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A Nonexistence Result to a Cauchy Problem in Nonlinear One Dimensional Thermoelasticity
Journal of Mathematical Analysis and Applications 254, 71᎐86 Ž2001. doi:10.1006rjmaa.2000.7203, available online at http:rrwww.idealibrary.com on
A Nonexistence Result to a Cauchy Problem in Nonlinear One Dimensional Thermoelasticity Mokhtar Kirane Faculte´ de Mathematiques et d’informatique, LAMFA UPRES A 6119, Uni¨ ersite´ de ´ Picardie Jules Verne, 33 Rue Saint Leu, 80039 Amiens Cedex 1, France
and Nasser-eddine Tatar Institut de Mathematiques, Uni¨ ersite´ Badji Mokhtar, B.P. 12, Annaba 23000, Algeria ´ Submitted by P. Broadbridge Received May 25, 1999
Considering the Cauchy problem for a nonlinear system arising in thermoelasticity, it is proved that its solution develops singularities in finite time depending on the size and regularity of the initial data. The present work is distinguished from a previous work by relaxing the requirements on the initial data and allowing for a slightly more general and nonautonomous forcing term besides permitting the insertion of gradient terms in both equations of the system. 䊚 2001 Academic Press
1. INTRODUCTION Over the last decades, there has been a large literature on results concerning local existence, global existence, uniqueness, and large time behavior of classical solutions to certain initial value problems of nonlinear thermoelasticity with smooth and small Žin some norms. initial data Žsee w11, 13᎐15x.. Whereas one fails to prove existence of global solutions for large initial data Žin some sense. Žsee w1᎐3, 7, 8, 10x., recently, S. Messaoudi w12x showed that, for a particular type of semi-linear thermoelastic equations, weak solutions collapse in finite time. Here our result improves Messaoudi’s in that we allow for a more general set of initial data. In fact, we eliminate a restrictive condition which was crucial in Messaoudi’s analysis Žsee comments below.. Moreover the class of systems we consider 71 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
72
KIRANE AND TATAR
is a bit larger since we insert gradient terms of the two unknowns in both equations. It is worth noting that the blow up result we obtain is proved for a, suitably constrained, nonlinear nonautonomous forcing term. This is established by choosing an appropriate functional satisfying all the requirements of a lemma in w4x. This lemma, which is crucial in our present treatment, is in fact a compact version of the concavity method of Levine w8x Žsee w6, 9x.. In w5x, the authors have applied this method to hyperbolic problems with dynamic boundary conditions. For convenience and completeness we shall report the lemma in question with its proof in the next section. Section 3 is devoted to our main result. In Section 4 a detailed study for a vanishing initial energy is given. Finally, in Section 5, we give another set of sufficient conditions assuring collapse in finite time. We should mention that these results have been largely motivated by the work of Levine w8x and Kalantarov and Ladyzhenskaya w4x.
2. PRELIMINARIES Let us first state the problem. We consider the following one dimensional semi-linear system
¡u Ž x, t . s au Ž x, t . q b Ž x, t . q du Ž x, t . y mu Ž x, t . ~ qf Ž t , u Ž x, t . . ¢c Ž x, t . s K Ž x, t . q bu Ž x, t . q pu Ž x, t . q q Ž x, t . tt
t
xx
x
xx
x
xt
t
x
Ž 1.
x
with x g R, t ) 0, and u Ž x, 0 . s u 0 Ž x . ,
u t Ž x, 0 . s u1 Ž x . ,
Ž x, 0 . s 0 Ž x . , x g R.
Ž 2. The coefficients are such that a, b, c, K ) 0
Ž 3.
and d, m, p, q G 0. The set R denotes the set of all real numbers. The case of assumption Ž3. and d s m s p s q s 0 has been studied in Messaoudi w12x. We are interested in finding sufficient conditions causing collapse in finite time for the solution of the above system. As for existence on a maximal time interval, uniqueness, and regularity we refer to Slemrod w14x.
73
A CAUCHY PROBLEM IN THERMOELASTICITY
Next we state and prove the basic lemma Ždue to Kalantarov and Ladyzhenskaya w4x. which provides the method of proof for our result. LEMMA 1. Assume that a twice differentiable, positi¨ e function ⌿ Ž t . satisfies for all t G 0 the inequality 2
⌿ Ž t . ⌿⬙ Ž t . y Ž 1 q ␥ . Ž ⌿⬘ . Ž t . G y2C1⌿ Ž t . ⌿⬘ Ž t . y C2⌿ 2 Ž t . , Ž 4 . where ␥ ) 0 and C1 , C2 G 0. Then Ža. if ⌿ Ž0. ) 0, ⌿⬘Ž0. q ␥ 2 ␥y1 ⌿ Ž0. ) 0, and C1 q C2 ) 0 we ha¨ e ⌿ Ž t . ª q⬁ as t ª t1 F t 2 s
1
ln
2 C12 q ␥ C2
'
ž
␥ 1⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 . ␥ 2⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 .
/
,
Ž 5.
where ␥ 1 s yC1 q C12 q ␥ C2 , ␥ 2 s yC1 y C12 q ␥ C2 , Žb. if ⌿ Ž0. ) 0, ⌿⬘Ž0. ) 0, and C1 s C2 s 0, then ⌿ Ž t . ª q⬁ as
'
'
t ª t1 F t 2 s
⌿ Ž 0.
␥ ⌿⬘ Ž 0 .
.
Proof. If we set ⌽ Ž t . s ⌿y␥ Ž t ., then it is easy to see using Ž4. and ⌽⬘ Ž t . s y␥ ⌽⬙ Ž t . s y
␥ ⌿
2q ␥
Ž t.
⌿⬘ Ž t . ⌿ 1q ␥ Ž t .
,
⌿⬙ Ž t . ⌿ Ž t . y Ž 1 q ␥ . Ž ⌿⬘ Ž t . .
2
that ⌽ is the solution of the second order differential equation ⌽⬙ Ž t . q 2C1⌽⬘ Ž t . y C2 ⌽ Ž t . s f Ž t . F 0
for all t G 0.
Ž 6.
Thus, for C1 q C2 ) 0 ⌽ Ž t . s  1 e␥ 1 t q  2 e␥ 2 t q Ž ␥ 1 y ␥ 2 . F  1 e␥ 1 t q  2 e␥ 2 t
y1
t
H0 f Ž . w e
for all t G 0,
␥ 1Ž ty .
y e␥ 2 Ž ty . x d
74
KIRANE AND TATAR
where  1 and  2 are defined by the system
½
1 q  2 s ⌽ Ž 0.  1 ␥ 1 q  2 ␥ 2 s ⌽⬘ Ž 0 . .
Clearly, we obtain y1
1 s yŽ ␥ 1 y ␥ 2 . 2 s Ž ␥ 1 y ␥ 2 .
y1
␥ 2⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 . ⌿y1y ␥ Ž 0 . - 0
␥ 1⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 . ⌿y1y ␥ Ž 0 . ) 0.
We infer from the signs of  1 ,  2 , ␥ 1 , ␥ 2 , and the fact ⌽ Ž t . F  1 e␥ 1 t q  2 e␥ 2 t that ⌽ Ž t . must vanish for some finite time t 1. Indeed, the function  1 e␥ 1 t q  2 e␥ 2 t vanishes at the time t 2 Žsee Ž6.. and becomes negative afterwards. This implies that ⌿ Ž t . ª q⬁ as t approaches t 1 estimated by Ž5.. For the second case Žb., Ž6. implies ⌽⬙ Ž t . F 0. By integration we have ⌽ Ž t . F ⌽ Ž 0 . q ⌽⬘ Ž 0 . t and by definition of ⌽ Ž t . ⌿ ␥ Ž t . G ⌿ ␥ Ž 0 . 1 y ␥ t ⌿y1 Ž 0 . ⌿⬘ Ž 0 .
y1
.
Since ⌿ Ž0. ) 0 and ⌿⬘Ž0. ) 0, the conclusion of the lemma follows easily. This completes the proof. Finally, we shall make use of the -inequality ab F a2 q
1 4
b2 ,
for all a, b g R, ) 0
Ž 7.
0 g H 1 Ž R.
Ž 8.
and the fact that for u0 g H 2 Ž R. ,
u1 g H 1 Ž R . ,
uŽ⭈, t . and its spatial derivatives up to order 2 are continuous and vanish at infinity.
A CAUCHY PROBLEM IN THERMOELASTICITY
75
3. BLOW UP FOR NEGATIVE INITIAL ENERGY Before stating our first result let us multiply the first equation of Ž1. by u t and the second equation by , then integrate over R = w0, t x q⬁
t
Hy⬁ H0 u u t
y au t u x x y du t u x q mu 2t y u t f Ž t , u . 4
tt
q⬁
t
Hy⬁ H0 c y K
q
t
xx
y p u x y qx 4 d dx s 0.
Ž 9.
We suppose that for fixed t, f Ž t, u. is the Frechet derivative of some ´ functional F Ž t, u., so that d dt
F Ž t , u . s Ft Ž t , u . q f Ž t , u . u t .
Ž 10 .
Taking this into account, Ž9. becomes q⬁
Hy⬁
½
1 2
q
a
u 2t q q⬁
2
u 2x y F Ž t , u . q
t
Hy⬁ H0 mu s
q⬁
Hy⬁
½
1 2
2 t
c 2
2 dx
5
q K x2 q F Ž , u . y du t u x y p u x 4 d dx
u12 q
a 2
u 20 x y F Ž 0, u 0 . q
c 2
02 dx.
5
Ž 11 .
Now let EŽ t . s
q⬁
Hy⬁
½
1 2
u 2t q
a 2
u 2x y F Ž t , u . q
c 2
2 dx.
5
Ž 12 .
Then Ž11. may be written EŽ t . q
q⬁
t
Hy⬁ H0 mu
2 t
q K x2 q F Ž , u . y du t u x y p u x 4 d dx s E Ž 0 . .
In addition to Ž10. we require f Ž t, u. to satisfy uf Ž t , u . G 2 Ž 1 q 2␥ . F Ž t , u .
Ž 13 .
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KIRANE AND TATAR
for some ␥ G Ž1r4.ŽŽ1 q b 2rac.1r2 y 1., Ft Ž t , u . G 2 Ž 2 y m . F Ž t , u .
Ž 14 .
with 2 y m greater than or equal to the positive root of
Ž 4 amc . 2 y Ž cd 2 . y p 2 m s 0
Ž 15 .
Ž Ft Ž t, u. G 0 in case m s p s d s 0., or Ft Ž t , u . G 2 Ž ˜2 y m . F Ž t , u . q
p2 4 3
u2
Ž 16 .
with ˜2 the positive root of
Ž 4 a. 2 y Ž 4 am . y d 2 s 0 Ž Ft Ž t, u. G Ž p 2r4K . u 2 in case m s d s 0.. The initial data are, for the moment, such that u 0 / 0 and q⬁
Hy⬁ u
dx F
2 0
2␥
q⬁
H uu m y⬁ 0
1
dx.
Ž 17 .
A simple example of f satisfying the above hypotheses is f Ž t, u. s e t u p , p ) 1 with an appropriate constant . THEOREM 1. Suppose that u 0 , u1 , and satisfy Ž8., Ž17. and are such that E Ž0. - 0. Suppose further that f Ž t, u. satisfies Ž10., Ž13., and Ž14. or Ž10., Ž13., and Ž16.. Then there exists a positi¨ e number T - q⬁ such that limy
tªT
q⬁
Hy⬁ u
2
Ž t , x . dx s q⬁.
2 Ž . 2 where  and t 0 Proof. Let us introduce ⌿ Ž t . s Hq⬁ y⬁ u dx q  t q t 0 are to be determined later. Then
⌿⬘ Ž t . s 2
q⬁
žH
y⬁
uu t dx q  Ž t q t 0 .
/
and ⌿⬙ Ž t . s 2
ž
q⬁
Hy⬁ Ž u
2 t
q uu t t . dx q  .
/
77
A CAUCHY PROBLEM IN THERMOELASTICITY
We insert the values of ⌿, ⌿⬘, and ⌿⬙ into the expression ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 s 2⌿
q⬁
½H
y⬁
Ž u 2t q uu t t . dx q 
y 4Ž 1 q ␥ .
q⬁
½H
y⬁
5 2
uu t dx q  Ž t q t 0 .
5
.
By the Cauchy᎐Schwarz inequality we have
½
2
q⬁
Hy⬁ uu F
t
dx q  Ž t q t 0 . q⬁
½ žH
y⬁
5
1r2 2
u dx
q⬁
/ žH
y⬁
2
1r2
u 2t
dx
q  Ž t q t0 . 
/
'
'
5
.
Applying Holder’s inequality to the right hand side we find q⬁
½H
y⬁
2
uu t dx q  Ž t q t 0 .
5
F⌿
q⬁
½H
y⬁
u 2t dx q  .
5
Therefore
⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G 2⌿ y Ž 1 q 2␥ .
½
ž
q⬁
Hy⬁ u
2 t
dx q  q
/
q⬁
Hy⬁ uu
tt
5
dx .
Next, from the first equation of Ž1. and an integration by parts we get ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G 2⌿ y Ž 1 q 2␥ .
½
ž
q⬁
Hy⬁ u
2 t
q ya
/
q⬁
Hy⬁ u
y
m 2
2 x
⌿⬘ Ž t . q
q⬁
Hy⬁ u dx
dx y b
q⬁
x
Hy⬁ uf Ž t , u . dx
5
.
78
KIRANE AND TATAR
The -inequality Ž7. implies ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2
½
G 2⌿ y Ž 1 q 2␥ . y 1
ž
q⬁
Hy⬁ u
q⬁
Hy⬁
2
2 t
q y aq
ž
/
m
dx y
2
⌿⬘ Ž t . q
b2 4 1
/
q⬁
Hy⬁ u
2 x
dx
q⬁
Hy⬁ uf Ž t , u . dx
5
.
Finally the definition Ž12. of E Ž t . yields ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2
½
G 2⌿ y Ž 1 q 2␥ . E Ž t . q 2 a␥ y
ž
q c Ž 1 q 2␥ . y 1 q
q⬁
Hy⬁
q⬁
Hy⬁
2
b2 4 1
/
q⬁
Hy⬁ u
2 x
dx
dx
uf Ž t , u . y 2 Ž 1 q 2␥ . F Ž t , u . dx y
m 2
5
⌿⬘ y  Ž 1 q 2␥ . . Ž 18 .
Assumption Ž13. simplifies the right hand side of Ž18. to ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2
½
G 2⌿ y2 Ž 1 q 2␥ . E Ž t . q 2 a␥ y
ž
q⬁
q c Ž 1 q 2␥ . y 1
Hy⬁
2
b2 4 1
dx y
m 2
/
q⬁
Hy⬁ u
2 x
dx
5
⌿⬘ y  Ž 1 q 2␥ . . Ž 19 .
q⬁ 2 We pick 1 s b 2r8 a␥ , so that the coefficient of Hy⬁ u x dx vanishes. It is now easy to see that the assumption on ␥ in Ž13. implies the positivity of cŽ1 q 2␥ . y 1. These considerations imply
⌿⌿⬙ y Ž 1 q ␥ . Ž ⌿⬘ .
2
G 2⌿ y2 Ž 1 q 2␥ . E Ž t . y
½
m 2
⌿⬘ y  Ž 1 q 2␥ . .
5
Ž 20 .
79
A CAUCHY PROBLEM IN THERMOELASTICITY
Case 1. If Ž14. is satisfied, we differentiate E Ž t . and use the inequality Ž7. to get E⬘ Ž t . s
q⬁
Hy⬁ ymu
F Ž 2 y m. q
1 4
ž
2 t
y K x2 y Ft Ž t , u . q du t u x q p u x 4 dx
q⬁
Hy⬁ u
d2
2
p2
q
3
dx y
2 t
/
q⬁
q⬁
Hy⬁ F Ž t , u . dx q Hy⬁
q⬁
Hy⬁ u
t
2 x
3
2
dx
dx
or E⬘ Ž t . F 2 Ž 2 y m . E Ž t . q q
q⬁
Hy⬁
1
d2
4
2
½ž
q
p2
3
/
y aŽ 2 y m .
5
q⬁
Hy⬁ u
2 x
dx
2 Ž 2 y m . F Ž t , u . y Ft Ž t , u . dx
q 3 y Ž 2 y m. c4
q⬁
Hy⬁
2
dx.
Ž 21 .
Ži. If m s 0 and p s d s 0, then it is easy to see from the first equality of Ž3.17. that when Ft Ž t, u. G 0 we have E⬘Ž t . F 0. Hence E Ž t . F E Ž0. for all t G 0. Žii. If m s 0 and p q d ) 0, then we can choose 3 s 2 c and pick 2 so that Ž1r4.Ž d 2r 2 q p 2r 3 . y a 2 - 0. Clearly 2 ) ŽŽ cd 2 q p 2 .r4 ac .1r2 will do. We obtain E⬘Ž t . F 2 2 E Ž t . and again E Ž t . F E Ž0. for all t G 0. Žiii. If m / 0, then we put 3 s Ž 2 y m. c and choose 2 such that 2 y m ) 0 and M[
1 4
ž
d2
2
q
p2
Ž 2 y m. c
/
y a Ž 2 y m . F 0.
This can be done as follows. For 2 ) m, since m / 0 M-
1 4
ž
d2 m
q
p2
Ž 2 y m. c 2
/
y aŽ 2 y m .
- Ž y4 amc . Ž 2 y m . q cd 2 Ž 2 y m . q p 2 m r 4 mc Ž 2 y m . ,
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KIRANE AND TATAR
it suffices to take 2 as in Ž15. to obtain E⬘Ž t . F 2Ž 2 y m. E Ž t . and consequently E Ž t . F E Ž0. - 0 for all t G 0. Case 2. If Ž16. holds, then we use the expression E⬘ Ž t . s
q⬁
Hy⬁ ymu
2 t
y K x2 y Ft Ž t , u . q du t u x y px u4 dx
F 2 Ž ˜2 y m . E Ž t . q qŽ 3 y K . q⬁
q
Hy⬁
½
q⬁
Hy⬁
2 x
d2 4˜2
y a Ž ˜2 y m .
dx y Ž ˜2 y m .
2 Ž ˜2 y m . F Ž t , u . q
p2 4 3
q⬁
Hy⬁ u
q⬁
Hy⬁
2
2 x
dx
dx
5
u 2 y Ft Ž t , u . dx.
Choosing 3 s K and ˜2 y m G 0 such that d 2r4˜2 y aŽ ˜2 y m. s 0, that is, ˜2 is the positive root of Ž4 a. 2 y Ž4 am. y d 2 s 0 and assuming Ž16., we again obtain E Ž t . F E Ž0. for all t G 0. Now back to Ž20., from which it results ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G 2⌿ y Ž 1 q 2␥ . Ž  q 2 E Ž 0 . . y
½
m 2
⌿⬘ .
5
We choose  ) 0 such that  q 2 E Ž0. F 0. Thus ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G ym⌿⌿⬘.
Ž 22 .
The inequality Ž4. in Lemma 1 is therefore satisfied with C2 s 0 and C1 s mr2. If m s 0, we choose t 0 so that ⌿⬘Ž0. ) 0 and apply part Žb. of Lemma 1. The condition ⌿ Ž0. ) 0 holds since u 0 / 0. It remains to check that ⌿⬘Ž0. ) m␥y1 ⌿ Ž0., that is, q⬁
Hy⬁ u
2 0
dx q  t 02 -
2␥ m
q⬁
žH
y⬁
u 0 u1 dx q  t 0
/
which may be written as a second order inequality in t 0
 t 02 y
2␥ m
t0 q
q⬁
žH
y⬁
u 02 dx y
2␥
q⬁
H uu m y⬁ 0
1
dx - 0.
/
Ž 23 .
An easy computation shows that we may choose t 0 satisfying Ž23. as long as assumption Ž14. holds. ŽSee Remark 1 below.. All the requirements of Lemma 1 are now fulfilled and the conclusion follows.
81
A CAUCHY PROBLEM IN THERMOELASTICITY
Remark 1. Observe from Ž23. that the condition Ž17. can be weakened. In fact we only need to choose  such that m2
␥
2
q⬁
Hy⬁ u
2 0
dx y
2␥
q⬁
H uu m y⬁ 0
1
dx F  F y2 E Ž 0 . .
Remark 2. The escape time is estimated according to Lemma 1 as
TyF t 2 s
1 m
2␥ ln 2␥
q⬁
žH
y⬁
q⬁
žH
y⬁
u 0 u1 dx q  t 0
u 0 u1 dx q  t 0 y m
q⬁
/ žH
y⬁
/ u 02 dx q  t 02
/
and when m s 0, the second part of Lemma 1 applies giving q⬁
TyF t 2 s
Hy⬁ u 2␥
ž
2 0
dx q  t 02 .
q⬁
Hy⬁ u u 0
1
dx q  t 0
/
Remark 3. It is of interest to note that even when E Ž0. s 0 Ž E Ž0. G 0 in case m s p s d s 0., if 5 u x 5 2 Žor 5 5 2 . grows unboundedly then 5 u 5 2 must blow up in finite time. This follows from Ž19. by taking 1 s Ž1 q 2 2␥ . c. The hypothesis on ␥ in Ž13. implies that the coefficient of Hq⬁ y⬁ u x dx is positive. After a certain time Žwhich then can be considered as the new starting time. the expression y Ž 1 q 2␥ . Ž  q 2 E Ž 0 . . q 2 a␥ y
ž
b2 4 1
/
q⬁
Hy⬁ u
2 x
dx
becomes Žand remains. positive yielding Ž22.. Comments. For f Ž t, u. s f Ž u. s u ␣ , ␣ ) 1, and d s m s p s q s 0 we find the system in w12x. The argument above works Žsee Remarks 1, 2, and Ž13.. for the same values of ␣ . But here we do not impose the w x condition Hq⬁ y⬁ u 0 u1 dx ) 0 as in 12 since it will always be possible to q⬁ choose t 0 satisfying Hy⬁ u 0 u1 dx q  t 0 ) 0. Observe that in this case assumption Ž17. disappears. The following proposition Žwhich is due to Levine and Sacks w9x. shows that we can find sufficient conditions under which initial data provide negative initial energy.
82
KIRANE AND TATAR
PROPOSITION 1. Suppose that lim uªq⬁ f Žuu . \ lim uªq⬁ g Ž u. s q⬁, u 0 g H 1 ŽR., u1 g L2 ŽR., and 0 g L2 ŽR.. Then there exists a sufficiently large such that E Ž u 0 , u1 , 0 . - 0. Proof. Suppose for contradiction that E Ž u 0 , u1 , 0 . ) 0 for all G 1. Then by the definition of the energy 2
2
q⬁
q⬁
Hy⬁ F Ž u . dx F Hy⬁ u 0
2 0x
dx q
1
2
q⬁
Hy⬁ Ž u
2 1
q c 02 . dx F M,
where M is a positive constant independent of . But 1
2
q⬁
Hy⬁ F Ž u . dx 0
s s
1
2
q⬁
u 0
ž
Hy⬁ H0
q⬁
Hy⬁ u
2 0
ž
1
2
/
f Ž s . ds dx u 0
u 20
H0
sg Ž s . ds dx ª q⬁,
/
when ª q⬁.
Here we have used F Ž u. s H0u f Ž s . ds. This is a contradiction. 4. BLOW UP FOR VANISHING INITIAL ENERGY In this section it will be clear that for vanishing initial energy we can obtain blow up in finite time even when condition Ž17. is violated. THEOREM 2. s 0.
Suppose that the hypotheses of Theorem 1 hold with E Ž0.
Ži. If ⌿⬘Ž0. ) m␥ ⌿ Ž0., then 5 u 5 2 blows up in a finite time. Žii. If ⌿⬘Ž0. s m␥ ⌿ Ž0., then either 5 u 5 2 blows up in a finite time or else 5 u5 2 s 5 u0 5 2 e Žiii.
If ⌿⬘Ž0. -
m ␥
m 2␥
t
for all t G 0.
Ž 24 .
⌿ Ž0., then either 5 u 5 2 blows up in a finite time or
else 5 u5 2 F 5 u0 5 2 e
m 2␥
t
for all t G 0.
Ž 25 .
Proof. For E Ž0. s 0 we can take  s 0. Ži. This follows readily from Ž20. and Lemma 1 with C1 s mr2 and C2 s 0.
83
A CAUCHY PROBLEM IN THERMOELASTICITY
Žii. Letting H Ž t . [ ln ⌿ Ž t . y s ⌿ Ž0. implies H⬘Ž0. s 0. m ␥
m ␥
t, we note that the condition ⌿⬘Ž0.
A glance at the proof of Lemma 1 gives  1 s 0,  2 s ⌿y␥ Ž0., and ⌿ Ž t . G ⌿ Ž0. e Ž m r␥ .t , which in turn yields H Ž t . G H Ž0.. Consequently, either H Ž t . ' H Ž0. or else H⬘Ž˜t . ) 0 for some ˜t ) 0. In the latter case we obtain blow up from Theorem 1 since the argument there is applicable for ˜t and the collapse occurs at a time t1 ) ˜t. In the former case, ln ⌿ Ž t . y m␥ t s ln ⌿ Ž0. for all t G 0, implying ⌿ Ž t . s ⌿ Ž0. e Ž m r␥ .t for all t G 0. Žiii. If ⌿⬘Ž0. - m␥ ⌿ Ž0., then H⬘Ž0. - 0. It follows, by continuity, that if there does not exist a finite time ˆt such that H⬘Žˆt . s 0, then we may assume that H⬘Ž t . - 0 for all t G 0. By Žii., H⬘Žˆt . s 0, for some finite ˆt, may lead to ⌿ Ž t . s ⌿ Ž0. e Ž m r␥ .t from which we infer ⌿⬘Ž0. s m␥ ⌿ Ž0.. This is a contradiction. Hence there only remains the collapse of 5 u 5 2 at a finite time. If H⬘Ž t . - 0 for all t G 0, then H Ž t . F H Ž0. for all t G 0. This means that 5 u 5 2 F 5 u 0 5 2 e Ž m r2␥ .t for all t G 0. The proof of the theorem is complete. Remark 4. If m s 0 then in case Žii., ⌿⬙ Ž t . ' 0 and on the other hand
½
⌿⬙ Ž t . G 2 y2 Ž 1 q 2␥ . E Ž t . q q 2 a␥ y
ž
b2 4
q 2Ž 1 q ␥ .
q⬁
/H
y⬁
q⬁
Hy⬁ u
2 t
q⬁
Hy⬁
uf Ž t , u . y 2 Ž 1 q 2␥ . F Ž t , u . dx
u 2x dx q c Ž 1 q 2␥ . y
q⬁
Hy⬁
2
dx
5
dx .
So that for s cŽ1 q 2␥ . and ␥ as in Ž13. we get from the previous q⬁ 2 q⬁ 2 inequality Hy⬁ u t dx s Hy⬁ u x dx and this implies u ' u 0 ' 0. Thus, either the solution blows up or else it coincides with its initial value which must vanish. If m s 0 in case Žiii., then we can see that either the solution blows up or else it is uniformly bounded in the L2-norm Žfor all time..
5. AN ALTERNATIVE TO CONDITIONS Ž14. AND Ž16. In this section we are going to see that conditions Ž14. and Ž13. may be replaced by the condition Ft Ž t , u . G 0,
for all t ) 0.
Ž 26 .
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KIRANE AND TATAR
Indeed, with minor changes in the hypothesis of Theorem 1, namely uf Ž t , u . G 2 Ž 1 q 2␥ . F Ž t , u .
Ž 27 .
for some ␥ G Ž1r4.Ž1 q b 2rac.1r2 ,
ž
1q
2 ␣␥ m
q⬁
/H
y⬁
u 20 dx -
2␥
q⬁
H uu m y⬁ 0
1
dx
Ž 28 .
for some ␣ to be determined later, and E˜Ž 0 . s
q⬁
Hy⬁
½Ž
m q 2␣ . 2
␣ u 20 q
1 2
u12 y ␣ u 0 u1 q F Ž 0, u 0 . q yF
a 2
u 02 x
c 2
5
02 F 0, Ž 29 .
we can prove the THEOREM 3. Assume that conditions Ž8., Ž10., and Ž26. ᎐ Ž29. are satisfied. Then the conclusions of Theorem 1 and Theorem 2 hold. Proof. We introduce the new functions ¨ Ž t . s ey␣ t u Ž t .
and
w Ž t . s ey␣ t Ž t . .
Then ¨ Ž t . and w Ž t . satisfy the system
½
¨ t t s a¨ x x q bwx q d¨ x y Ž m q 2 ␣ . ¨ t y Ž m q ␣ . ␣ ¨ q f˜Ž t , ¨ .
cwt s Kwx x q b¨ x t q Ž ␣ b q p . ¨ x q qwx y ␣ w
Ž 30 .
with ¨ 0 Ž x . s u0 Ž x . ,
and
¨ 1 Ž x . s u1 Ž x . y ␣ u 0 Ž x . ,
Ž 31 .
w0 Ž x . s 0 Ž x . ,
where f˜Ž t, ¨ . s ey␣ t f Ž t, e ␣ t ¨ .. We want to show that Theorem 1 applies for this system. In fact the proof of Theorem 1 carries through almost unchanged. Thus we only mention the main differences. The appropriate energy is E˜Ž t . s
q⬁
Hy⬁
½Ž
mq␣. 2
␣¨2 q
1 2
¨ 12 q
a 2
¨ x2 y F˜Ž t , ¨ . q
c 2
5
w 2 dx ,
85
A CAUCHY PROBLEM IN THERMOELASTICITY
where F˜Ž t, ¨ . s ey2 ␣ tF Ž t, e ␣ t ¨ .. Next we want to check the requirements of Theorem 1 for system Ž22. Ža. ¨ f˜Ž t, ¨ . s ey␣ t uey ␣ t f Ž t, u. s ey2 ␣ t uf Ž t, u. G 2Ž1 q 2␥ . F˜Ž t, ¨ . Žb. from Ž26. and Ža., we have F˜t Ž t , ¨ . s ey2 ␣ tFt Ž t , e ␣ t ¨ . q ␣ ¨ f˜Ž t , ¨ . y 2 ␣ F˜Ž t , ¨ . G ␣ ¨ f˜Ž t , ¨ . y 2 ␣ F˜Ž t , ¨ . G 2 ␣ Ž 1 q 2␥ . F˜Ž t , ¨ . y 2 ␣ F˜Ž t , ¨ . G 4␣␥ F˜Ž t , ¨ . . To fulfill assumption Ž14. it suffices to pick ␣ such that
␣ G Ž X2 y m . r2␥ ,
Ž 32 .
where Ž X2 y m. is greater than or equal to the positive root of the second order polynomial 2 Ž 4 mca. 2 y Ž d 2 c . y m Ž ␣ b q p . .
Note that this polynomial depends on ␣ , yet Ž32. is possible. Indeed, from Ž27. it suffices to choose ␣ greater than or equal to the positive root of the polynomial m Ž b 2 y 16␥ 2 ac . 2 q 2 Ž bpm q ␥ d 2 c . q p 2 m. This is found after an easy computation. The conclusion of Theorem 3 holds with 5 u 5 2 s 5 u 0 5 2 e Ž m r2␥q ␣ .t in Žii. and 5 u 5 2 F 5 u 0 5 2 e Ž m r2␥q ␣ .t in Žiii.. This completes the proof.
REFERENCES 1. B. D. Coleman and M. E. Gurtin, Waves in materials with memory. III. Thermodynamic influences on the growth and decay of acceleration waves, Arch. Rational Mech. Anal. 19 Ž1965., 266᎐298. 2. C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equation of non linear thermoelasticity, Quart. Appl. Math. 44 Ž1966., 463᎐474. 3. W. J. Hrusa and S. Messaoudi, On formation of singularities in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 111 Ž1990., 135᎐151. 4. V. K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, J. So¨ iet Math. 10 Ž1978., 53᎐70. 5. M. Kirane, S. Kouachi, and N. Tatar, Nonexistence of global solutions of some quasilinear hyperbolic equations with dynamic boundary conditions, Math. Nachr. 176 Ž1995., 139᎐147.
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6. R. J. Knops, H. A. Levine, and L. E. Payne, Nonexistence, instability and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal. 55 Ž1974., 52᎐72. 7. P. D. Lax, Development of singularities in solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5 Ž1964., 611᎐613. 8. H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu t t s Au q F Ž u., Trans. Amer. Math. Soc. 192 Ž1974., 1᎐21. 9. H. A. Levine and P. E. Sacks, Some existence and non existence theorems for solutions of degenerate parabolic equations, J. Differential Equations 52 Ž1984., 135᎐161. 10. R. C. Mac Camy and V. J. Mizel, Existence and non existence in the large of solutions of quasilinear wave equations, Arch. Rational Mech. Anal. 25 Ž1967., 299᎐320. 11. S. Messaoudi, ‘‘On Local Existence and Formation of Singularities in Non linear Thermoelasticity,’’ U.M.I., Order No. 9006215, Ann Arbor, MI 48106. 12. S. Messaoudi, On weak solutions of semi-linear thermoelastic equations, Maghreb Math. Re¨ . 1 Ž1992., 31᎐40. 13. R. Racke, On the Cauchy problem in non linear three dimensional thermoelasticity, Math. Z. 203 Ž1990., 649᎐682. 14. M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non linear thermoelasticity, Arch. Rational Mech. Anal. 76 Ž1981., 97᎐133. 15. S. Zheng and W. Shen, Global solutions to the Cauchy problem of a class of hyperbolicparabolic coupled systems, Sci. Sinica Ser. A 30 Ž1987., 1133᎐1149.
A Nonexistence Result to a Cauchy Problem in Nonlinear One Dimensional Thermoelasticity Mokhtar Kirane Faculte´ de Mathematiques et d’informatique, LAMFA UPRES A 6119, Uni¨ ersite´ de ´ Picardie Jules Verne, 33 Rue Saint Leu, 80039 Amiens Cedex 1, France
and Nasser-eddine Tatar Institut de Mathematiques, Uni¨ ersite´ Badji Mokhtar, B.P. 12, Annaba 23000, Algeria ´ Submitted by P. Broadbridge Received May 25, 1999
Considering the Cauchy problem for a nonlinear system arising in thermoelasticity, it is proved that its solution develops singularities in finite time depending on the size and regularity of the initial data. The present work is distinguished from a previous work by relaxing the requirements on the initial data and allowing for a slightly more general and nonautonomous forcing term besides permitting the insertion of gradient terms in both equations of the system. 䊚 2001 Academic Press
1. INTRODUCTION Over the last decades, there has been a large literature on results concerning local existence, global existence, uniqueness, and large time behavior of classical solutions to certain initial value problems of nonlinear thermoelasticity with smooth and small Žin some norms. initial data Žsee w11, 13᎐15x.. Whereas one fails to prove existence of global solutions for large initial data Žin some sense. Žsee w1᎐3, 7, 8, 10x., recently, S. Messaoudi w12x showed that, for a particular type of semi-linear thermoelastic equations, weak solutions collapse in finite time. Here our result improves Messaoudi’s in that we allow for a more general set of initial data. In fact, we eliminate a restrictive condition which was crucial in Messaoudi’s analysis Žsee comments below.. Moreover the class of systems we consider 71 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
72
KIRANE AND TATAR
is a bit larger since we insert gradient terms of the two unknowns in both equations. It is worth noting that the blow up result we obtain is proved for a, suitably constrained, nonlinear nonautonomous forcing term. This is established by choosing an appropriate functional satisfying all the requirements of a lemma in w4x. This lemma, which is crucial in our present treatment, is in fact a compact version of the concavity method of Levine w8x Žsee w6, 9x.. In w5x, the authors have applied this method to hyperbolic problems with dynamic boundary conditions. For convenience and completeness we shall report the lemma in question with its proof in the next section. Section 3 is devoted to our main result. In Section 4 a detailed study for a vanishing initial energy is given. Finally, in Section 5, we give another set of sufficient conditions assuring collapse in finite time. We should mention that these results have been largely motivated by the work of Levine w8x and Kalantarov and Ladyzhenskaya w4x.
2. PRELIMINARIES Let us first state the problem. We consider the following one dimensional semi-linear system
¡u Ž x, t . s au Ž x, t . q b Ž x, t . q du Ž x, t . y mu Ž x, t . ~ qf Ž t , u Ž x, t . . ¢c Ž x, t . s K Ž x, t . q bu Ž x, t . q pu Ž x, t . q q Ž x, t . tt
t
xx
x
xx
x
xt
t
x
Ž 1.
x
with x g R, t ) 0, and u Ž x, 0 . s u 0 Ž x . ,
u t Ž x, 0 . s u1 Ž x . ,
Ž x, 0 . s 0 Ž x . , x g R.
Ž 2. The coefficients are such that a, b, c, K ) 0
Ž 3.
and d, m, p, q G 0. The set R denotes the set of all real numbers. The case of assumption Ž3. and d s m s p s q s 0 has been studied in Messaoudi w12x. We are interested in finding sufficient conditions causing collapse in finite time for the solution of the above system. As for existence on a maximal time interval, uniqueness, and regularity we refer to Slemrod w14x.
73
A CAUCHY PROBLEM IN THERMOELASTICITY
Next we state and prove the basic lemma Ždue to Kalantarov and Ladyzhenskaya w4x. which provides the method of proof for our result. LEMMA 1. Assume that a twice differentiable, positi¨ e function ⌿ Ž t . satisfies for all t G 0 the inequality 2
⌿ Ž t . ⌿⬙ Ž t . y Ž 1 q ␥ . Ž ⌿⬘ . Ž t . G y2C1⌿ Ž t . ⌿⬘ Ž t . y C2⌿ 2 Ž t . , Ž 4 . where ␥ ) 0 and C1 , C2 G 0. Then Ža. if ⌿ Ž0. ) 0, ⌿⬘Ž0. q ␥ 2 ␥y1 ⌿ Ž0. ) 0, and C1 q C2 ) 0 we ha¨ e ⌿ Ž t . ª q⬁ as t ª t1 F t 2 s
1
ln
2 C12 q ␥ C2
'
ž
␥ 1⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 . ␥ 2⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 .
/
,
Ž 5.
where ␥ 1 s yC1 q C12 q ␥ C2 , ␥ 2 s yC1 y C12 q ␥ C2 , Žb. if ⌿ Ž0. ) 0, ⌿⬘Ž0. ) 0, and C1 s C2 s 0, then ⌿ Ž t . ª q⬁ as
'
'
t ª t1 F t 2 s
⌿ Ž 0.
␥ ⌿⬘ Ž 0 .
.
Proof. If we set ⌽ Ž t . s ⌿y␥ Ž t ., then it is easy to see using Ž4. and ⌽⬘ Ž t . s y␥ ⌽⬙ Ž t . s y
␥ ⌿
2q ␥
Ž t.
⌿⬘ Ž t . ⌿ 1q ␥ Ž t .
,
⌿⬙ Ž t . ⌿ Ž t . y Ž 1 q ␥ . Ž ⌿⬘ Ž t . .
2
that ⌽ is the solution of the second order differential equation ⌽⬙ Ž t . q 2C1⌽⬘ Ž t . y C2 ⌽ Ž t . s f Ž t . F 0
for all t G 0.
Ž 6.
Thus, for C1 q C2 ) 0 ⌽ Ž t . s  1 e␥ 1 t q  2 e␥ 2 t q Ž ␥ 1 y ␥ 2 . F  1 e␥ 1 t q  2 e␥ 2 t
y1
t
H0 f Ž . w e
for all t G 0,
␥ 1Ž ty .
y e␥ 2 Ž ty . x d
74
KIRANE AND TATAR
where  1 and  2 are defined by the system
½
1 q  2 s ⌽ Ž 0.  1 ␥ 1 q  2 ␥ 2 s ⌽⬘ Ž 0 . .
Clearly, we obtain y1
1 s yŽ ␥ 1 y ␥ 2 . 2 s Ž ␥ 1 y ␥ 2 .
y1
␥ 2⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 . ⌿y1y ␥ Ž 0 . - 0
␥ 1⌿ Ž 0 . q ␥ ⌿⬘ Ž 0 . ⌿y1y ␥ Ž 0 . ) 0.
We infer from the signs of  1 ,  2 , ␥ 1 , ␥ 2 , and the fact ⌽ Ž t . F  1 e␥ 1 t q  2 e␥ 2 t that ⌽ Ž t . must vanish for some finite time t 1. Indeed, the function  1 e␥ 1 t q  2 e␥ 2 t vanishes at the time t 2 Žsee Ž6.. and becomes negative afterwards. This implies that ⌿ Ž t . ª q⬁ as t approaches t 1 estimated by Ž5.. For the second case Žb., Ž6. implies ⌽⬙ Ž t . F 0. By integration we have ⌽ Ž t . F ⌽ Ž 0 . q ⌽⬘ Ž 0 . t and by definition of ⌽ Ž t . ⌿ ␥ Ž t . G ⌿ ␥ Ž 0 . 1 y ␥ t ⌿y1 Ž 0 . ⌿⬘ Ž 0 .
y1
.
Since ⌿ Ž0. ) 0 and ⌿⬘Ž0. ) 0, the conclusion of the lemma follows easily. This completes the proof. Finally, we shall make use of the -inequality ab F a2 q
1 4
b2 ,
for all a, b g R, ) 0
Ž 7.
0 g H 1 Ž R.
Ž 8.
and the fact that for u0 g H 2 Ž R. ,
u1 g H 1 Ž R . ,
uŽ⭈, t . and its spatial derivatives up to order 2 are continuous and vanish at infinity.
A CAUCHY PROBLEM IN THERMOELASTICITY
75
3. BLOW UP FOR NEGATIVE INITIAL ENERGY Before stating our first result let us multiply the first equation of Ž1. by u t and the second equation by , then integrate over R = w0, t x q⬁
t
Hy⬁ H0 u u t
y au t u x x y du t u x q mu 2t y u t f Ž t , u . 4
tt
q⬁
t
Hy⬁ H0 c y K
q
t
xx
y p u x y qx 4 d dx s 0.
Ž 9.
We suppose that for fixed t, f Ž t, u. is the Frechet derivative of some ´ functional F Ž t, u., so that d dt
F Ž t , u . s Ft Ž t , u . q f Ž t , u . u t .
Ž 10 .
Taking this into account, Ž9. becomes q⬁
Hy⬁
½
1 2
q
a
u 2t q q⬁
2
u 2x y F Ž t , u . q
t
Hy⬁ H0 mu s
q⬁
Hy⬁
½
1 2
2 t
c 2
2 dx
5
q K x2 q F Ž , u . y du t u x y p u x 4 d dx
u12 q
a 2
u 20 x y F Ž 0, u 0 . q
c 2
02 dx.
5
Ž 11 .
Now let EŽ t . s
q⬁
Hy⬁
½
1 2
u 2t q
a 2
u 2x y F Ž t , u . q
c 2
2 dx.
5
Ž 12 .
Then Ž11. may be written EŽ t . q
q⬁
t
Hy⬁ H0 mu
2 t
q K x2 q F Ž , u . y du t u x y p u x 4 d dx s E Ž 0 . .
In addition to Ž10. we require f Ž t, u. to satisfy uf Ž t , u . G 2 Ž 1 q 2␥ . F Ž t , u .
Ž 13 .
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KIRANE AND TATAR
for some ␥ G Ž1r4.ŽŽ1 q b 2rac.1r2 y 1., Ft Ž t , u . G 2 Ž 2 y m . F Ž t , u .
Ž 14 .
with 2 y m greater than or equal to the positive root of
Ž 4 amc . 2 y Ž cd 2 . y p 2 m s 0
Ž 15 .
Ž Ft Ž t, u. G 0 in case m s p s d s 0., or Ft Ž t , u . G 2 Ž ˜2 y m . F Ž t , u . q
p2 4 3
u2
Ž 16 .
with ˜2 the positive root of
Ž 4 a. 2 y Ž 4 am . y d 2 s 0 Ž Ft Ž t, u. G Ž p 2r4K . u 2 in case m s d s 0.. The initial data are, for the moment, such that u 0 / 0 and q⬁
Hy⬁ u
dx F
2 0
2␥
q⬁
H uu m y⬁ 0
1
dx.
Ž 17 .
A simple example of f satisfying the above hypotheses is f Ž t, u. s e t u p , p ) 1 with an appropriate constant . THEOREM 1. Suppose that u 0 , u1 , and satisfy Ž8., Ž17. and are such that E Ž0. - 0. Suppose further that f Ž t, u. satisfies Ž10., Ž13., and Ž14. or Ž10., Ž13., and Ž16.. Then there exists a positi¨ e number T - q⬁ such that limy
tªT
q⬁
Hy⬁ u
2
Ž t , x . dx s q⬁.
2 Ž . 2 where  and t 0 Proof. Let us introduce ⌿ Ž t . s Hq⬁ y⬁ u dx q  t q t 0 are to be determined later. Then
⌿⬘ Ž t . s 2
q⬁
žH
y⬁
uu t dx q  Ž t q t 0 .
/
and ⌿⬙ Ž t . s 2
ž
q⬁
Hy⬁ Ž u
2 t
q uu t t . dx q  .
/
77
A CAUCHY PROBLEM IN THERMOELASTICITY
We insert the values of ⌿, ⌿⬘, and ⌿⬙ into the expression ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 s 2⌿
q⬁
½H
y⬁
Ž u 2t q uu t t . dx q 
y 4Ž 1 q ␥ .
q⬁
½H
y⬁
5 2
uu t dx q  Ž t q t 0 .
5
.
By the Cauchy᎐Schwarz inequality we have
½
2
q⬁
Hy⬁ uu F
t
dx q  Ž t q t 0 . q⬁
½ žH
y⬁
5
1r2 2
u dx
q⬁
/ žH
y⬁
2
1r2
u 2t
dx
q  Ž t q t0 . 
/
'
'
5
.
Applying Holder’s inequality to the right hand side we find q⬁
½H
y⬁
2
uu t dx q  Ž t q t 0 .
5
F⌿
q⬁
½H
y⬁
u 2t dx q  .
5
Therefore
⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G 2⌿ y Ž 1 q 2␥ .
½
ž
q⬁
Hy⬁ u
2 t
dx q  q
/
q⬁
Hy⬁ uu
tt
5
dx .
Next, from the first equation of Ž1. and an integration by parts we get ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G 2⌿ y Ž 1 q 2␥ .
½
ž
q⬁
Hy⬁ u
2 t
q ya
/
q⬁
Hy⬁ u
y
m 2
2 x
⌿⬘ Ž t . q
q⬁
Hy⬁ u dx
dx y b
q⬁
x
Hy⬁ uf Ž t , u . dx
5
.
78
KIRANE AND TATAR
The -inequality Ž7. implies ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2
½
G 2⌿ y Ž 1 q 2␥ . y 1
ž
q⬁
Hy⬁ u
q⬁
Hy⬁
2
2 t
q y aq
ž
/
m
dx y
2
⌿⬘ Ž t . q
b2 4 1
/
q⬁
Hy⬁ u
2 x
dx
q⬁
Hy⬁ uf Ž t , u . dx
5
.
Finally the definition Ž12. of E Ž t . yields ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2
½
G 2⌿ y Ž 1 q 2␥ . E Ž t . q 2 a␥ y
ž
q c Ž 1 q 2␥ . y 1 q
q⬁
Hy⬁
q⬁
Hy⬁
2
b2 4 1
/
q⬁
Hy⬁ u
2 x
dx
dx
uf Ž t , u . y 2 Ž 1 q 2␥ . F Ž t , u . dx y
m 2
5
⌿⬘ y  Ž 1 q 2␥ . . Ž 18 .
Assumption Ž13. simplifies the right hand side of Ž18. to ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2
½
G 2⌿ y2 Ž 1 q 2␥ . E Ž t . q 2 a␥ y
ž
q⬁
q c Ž 1 q 2␥ . y 1
Hy⬁
2
b2 4 1
dx y
m 2
/
q⬁
Hy⬁ u
2 x
dx
5
⌿⬘ y  Ž 1 q 2␥ . . Ž 19 .
q⬁ 2 We pick 1 s b 2r8 a␥ , so that the coefficient of Hy⬁ u x dx vanishes. It is now easy to see that the assumption on ␥ in Ž13. implies the positivity of cŽ1 q 2␥ . y 1. These considerations imply
⌿⌿⬙ y Ž 1 q ␥ . Ž ⌿⬘ .
2
G 2⌿ y2 Ž 1 q 2␥ . E Ž t . y
½
m 2
⌿⬘ y  Ž 1 q 2␥ . .
5
Ž 20 .
79
A CAUCHY PROBLEM IN THERMOELASTICITY
Case 1. If Ž14. is satisfied, we differentiate E Ž t . and use the inequality Ž7. to get E⬘ Ž t . s
q⬁
Hy⬁ ymu
F Ž 2 y m. q
1 4
ž
2 t
y K x2 y Ft Ž t , u . q du t u x q p u x 4 dx
q⬁
Hy⬁ u
d2
2
p2
q
3
dx y
2 t
/
q⬁
q⬁
Hy⬁ F Ž t , u . dx q Hy⬁
q⬁
Hy⬁ u
t
2 x
3
2
dx
dx
or E⬘ Ž t . F 2 Ž 2 y m . E Ž t . q q
q⬁
Hy⬁
1
d2
4
2
½ž
q
p2
3
/
y aŽ 2 y m .
5
q⬁
Hy⬁ u
2 x
dx
2 Ž 2 y m . F Ž t , u . y Ft Ž t , u . dx
q 3 y Ž 2 y m. c4
q⬁
Hy⬁
2
dx.
Ž 21 .
Ži. If m s 0 and p s d s 0, then it is easy to see from the first equality of Ž3.17. that when Ft Ž t, u. G 0 we have E⬘Ž t . F 0. Hence E Ž t . F E Ž0. for all t G 0. Žii. If m s 0 and p q d ) 0, then we can choose 3 s 2 c and pick 2 so that Ž1r4.Ž d 2r 2 q p 2r 3 . y a 2 - 0. Clearly 2 ) ŽŽ cd 2 q p 2 .r4 ac .1r2 will do. We obtain E⬘Ž t . F 2 2 E Ž t . and again E Ž t . F E Ž0. for all t G 0. Žiii. If m / 0, then we put 3 s Ž 2 y m. c and choose 2 such that 2 y m ) 0 and M[
1 4
ž
d2
2
q
p2
Ž 2 y m. c
/
y a Ž 2 y m . F 0.
This can be done as follows. For 2 ) m, since m / 0 M-
1 4
ž
d2 m
q
p2
Ž 2 y m. c 2
/
y aŽ 2 y m .
- Ž y4 amc . Ž 2 y m . q cd 2 Ž 2 y m . q p 2 m r 4 mc Ž 2 y m . ,
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KIRANE AND TATAR
it suffices to take 2 as in Ž15. to obtain E⬘Ž t . F 2Ž 2 y m. E Ž t . and consequently E Ž t . F E Ž0. - 0 for all t G 0. Case 2. If Ž16. holds, then we use the expression E⬘ Ž t . s
q⬁
Hy⬁ ymu
2 t
y K x2 y Ft Ž t , u . q du t u x y px u4 dx
F 2 Ž ˜2 y m . E Ž t . q qŽ 3 y K . q⬁
q
Hy⬁
½
q⬁
Hy⬁
2 x
d2 4˜2
y a Ž ˜2 y m .
dx y Ž ˜2 y m .
2 Ž ˜2 y m . F Ž t , u . q
p2 4 3
q⬁
Hy⬁ u
q⬁
Hy⬁
2
2 x
dx
dx
5
u 2 y Ft Ž t , u . dx.
Choosing 3 s K and ˜2 y m G 0 such that d 2r4˜2 y aŽ ˜2 y m. s 0, that is, ˜2 is the positive root of Ž4 a. 2 y Ž4 am. y d 2 s 0 and assuming Ž16., we again obtain E Ž t . F E Ž0. for all t G 0. Now back to Ž20., from which it results ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G 2⌿ y Ž 1 q 2␥ . Ž  q 2 E Ž 0 . . y
½
m 2
⌿⬘ .
5
We choose  ) 0 such that  q 2 E Ž0. F 0. Thus ⌿⌿⬙ y Ž 1 q ␥ . ⌿⬘ 2 G ym⌿⌿⬘.
Ž 22 .
The inequality Ž4. in Lemma 1 is therefore satisfied with C2 s 0 and C1 s mr2. If m s 0, we choose t 0 so that ⌿⬘Ž0. ) 0 and apply part Žb. of Lemma 1. The condition ⌿ Ž0. ) 0 holds since u 0 / 0. It remains to check that ⌿⬘Ž0. ) m␥y1 ⌿ Ž0., that is, q⬁
Hy⬁ u
2 0
dx q  t 02 -
2␥ m
q⬁
žH
y⬁
u 0 u1 dx q  t 0
/
which may be written as a second order inequality in t 0
 t 02 y
2␥ m
t0 q
q⬁
žH
y⬁
u 02 dx y
2␥
q⬁
H uu m y⬁ 0
1
dx - 0.
/
Ž 23 .
An easy computation shows that we may choose t 0 satisfying Ž23. as long as assumption Ž14. holds. ŽSee Remark 1 below.. All the requirements of Lemma 1 are now fulfilled and the conclusion follows.
81
A CAUCHY PROBLEM IN THERMOELASTICITY
Remark 1. Observe from Ž23. that the condition Ž17. can be weakened. In fact we only need to choose  such that m2
␥
2
q⬁
Hy⬁ u
2 0
dx y
2␥
q⬁
H uu m y⬁ 0
1
dx F  F y2 E Ž 0 . .
Remark 2. The escape time is estimated according to Lemma 1 as
TyF t 2 s
1 m
2␥ ln 2␥
q⬁
žH
y⬁
q⬁
žH
y⬁
u 0 u1 dx q  t 0
u 0 u1 dx q  t 0 y m
q⬁
/ žH
y⬁
/ u 02 dx q  t 02
/
and when m s 0, the second part of Lemma 1 applies giving q⬁
TyF t 2 s
Hy⬁ u 2␥
ž
2 0
dx q  t 02 .
q⬁
Hy⬁ u u 0
1
dx q  t 0
/
Remark 3. It is of interest to note that even when E Ž0. s 0 Ž E Ž0. G 0 in case m s p s d s 0., if 5 u x 5 2 Žor 5 5 2 . grows unboundedly then 5 u 5 2 must blow up in finite time. This follows from Ž19. by taking 1 s Ž1 q 2 2␥ . c. The hypothesis on ␥ in Ž13. implies that the coefficient of Hq⬁ y⬁ u x dx is positive. After a certain time Žwhich then can be considered as the new starting time. the expression y Ž 1 q 2␥ . Ž  q 2 E Ž 0 . . q 2 a␥ y
ž
b2 4 1
/
q⬁
Hy⬁ u
2 x
dx
becomes Žand remains. positive yielding Ž22.. Comments. For f Ž t, u. s f Ž u. s u ␣ , ␣ ) 1, and d s m s p s q s 0 we find the system in w12x. The argument above works Žsee Remarks 1, 2, and Ž13.. for the same values of ␣ . But here we do not impose the w x condition Hq⬁ y⬁ u 0 u1 dx ) 0 as in 12 since it will always be possible to q⬁ choose t 0 satisfying Hy⬁ u 0 u1 dx q  t 0 ) 0. Observe that in this case assumption Ž17. disappears. The following proposition Žwhich is due to Levine and Sacks w9x. shows that we can find sufficient conditions under which initial data provide negative initial energy.
82
KIRANE AND TATAR
PROPOSITION 1. Suppose that lim uªq⬁ f Žuu . \ lim uªq⬁ g Ž u. s q⬁, u 0 g H 1 ŽR., u1 g L2 ŽR., and 0 g L2 ŽR.. Then there exists a sufficiently large such that E Ž u 0 , u1 , 0 . - 0. Proof. Suppose for contradiction that E Ž u 0 , u1 , 0 . ) 0 for all G 1. Then by the definition of the energy 2
2
q⬁
q⬁
Hy⬁ F Ž u . dx F Hy⬁ u 0
2 0x
dx q
1
2
q⬁
Hy⬁ Ž u
2 1
q c 02 . dx F M,
where M is a positive constant independent of . But 1
2
q⬁
Hy⬁ F Ž u . dx 0
s s
1
2
q⬁
u 0
ž
Hy⬁ H0
q⬁
Hy⬁ u
2 0
ž
1
2
/
f Ž s . ds dx u 0
u 20
H0
sg Ž s . ds dx ª q⬁,
/
when ª q⬁.
Here we have used F Ž u. s H0u f Ž s . ds. This is a contradiction. 4. BLOW UP FOR VANISHING INITIAL ENERGY In this section it will be clear that for vanishing initial energy we can obtain blow up in finite time even when condition Ž17. is violated. THEOREM 2. s 0.
Suppose that the hypotheses of Theorem 1 hold with E Ž0.
Ži. If ⌿⬘Ž0. ) m␥ ⌿ Ž0., then 5 u 5 2 blows up in a finite time. Žii. If ⌿⬘Ž0. s m␥ ⌿ Ž0., then either 5 u 5 2 blows up in a finite time or else 5 u5 2 s 5 u0 5 2 e Žiii.
If ⌿⬘Ž0. -
m ␥
m 2␥
t
for all t G 0.
Ž 24 .
⌿ Ž0., then either 5 u 5 2 blows up in a finite time or
else 5 u5 2 F 5 u0 5 2 e
m 2␥
t
for all t G 0.
Ž 25 .
Proof. For E Ž0. s 0 we can take  s 0. Ži. This follows readily from Ž20. and Lemma 1 with C1 s mr2 and C2 s 0.
83
A CAUCHY PROBLEM IN THERMOELASTICITY
Žii. Letting H Ž t . [ ln ⌿ Ž t . y s ⌿ Ž0. implies H⬘Ž0. s 0. m ␥
m ␥
t, we note that the condition ⌿⬘Ž0.
A glance at the proof of Lemma 1 gives  1 s 0,  2 s ⌿y␥ Ž0., and ⌿ Ž t . G ⌿ Ž0. e Ž m r␥ .t , which in turn yields H Ž t . G H Ž0.. Consequently, either H Ž t . ' H Ž0. or else H⬘Ž˜t . ) 0 for some ˜t ) 0. In the latter case we obtain blow up from Theorem 1 since the argument there is applicable for ˜t and the collapse occurs at a time t1 ) ˜t. In the former case, ln ⌿ Ž t . y m␥ t s ln ⌿ Ž0. for all t G 0, implying ⌿ Ž t . s ⌿ Ž0. e Ž m r␥ .t for all t G 0. Žiii. If ⌿⬘Ž0. - m␥ ⌿ Ž0., then H⬘Ž0. - 0. It follows, by continuity, that if there does not exist a finite time ˆt such that H⬘Žˆt . s 0, then we may assume that H⬘Ž t . - 0 for all t G 0. By Žii., H⬘Žˆt . s 0, for some finite ˆt, may lead to ⌿ Ž t . s ⌿ Ž0. e Ž m r␥ .t from which we infer ⌿⬘Ž0. s m␥ ⌿ Ž0.. This is a contradiction. Hence there only remains the collapse of 5 u 5 2 at a finite time. If H⬘Ž t . - 0 for all t G 0, then H Ž t . F H Ž0. for all t G 0. This means that 5 u 5 2 F 5 u 0 5 2 e Ž m r2␥ .t for all t G 0. The proof of the theorem is complete. Remark 4. If m s 0 then in case Žii., ⌿⬙ Ž t . ' 0 and on the other hand
½
⌿⬙ Ž t . G 2 y2 Ž 1 q 2␥ . E Ž t . q q 2 a␥ y
ž
b2 4
q 2Ž 1 q ␥ .
q⬁
/H
y⬁
q⬁
Hy⬁ u
2 t
q⬁
Hy⬁
uf Ž t , u . y 2 Ž 1 q 2␥ . F Ž t , u . dx
u 2x dx q c Ž 1 q 2␥ . y
q⬁
Hy⬁
2
dx
5
dx .
So that for s cŽ1 q 2␥ . and ␥ as in Ž13. we get from the previous q⬁ 2 q⬁ 2 inequality Hy⬁ u t dx s Hy⬁ u x dx and this implies u ' u 0 ' 0. Thus, either the solution blows up or else it coincides with its initial value which must vanish. If m s 0 in case Žiii., then we can see that either the solution blows up or else it is uniformly bounded in the L2-norm Žfor all time..
5. AN ALTERNATIVE TO CONDITIONS Ž14. AND Ž16. In this section we are going to see that conditions Ž14. and Ž13. may be replaced by the condition Ft Ž t , u . G 0,
for all t ) 0.
Ž 26 .
84
KIRANE AND TATAR
Indeed, with minor changes in the hypothesis of Theorem 1, namely uf Ž t , u . G 2 Ž 1 q 2␥ . F Ž t , u .
Ž 27 .
for some ␥ G Ž1r4.Ž1 q b 2rac.1r2 ,
ž
1q
2 ␣␥ m
q⬁
/H
y⬁
u 20 dx -
2␥
q⬁
H uu m y⬁ 0
1
dx
Ž 28 .
for some ␣ to be determined later, and E˜Ž 0 . s
q⬁
Hy⬁
½Ž
m q 2␣ . 2
␣ u 20 q
1 2
u12 y ␣ u 0 u1 q F Ž 0, u 0 . q yF
a 2
u 02 x
c 2
5
02 F 0, Ž 29 .
we can prove the THEOREM 3. Assume that conditions Ž8., Ž10., and Ž26. ᎐ Ž29. are satisfied. Then the conclusions of Theorem 1 and Theorem 2 hold. Proof. We introduce the new functions ¨ Ž t . s ey␣ t u Ž t .
and
w Ž t . s ey␣ t Ž t . .
Then ¨ Ž t . and w Ž t . satisfy the system
½
¨ t t s a¨ x x q bwx q d¨ x y Ž m q 2 ␣ . ¨ t y Ž m q ␣ . ␣ ¨ q f˜Ž t , ¨ .
cwt s Kwx x q b¨ x t q Ž ␣ b q p . ¨ x q qwx y ␣ w
Ž 30 .
with ¨ 0 Ž x . s u0 Ž x . ,
and
¨ 1 Ž x . s u1 Ž x . y ␣ u 0 Ž x . ,
Ž 31 .
w0 Ž x . s 0 Ž x . ,
where f˜Ž t, ¨ . s ey␣ t f Ž t, e ␣ t ¨ .. We want to show that Theorem 1 applies for this system. In fact the proof of Theorem 1 carries through almost unchanged. Thus we only mention the main differences. The appropriate energy is E˜Ž t . s
q⬁
Hy⬁
½Ž
mq␣. 2
␣¨2 q
1 2
¨ 12 q
a 2
¨ x2 y F˜Ž t , ¨ . q
c 2
5
w 2 dx ,
85
A CAUCHY PROBLEM IN THERMOELASTICITY
where F˜Ž t, ¨ . s ey2 ␣ tF Ž t, e ␣ t ¨ .. Next we want to check the requirements of Theorem 1 for system Ž22. Ža. ¨ f˜Ž t, ¨ . s ey␣ t uey ␣ t f Ž t, u. s ey2 ␣ t uf Ž t, u. G 2Ž1 q 2␥ . F˜Ž t, ¨ . Žb. from Ž26. and Ža., we have F˜t Ž t , ¨ . s ey2 ␣ tFt Ž t , e ␣ t ¨ . q ␣ ¨ f˜Ž t , ¨ . y 2 ␣ F˜Ž t , ¨ . G ␣ ¨ f˜Ž t , ¨ . y 2 ␣ F˜Ž t , ¨ . G 2 ␣ Ž 1 q 2␥ . F˜Ž t , ¨ . y 2 ␣ F˜Ž t , ¨ . G 4␣␥ F˜Ž t , ¨ . . To fulfill assumption Ž14. it suffices to pick ␣ such that
␣ G Ž X2 y m . r2␥ ,
Ž 32 .
where Ž X2 y m. is greater than or equal to the positive root of the second order polynomial 2 Ž 4 mca. 2 y Ž d 2 c . y m Ž ␣ b q p . .
Note that this polynomial depends on ␣ , yet Ž32. is possible. Indeed, from Ž27. it suffices to choose ␣ greater than or equal to the positive root of the polynomial m Ž b 2 y 16␥ 2 ac . 2 q 2 Ž bpm q ␥ d 2 c . q p 2 m. This is found after an easy computation. The conclusion of Theorem 3 holds with 5 u 5 2 s 5 u 0 5 2 e Ž m r2␥q ␣ .t in Žii. and 5 u 5 2 F 5 u 0 5 2 e Ž m r2␥q ␣ .t in Žiii.. This completes the proof.
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6. R. J. Knops, H. A. Levine, and L. E. Payne, Nonexistence, instability and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal. 55 Ž1974., 52᎐72. 7. P. D. Lax, Development of singularities in solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5 Ž1964., 611᎐613. 8. H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu t t s Au q F Ž u., Trans. Amer. Math. Soc. 192 Ž1974., 1᎐21. 9. H. A. Levine and P. E. Sacks, Some existence and non existence theorems for solutions of degenerate parabolic equations, J. Differential Equations 52 Ž1984., 135᎐161. 10. R. C. Mac Camy and V. J. Mizel, Existence and non existence in the large of solutions of quasilinear wave equations, Arch. Rational Mech. Anal. 25 Ž1967., 299᎐320. 11. S. Messaoudi, ‘‘On Local Existence and Formation of Singularities in Non linear Thermoelasticity,’’ U.M.I., Order No. 9006215, Ann Arbor, MI 48106. 12. S. Messaoudi, On weak solutions of semi-linear thermoelastic equations, Maghreb Math. Re¨ . 1 Ž1992., 31᎐40. 13. R. Racke, On the Cauchy problem in non linear three dimensional thermoelasticity, Math. Z. 203 Ž1990., 649᎐682. 14. M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non linear thermoelasticity, Arch. Rational Mech. Anal. 76 Ž1981., 97᎐133. 15. S. Zheng and W. Shen, Global solutions to the Cauchy problem of a class of hyperbolicparabolic coupled systems, Sci. Sinica Ser. A 30 Ž1987., 1133᎐1149.