Ž10.. uniqueness for Ž4. ] Ž5.. Let us note that conditions ensuring the solvability of the b.v.p. Ž14. ] Ž15., in the strong or in the weak sense, are well known and can be found, for instance, in Gilbarg and Trudinger w7x. Further, for many domains V, and for many examples of boundary data, explicit analytic solutions are known, many of which can be found, for instance, in Carslaw and Jaeger w2x.
3. CONVERGENCE TO ASYMPTOTIC STATES For the remainder of this paper we shall make the following assumption Žunless otherwise stated.: Assumption. For u g R let F X Ž u . G m,
Ž 18 .
PROPERTIES OF A DIFFUSION MODEL
123
where m is a positive constant. ŽIn the context of standard heat conduction this means that the thermal diffusivity is bounded below by m.. For an easy proof of theorems related to the asymptotic states, we need the following lemmas, which are to a large extent a generalization of those contained in w5x. LEMMA 1.
Supposing that ¨ g R and setting GŽ ¨ , U . s
¨
H0 L Ž ¨ , U . d¨ ,
Ž 19 .
one has
G
G Ž 0, U . s
Ž i.
¨
2G
Ž ii .
¨2
s 0,
Ž 20 .
¨ s0
) m,
Ž 21 .
G Ž ¨ , U . G 12 m¨ 2 ,
Ž iii . G
2
s L2 Ž ¨ , U . G 2 mG Ž ¨ , U . .
ž /
Ž iv.
Ž 22 .
¨
Ž 23 .
Proof. Note that Ž11., Ž18., and Ž19. immediately imply Ž20. ] Ž21., while Ž22. comes from the Taylor expansion and Ž20. ] Ž21.. Finally, from
L
L Ž 0, U . s 0,
¨
L2
) m,
Ž 24 .
L G 2 Lm, s 2L ¨ ¨ F 2 Lm,
¨ ) 0, ¨ - 0,
½
one obtains Ž23.. LEMMA 2.
Let ¨ g R and let Ž19. hold. On setting Gn s
¨
2 nq1
H0 L
Ž ¨ , U . d¨ ,
G0 s G
Ž 25 .
Ž n being a positi¨ e integer or zero., it follows that 0 F Gn - ¨ L2 nq1 , Gn G
m2 nq1 2 Ž n q 1.
¨ 2Ž nq1. ,
L2 nq2 G mGn .
Ž 26 .
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FLAVIN AND RIONERO
Proof. Because Ž19. ] Ž21. imply LŽ ¨ , U . ¨
) m,
; ¨ / 0,
Ž 27 .
it follows that L2 nq1 Ž ¨ , ? . G m2 nq1 ¨ 2 nq1 , 2 nq1
L
Ž ¨ , ?. F m
2 nq1
¨
2 nq1
¨ G 0,
Ž 28 .
¨ F 0.
,
Integration of Ž28. yields Ž26 2 .. Turning to Ž26 1 ., let us notice that
¨
L2 nq1 ) Ž 2 n q 1 . mL2 n ) 0,
; ¨ g R, ¨ / 0.
Therefore, because of Ž18., it follows that ¨ )0 ¨ -0
« «
¨
2 nq1
H0 L
0
0 2 nq1
HL ¨
¨
Ž ¨ , ? . d¨ - L2 nq1 Ž ¨ , ? . H d¨ s ¨ L2 nq1 Ž ¨ , ? . , 0
Ž ¨ , ? . d¨ ) L2 nq1 Ž ¨ , ? . H d¨ s y¨ L2 nq1 Ž ¨ , ? . , ¨
which together imply Ž26 1 .. The proof of Ž26 3 . is now trivial. In fact, for ¨ / 0, Ž26 1 . and Ž27. imply L2 nq1 Ž ¨ , ? . ¨
2 nq1
H0 L
Ž ¨ , ? . d¨
)
L2 nq2 Ž ¨ , ? . ¨L
2 nq1
Ž ¨ , ?.
s
L Ž ¨ , ?. ¨
) m.
We are now in a position to prove some theorems concerning asymptotic states, the first of these being: THEOREM 1. Let Ž18. hold and let the b.¨ . p. Ž4. ] Ž5. be sol¨ able. Then U is exponentially asymptotically stable in the L2-norm and is the asymptotic state of any solution of the i.b.¨ . p. Ž1. ] Ž3. in this norm. Proof. Let us introduce the Liapunov functional VŽ t. s
HVG
¨ Ž x, t . , U Ž x . dV
Ž 29 .
PROPERTIES OF A DIFFUSION MODEL
125
and notice that, by Ž20. and Ž22., it is positive definite with respect to ¨ . Along the relevant solutions, it turns out that V˙ s s
HVL¨
t
dV s
HVL D L dV s HVL= ? Ž =L . dV
H V L=L ? n dS y HV Ž =L .
2
dV ,
Ž 30 .
where n is the outward unit normal vector to V. On taking into account Ž8 2 ., Ž11., and the Poincare ´ inequality, it follows that V˙ F yg
2
HVL
dV
Ž 31 .
where g ) 0 is the positive constant in the Poincare ´ inequality Žsee Appendix 1.. Using Ž23., Ž29., and Ž31., it turns out that V˙ F y2 mg V ,
Ž 32 .
V F V Ž 0 . ey2 mg t .
Ž 33 .
which implies
Whereas this latter result means exponential asymptotic stability in the measure V, we proceed to the weaker, but more con¨ entional, result embodied in Theorem 1: this follows from Ž33. on noting that Ž22. implies
HV¨
2
dV F
2 m
V Ž 0 . ey2 mg t .
Ž 34 .
Remark 3. The decay constants occurring in the foregoing results Ž33. ] Ž34. are not quite as good as in their counterparts in an earlier paper w5x on cold ice Žwhich has a particular constitutive equation.. The reason is that it was possible to use stronger assumptions in the circumstances of the latter paper. We now proceed to a generalization of Theorem 1. THEOREM 2. Let Ž18. hold and let the b.¨ . p. Ž4. ] Ž5. be sol¨ able. Then U is asymptotically exponentially stable in the L2 n-norm, n g Rq, and is the asymptotic state of each solution of Ž1. ] Ž3. in the L2 n-norm as t ª `. Proof. Let us introduce the Liapunov functional Vn s
HVG
n
¨ Ž x, t . , U Ž x . dV
Ž 35 .
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FLAVIN AND RIONERO
and notice that, by Ž25. and Ž26 2 ., it is positive definite in ¨ . Along the relevant solutions, it turns out that V˙n s s
2 nq1
HVL
¨ t dV s
2 nq1
H V L
2 nq1
HVL
D L dV s
=L ? n dS y Ž 2 n q 1 .
2 nq1
HVL 2n
HV L
= ? Ž =L . dV
2 Ž =L . dV ,
Ž 36 .
and hence by Ž8 2 . and Ž11. V˙n s y
Ž 2 n q 1. Ž n q 1.
nq1 2
HV Ž =L
2
. dV .
Ž 37 .
From Ž37. and the relevant Poincare ´ inequality Žsee Appendix 1., one obtains V˙n F y
Ž 2 n q 1. Ž n q 1.
2
g
2Ž nq1.
HVL
dV ,
and in view of Ž26 3 . it follows that V˙n F y
Ž 2 n q 1. Ž n q 1.
2
g mVn
and hence Vn F Vn Ž 0 . exp y Ž 2 n q 1 . Ž n q 1 .
y2
g mt ,
Ž 38 .
where VnŽ0. s VnŽ ¨ 0 , UŽ x ... Finally Ž26 2 . gives
HV¨
2Ž nq1.
dV F
2 Ž n q 1. m2 nq1
Vn Ž 0 . exp y Ž 2 n q 1 . Ž n q 1 .
y2
g mt . Ž 39 .
On raising both sides of Ž39. to the power of 2Ž n q 1.4 y1 and, on taking the limit as n ª `, one finds that sup ¨ Ž x, t . F my1 sup L Ž ¨ 0 , U . xgV
xgV
F sup v0 Ž x . ,
Ž x, t . g V = Rq,
Ž 40 .
xgV
where use has been made of Ž26 3 .. Thus one has THEOREM 3a. Let Ž18. hold and let Ž4. ] Ž5. be sol¨ able: Then the steady state U is stable in the L`-norm.
127
PROPERTIES OF A DIFFUSION MODEL
Greater effort is required to establish convergence in the pointwise norm. To this end, let us note: if r, s, t 1 ) 0, are constants such that r ) s ) t 1 ) 0, it follows from Ž37. that my1 L2 nq2 Ž x, s . G Vn Ž s . G Ž 2 n q 1 . Ž n q 1 .
y2
r
nq1 2
Hs HV Ž =L
. dV dt, Ž 411 .
where Ž26 3 . has also been used. Proceeding as in Evans w3, pp. 600, 601x, this together with a Sobolev inequality implies the following inequality connecting two norms at different times: L Ž ?, r .
6 Ž nq1 .
F C Ž2 nq2.
y1
1 q Ž r y s.
y1 y1 Ž2 nq2 .
L Ž ?, s .
2 nq2 ,
Ž 41 2 .
where C is a constant depending on V. Using a Moser iteration of this, one may prove, as in Evans w3, pp. 600, 601x, that for t ) t 1 ) 0 L Ž x, t .
`
F cty3 r4 L Ž x, tr2 .
2,
Ž 41 3 .
where c is a constant depending on V. Suppose now that, in addition to assumption Ž18., one assumes F X Ž u . - M,
Ž 42 1 .
where M is a positive constant; it is obvious in this case that one can derive upper and lower bounds mirroring the lower and upper ones occurring in Ž26 2 . and Ž26 3 ., respectively. Now Ž18. and Ž42 1 . together with Ž33. imply that L Ž x, t .
2
F Ž Mrm .
1r2
L Ž x, 0 . 2exp Ž ymg t . .
Ž 42 2 .
Using Ž41 3 ., Ž42 2 ., and Ž26 3 . one finds that for any t ) t 0 , t 0 being a constant, one has Žcf. w8x. m ¨ Ž x, t .
`
F L Ž x, t .
`
F N exp yg m Ž t y t 0 . r2 4 ,
Ž 43 .
where N is a constant depending inter alia on V. Consequently we have THEOREM 3b. Let Ž18. and Ž42 1 . hold and let Ž4. ] Ž5. be sol¨ able. Then the steady state U is exponentially asymptotically stable in the L`-norm, and it is the asymptotic state in this norm. Remark 4. Kawanago w8x also requires an assumption additional to Ž18. when deriving some of his results on exponential asymptotic stability.
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FLAVIN AND RIONERO
Remark 5. Let Ž18. hold and let Ž4. ] Ž5. be solvable. We will consider here the stability of U, when perturbations in the boundary data are also allowed. Denoting these by f Žx, t ., Ž8 2 . becomes
Ž x, t . g V = Rq.
¨ Ž x, t . s f Ž x, t . ,
Ž 44 .
Then from Ž30., by using the Friedrichs inequality and Ž18., we obtain V˙ F y2 mg 1V q C Ž t . ,
Ž 45 .
where CŽ t. s
H V
ž
L
L
q g 2 L2 dS,
/
n
Ž 46 .
in which g 1 , g 2 are the positive constants in the Friedrichs inequality Žsee Appendix 1., the second term in the integrand being the outward normal derivative. On integrating, it turns out that V Ž t . F V Ž 0 . ey2 mg t q ey2 mg t
t
H0 c Ž t . e
2 mgt
dt .
Ž 47 .
On setting S Ž t . s sup C Ž t . w0, t x
Ž 48 .
it turns out that V Ž t . F V Ž 0 . ey2 mg t q
SŽ t . 2 mg
Ž 1 y ey2 mg t . .
Ž 49 .
Therefore, because of Ž22., V Ž 0. -
m 2
«,
Ž 50 .
S Ž t . - m g« 2
imply
HV¨
2
dV F 2 « ,
Ž 51 .
i.e., the continuous dependence in the L2-norm on w0, t x. Moreover, lim S Ž t . s 0
tª`
Ž 52 1 .
129
PROPERTIES OF A DIFFUSION MODEL
implies that U is the asymptotic state in the L2-norm of any solution of Ž1. ] Ž2. and on V .
u Ž x, t . s u1 Ž x . q f Ž x, t .
Ž 52 2 .
By using the methodology used in proving Theorem 2, conditions ensuring the continuous dependence and the asymptotic behaviour in the L2Ž nq1. norm can be obtained. Analogously, conditions ensuring continuous dependence in the pointwise norm can be found. Remark 6. We notice that the Poincare ´ inequality holds also on noncompact domains, bounded at least in one direction. Therefore Theorems 1]3 and Remark 2 continue to hold also for these domains, at least with respect to perturbations spatially periodic in the direction in which the domains are unbounded.
4. LOGARITHMIC CONVEXITY CONSIDERATIONS In this section we establish, under appropriate assumptions, the logarithmic convexity of the Liapunov functional V Ž t . defined by Ž29. in the context previously described, both for the forward and backward in time equation. Recalling that ŽŽ19., Ž30.. V˙ s
HVG ¨
¨ t
dV s y
HV Ž =G . ¨
2
dV ,
Ž 53 .
where G¨ s Gr ¨ etc. Differentiating Ž532 . we obtain V¨Ž t . s y2
HV =G
¨
? =G¨ t d¨ s 2
HV DG G ¨
¨t
dV
s 2 G¨ ¨ ¨ t2 dV ,
Ž 54 .
H
where Ž7., Ž8 2 ., Ž20., etc. have been used together with the divergence theorem. Now using Ž53. ] Ž54. together with Schwarz’s inequality, we obtain 2
VV¨ y V˙ 2 s 2
G dV
HV
G¨ ¨ ¨ t2 dV y
HV
G 2 G¨ ¨¨ t2 dV
H
HV Ž G
¨¨
.
½H
y1
G¨ ¨ t dV
V
5
Ž GG¨ ¨ y 12 G¨2 . dV
,
Ž 55 .
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FLAVIN AND RIONERO
recalling that G¨ ¨ ) 0 in view of Ž18. and Ž21.. From this it follows that VV¨ y V˙ 2 G 0
Ž 56 .
GG¨ ¨ y 12 G¨2 G 0
Ž 57 .
provided that
Ži.e., that 'G is convex w.r.t. ¨ }assuming G / 0.. It is easily shown that the foregoing result continues to hold in the context of the backward in time equation, i.e., when y¨ t replaces ¨ t in Ž7.. Thus we have: THEOREM 4. In the context of the initial boundary ¨ alue problem defined by Ž7. ] Ž8., subject to Ž18., or for its counterpart for the backward in time equation Ž wherein y¨ t replaces ¨ t in Ž7.., suppose that there is no ¨ alue of t for which ¨ ' 0, ;x g V, and that G1r2 Ž ¨ , ? . s G Ž ¨ , ? .
'
Ž ) 0.
is a con¨ ex function of ¨ g Rq. Then VŽ t. s
HVG Ž ¨ , ?. dV
is a logarithmically con¨ ex function1 of t. Remark 7. Condition Ž57., concerning G, arising in the foregoing, is satisfied, for example, if the diffusivity k Ž u. s k 0 Ž 1 q « u2 n . , where k 0 , « are positive constants and n is a positive integer. Two implications of the foregoing result are now discussed. If the conditions of the foregoing theorem hold, logarithmic convexity implies Žcf. w4x. that V Ž t . F V Ž t1 .
Ž t 2 yt .r Ž t 2 yt 1 .
V Ž t2 .
Ž tyt 1 .r Ž t 2 yt 1 .
Ž 58 .
for 0 F t 1 F t F t 2 . Suppose that, for some fixed T, ¨ Ž x, t . ' 0, ¨ Ž x, T . ' 0. 1
Its logarithm is convex.
;x g V , for each t , 0 F t F T ,
Ž 59 .
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PROPERTIES OF A DIFFUSION MODEL
In Ž58. write t 1 s 0, t 2 s T y « , where « is a constant. T ) « ) 0 to obtain V Ž t . F V Ž 0.
Ž Ty « yt .Ž Ty « .y1
VŽT y « .
t Ž Ty « .y 1
.
Ž 60 .
Letting « ª 0, one obtains, in view of Ž59. and the positi¨ e definiteness of G, that for 0 F t F T VŽ t. ' 0 and that ¨ ' 0.
Thus supposition Ž59. cannot hold. Similarly, the following cannot hold: For some fixed T, T1 , ¨ Ž x, t . ' 0,
;x g V for each t , T F t F T1 ,
¨ Ž x, T . ' 0.
Thus one has the following THEOREM 5. In the context of the initial boundary ¨ alue problem gi¨ en by Ž7. ] Ž8. etc., or that of its backward in time counterpart, and subject to the restrictions Ž18. and Ž57., one has ¨ Ž x, T . ' 0,
;x g V ,
for some Ž finite. time T ŽG 0., implies that ¨ ' 0,
;x g V , ; t ,
i.e., one has uniqueness of the zero solution. Remark 8. This provides, inter alia, a complement to our earlier results which imply asymptotic decay to zero, in various measures, of the solution Žfor the forward in time problem.: the foregoing proposition establishes that the solution cannot decay identically to zero in finite time. A continuous dependence result also follows for the backward in time equation, using Ž58. etc. THEOREM 6. For the backward in time counterpart of the i.b.¨ . p. Ž7. ] Ž8. etc., subject to the restrictions Ž18. and Ž57., one has the following: pro¨ ided that, for some fixed T, V Ž T . F M,
Ž 61 .
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FLAVIN AND RIONERO
where M is a gi¨ en positi¨ e constant, then V Ž t . F M t r T V Ž 0.
1y trT
Ž 62 .
for 0 F t F T, establishing Holder continuous dependence, on the initial ¨ data, for the Ž zero. solution of Ž7. ] Ž8.. In the foregoing, the constant M may be related to the relevant melting temperature Žcf. w10x.. It is, of course, known that, in the absence of a restriction of type Ž61., the relevant i.b.v.p. is ill-posed w10x.
5. ANOTHER APPROACH TO ASYMPTOTIC STATES In this section we introduce a different Liapunov functional Žof type 2., yet one which is related to the first one used in Section 3, and exponential asymptotic stability etc. is established in this measure. The proof is especially simple and it is this simplicity which makes it attractive. Moreover, in one dimension, exponential asymptotic stability in the pointwise norm is implied. In the original context ŽŽ1. ] Ž9. etc.. we introduce the Liapunov functional EŽ t . s
1 2
HV Ž =L .
2
dV .
Ž 63 .
In view of Ž8 2 ., Ž9., and the Poincare ´ inequality Žcf. Appendix 1. there is a positive constant g such that E Ž t . G 12 g
2
HVL
dV .
Ž 64 .
In view of this and Ž11., E is positive definite with respect to ¨ , and thus serves as a Liapunov stability measure for the context in question. Along the solutions of Ž7. ] Ž8. we obtain, using the divergence theorem etc., E˙Ž t . s
HV =L ? =L
t
dV s
L
H V L =L ? n dS y HV n ¨ t
t
D L dV . Ž 65 .
But L s 0 on V « L t s 0 on V , L s FX Ž U q ¨ . ) m n
Ž 66 .
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PROPERTIES OF A DIFFUSION MODEL
on recalling Ž9. and Ž18.. Using Ž7., Ž65., and Ž66. one obtains E˙ s ym
HV Ž D L .
2
dV .
Ž 67 .
But L s 0 on V implies
g
HV Ž =L .
2
dV F
HV Ž D L .
2
dV ,
Ž 68 .
where g is the constant in the Poincare ´ inequality arising previously Žsee . Ž . Ž . Ž . Appendix 1 . Thus 63 , 67 , and 68 yield E˙Ž t . F y2 mg E Ž t . .
Ž 69 .
E Ž t . F E Ž 0 . ey2 mg t ,
Ž 70 .
Integration of this yields
wherein EŽ0. is given in terms of data by E Ž 0. s
HV
F X Ž ¨ 0 q U . = Ž U q ¨ 0 . y F X Ž U . =U
s
HV
F X Ž ¨ 0 q U . y F X Ž U . 4 =U q F X Ž ¨ 0 q U . =¨ 0
2
dV 2
dV . Ž 71 .
The foregoing result may be summarized thus: THEOREM 7. Let Ž18. hold and let Ž4. ] Ž5. be sol¨ able. Then U is asymptotically exponentially stable in the measure Ž65. Ž as con¨ eyed by Ž70. ] Ž71.. and is the asymptotic state of any solution of Ž1. ] Ž3. in this measure. It is readily verified that Ž70. implies that
HV
L2 q Ž =L .
2
dV F KE Ž 0 . ey2 mg t ,
Ž 72 .
where K is a computable constant. This implies a slight variant of the previous theorem, under the same underlying assumptions: THEOREM 7a. The steady state U is asymptotically exponentially stable in the H 1-norm of L and is the asymptotic state of any solution of Ž1. ] Ž3. in this norm.
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FLAVIN AND RIONERO
We conclude this section with two remarks: Remark 9. Consider a fixed, one-dimensional domain 0 F x F l, to which the previous analysis is applicable mutatis mutandis. The steady state is exponentially asymptotically stable in the pointwise norm. That this is so is a consequence of Ž70. together with the readily verified inequalities m2 ¨ 2 Ž x, t . F L2 Ž x, t . F
žH
2
x
L x dx
0
/
F x2
l 2
H0 L
x
F 2 l 2EŽ t . .
Remark 10. Using the considerations of this section, it is also possible to deduce pointwise decay to zero as t ª ` of ¨ , in R 3 : Note that Žin standard norm notation. Ži. it follows from Ž67. that t
H0
D L Ž x, t .
2 2
dt F my1 E Ž 0 . ;
Žii. one has the Sobolev inequality sup L Ž x, ? . F C D L Ž x, ? . V
2,
where C is a constant depending on V Že.g., w13x.. Using these and Ž26 2 . ] Ž26 3 . one finds that t
H0 ½ sup V
¨ Ž ?, t .
2
5
dt F c 2 my3 E Ž 0 . .
It follows from this, bearing in mind the assumed continuity of ¨ , that sup ¨ Ž ?, t . ª 0 V
as t ª `.
6. CASE OF MOVING BOUNDARIES In this section we consider an i.b.v.p. in one spatial dimension with moving boundaries and zero boundary values Že.g., of temperature.. Some stability and decay results are established for certain measures Žor norms. of type 2. These depend on a lemma given in Appendix 2 and also in w4x. Suppose ut s F Ž u.
xx,
xy Ž t . - x - xq Ž t . , t g Rq,
Ž 73 .
u Ž x, 0 . s u 0 Ž x . ,
Ž 74 .
u Ž xq Ž t . , t . s u Ž xy Ž t . , t . s 0.
Ž 75 .
PROPERTIES OF A DIFFUSION MODEL
135
It is supposed that xq Ž t ., xy Ž t . are single-valued, continuously differentiable functions such that xq Ž t . ) xy Ž t ., and that xq? Ž t . G 0,
xy? Ž t . F 0
Ž 76 .
Ž‘‘expanding’’ region.. Moreover, assumption Ž18. again obtains. Suppose that f Ž w . }arising in the Lemma in Appendix 2}is a twice continuously differentiable con¨ ex function Ži.e., f Y G 0. such that f Ž0. s 0; for such a function wf X Ž w . y f Ž w . G 0.
Ž 77 .
In what follows it is natural}though not necessary}to think of f Ž w . as a positive definite function of its argument. Defining JŽ t. s
HG f Ž F . dx
Ž 78 .
x
t
it follows from the Lemma in Appendix 2, Ž77., and Ž18. that J ?Ž t . s
HG f
Y
Ž Fx .
t
HG f
F ym
Y
F u
Fx2x dx
Ž Fx . Fx2x dx.
Ž 79 .
t
Thus we have THEOREM 8. For the i.b.¨ . p. with mo¨ ing boundaries, defined by Ž73. ] Ž75., subject to the restrictions on the boundary motion stated in the second paragraph of this section, the measure defined by Ž78. Ž wherein f Ž w . is a con¨ ex function of its argument such that f Ž0. s 0. is a nonincreasing function of the time t. If f is positive definite in its argument the foregoing proposition may be viewed as a stability estimate. As an example, take
f Ž w. s w2n, n being a positive integer. The foregoing implies 1r2 n
HG
t
Fx2 n
dx
1r2 n
F
HG
0
Fx2 n
dx
,
t G 0.
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FLAVIN AND RIONERO
Letting n ª `, we obtain THEOREM 9. restrictions
For the i.b.¨ . p. defined by Ž73. ] Ž76. subject to the stated sup Fx Ž x, t . F sup Fx Ž x, 0 . , x
t G 0.
Ž 80 .
x
As Fx is interpretable as heat flux in the context of standard heat flow, this means that the absolute value of the heat flux is a nonincreasing function of time. As another example, take
f Ž w. s w2 and Ž79. takes the form J ? Ž t . F y2 m
2 xx
HG F
dx.
Ž 81 .
t
Hence J ? Ž t . F y2 mp 2 ly2 Ž t .
2 x
HG F
dx,
Ž 82 .
t
where l Ž t . denotes the width of the region at time t; this follows on noting that HGt Fx dx s 0 and on using the well-known Poincare ´ inequality L
H0 F
2 xx
dx dx G p 2 Ly2
L
H0 F
2 x
dx
for smooth functions F Ž x . such that L
H0 F
x
dx s 0.
Integration of Ž82. yields a global decay Žasymptotic stability . result, whence pointwise decay Žasymptotic stability . also follows: THEOREM 10. restrictions
For the i.b.¨ . p. defined by Ž73. ] Ž76. subject to the stated JŽ t. s
2 x
HG F
dx
Ž 83 .
t
satisfies J Ž t . F J Ž 0 . exp y2p 2 m
t y2
H0 l
Ž tX . dtX .
Ž 84 .
137
PROPERTIES OF A DIFFUSION MODEL
Further, ' a positi¨ e constant C. u Ž x, t . F C J 1r2 Ž 0 . exp yp 2 m
t y2
H0 l
Ž tX . dtX .
Ž 85 .
That Ž85. follows from Ž84. is a simple consequence of Schwarz’s inequality and the zero boundary conditions satisfied by u. It is possible to enhance the decay rate in Ž84. if one retains the boundary terms on applying the Lemma of Appendix 2 to Ž83.. Specifically one obtains xsxq
J ? Ž t . F y2 m
HG
Fx2x dx y Fx2 x ? xsxy
t
F y2 m l Ž t . J Ž t . ,
Ž 86 .
where lŽ t . is the greatest value for which the inequality 2 xx
HG F
dx q Ž 2 m .
y1
xsxq
Ž x ? . Fx2
xsxy
t
G lŽ t .
2 x
HG F
dx
Ž 87 .
t
holds in the class of smooth F Ž x .. F Ž xq . s F Ž xy . s 0. This value may be obtained, by standard variational means Že.g., w11x., as the lowest Žpositive. eigenvalue of a problem consisting of a fourth order differential equation subject to the two zero boundary conditions and two natural ones. Integration of Ž86. yields an enhancement of Ž84. ] Ž85. with lŽ t . ) p 2 ly2 Ž t .. It may also be easily proven that lŽ t . cannot exceed 4p 2 ly2 Ž t ..
´ AND FRIEDRICHS INEQUALITIES APPENDIX 1: POINCARE For three-dimensional regions V with sufficiently smooth boundary V one has the following inequalities: Ži. For arbitrary functions F regular in the region V, vanishing on the boundary V, and such that =F is square integrable, one has
HV Ž =F .
2
dV G g
HVF
2
dV ,
where g is the lowest Žpositive. eigenvalue of DC q lC s 0
in V ,
Cs0
on V .
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FLAVIN AND RIONERO
Žii. One also has
HV Ž D x .
2
dV G g
HV Ž =x .
2
dV
for arbitrary regular functions such that x s 0 on V. This follows from the previous inequality, Schwarz’s inequality, etc. as follows: 1r2
H Ž =x . 2 dV s yH x D x dV F V
HV
V
H Ž D x . 2 dV
x 2 dV
V
1r2
F gy1
HV Ž =x .
2
HV Ž D x .
dV
2
dV
.
Analogues of the foregoing inequalities hold in all dimensions. Žiii. Let V be a domain with Lipschitz boundary. Then for every function u g C 1 Ž V . there exist positive constants g i Ž V . Ž i s 1, 2. such that w11x
HVu
2
dV F gy1 1
HV Ž =u .
2
dV q g 2 gy1 1
H Vu
2
dV .
APPENDIX 2 LEMMA. Consider a region R in the t]x plane bounded Ži. Žii.
by a straight line segment with abscissa t s 0, by two cur¨ es Cq, Cy defined respecti¨ ely by x s xq Ž t . ,
x s xy Ž t . ,
which are Ž a . single-¨ alued continuously differentiable functions, Ž b . such that xq Ž t . ) xy Ž t .. Let Gt denote the straight line segment contained in the region ha¨ ing abscissa t. Let c Ž t, x . be a twice continuously differentiable function defined in the region which takes zero ¨ alues on Cq, Cy. Let f Ž w . be a twice continuously differentiable function of its argument w.
139
PROPERTIES OF A DIFFUSION MODEL
Then in the context and notation of the last two paragraphs the quantity JŽ t. s
HG f Ž c . dx x
t
satisfies J ?Ž t . s
HG
f Y Ž c x . c t c x x dx y c x f X Ž c x . y f Ž c x . 4 x ? Ž t .
xsxq Ž t .
. xsxy Ž t .
t
The result is a simple consequence of the Leibnitz theorem and differentiation of the boundary conditions. A more general version is given in w4x.
ACKNOWLEDGMENTS This research has largely been performed during visits of S. Rionero to the Department of Mathematical Physics, National University of Ireland, Galway, in May 1996 and June 1997. This author ŽS.R.. thanks Ži. the warm hospitality extended to him by N.U.I Galway and Žii. the Italian M.U.R.S.T. Ž40% and 60% contracts.. Both authors acknowledge the G.N.F.M. of the Italian C.N.R. for the help given toward their scientific cooperation. The authors thank the referee for his remarks which have led them to obtain additional results.
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10. L. E. Payne, ‘‘Improperly Posed Problems in Partial Differential Equations,’’ SIAM Regional Conference Series in Applied Mathematics, Vol. 22, 1975. 11. K. Rektorys, ‘‘Variational Methods in Mathematics, Science and Engineering,’’ Reidel, Dordrecht, 1980. 12. S. Rionero and M. Maiellaro, On the stability of Couette]Poiseuille flows in the anisotropic MHD via the Liapunov direct method, Rend. Acad. Sci. Fis. Mat. Napoli 62 Ž1995., 315]332. 13. B. Straughan, ‘‘The Energy Method, Stability and Natural Convection,’’ Springer-Verlag, BerlinrNew York, 1993.