TR/10/82 November 1982 The Cauchy and Backward-Cauchy Problem for a Nonlinearly H y p e r e l a s t i c / V i s c o e l a s t i c I n f i n i t e R o d . by R. Saxton

TR/10/82 November 1982 The Cauchy and Backward-Cauchy Problem

for a Nonlinearly

H y p e r e l a s t i c / V i s c o e l a s t i c I n f i n i t e R o d . by

R. Saxton

I n t h i s p a p e r w e c o n s i d e r b o t h s i m p l e r a n d m o r e g e n e r a l f o r m s o f a n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n a r i s i n g f r o m a p a r t i c u l a r m o d e l o f a r o d v i b r a t i n g l o n g i t u d i n a l l y ( c f . J a u n z e m i s [ 4 ] , L o v e [ 6 ] ) . T h i s i s t h e E u l e r e q u a t i o n d e r i v e d f r o m a L a g r a n g i a n .

. IR Ω , dt dx f)]

, w(u )u

f(u u

[

L

₂¹ _x ²_{xt x}

t

t Ω 2t 2

2 1

1

⊆

− +

= ∫ ∫ ^1.

S e t t i n g δ L = 0 p r o v i d e s t h e e q u a t i o n o f m o t i o n . H e r e f , Ŵ a r e

g e n e r a l l y n o n l i n e a r f u n c t i o n s ( Ŵ i s t h e s t r a i n - e n e r g y d e n s i t y o f t h e m a t e r i a l a n d f i s a f u n c t i o n c o r r e s p o n d i n g t o t h e p r o d u c t o f a n o n - l i n e a r P o i s s o n ' s r a t i o w i t h a v a r i a b l e r a d i u s o f g y r a t i o n a b o u t t h e c e n t r a l a x i s o f t h e r o d ) . E l s e w h e r e ( [ 9 ] ) w e c o n s i d e r f ( u

_x

) t o

b e c o me s i n g u l a r a s u

_x

→ − 1 f or physica l re a s on s of ma te ri a l i nv e r s io n , b u t n o w w e g e n e r a l i z e i n a n o t h e r d i r e c t i o n a n d l e t f ( φ) = φ .

I f w e d e f i n e

, ) ) σ(u

( du ), dw w(u )

u , (u

W

x

x x

x

x

= u

^x

=

) 2.

t h e n t h e e q u a t i o n o f mo t i o n o b t a i n e d i s .

0 ) u ( u

u

_tt

−

_xxtt

− σ

_{x x}

= 3.

We may consider more generally the equation 0 ) u u, , u , u t, (x, T u

u

_tt

−

_xxtt

−

_{x x xt t}

= 4.

(see [8]) where the nonlinear function T depends on the two dependent and independent variables as well as on the first spatial derivatives of u and

u . H o w e v e r i n t h e c a s e o f e l a s t i c i t y i t i s a p p r o p r i a t e f o r T t o h a v e

t

dependence only on x , u

_x

and u

_xt

, and we consider primarily homogeneous materials for which T has the form

. ) (u τ ) σ(u

T =

_x

+

_xt

5.

In places, we indicate also how one treats the more general problem 4.

The paper proves local and global existence results for various spaces with ,

0 t , IR

x ∈ ≥ S o me o f t h e s e s p a c e s a r e c h o s e n t o d e mo n s t r a t e p o s s i b l e b e h a v i o u r o f s o l u t i o n s a s x

^→

± ∞ , a n d a r e i n t e r e s t i n g i n v i e w o f solitary wave solutions (see [10]) which are known to exist.

First several restrictions are imposed on σ, and W to permit sufficient τ tractability in investigating local and global existence.

P r e s e n c e o r a b s e n c e o f v i s c o u s t e r ms d o e s n o t a f f e c t t h e question of global existence since the equation will be shown to possess g l o b a l l y u n i q u e s o l u t i o n s e v e n i n t h e h y p e r e l a s t i c c a s e , w h e n τ ≡ 0 . A n y a s s u m p t i o n s o f v i s c o u s d a m p i n g s u c h a s φτ ( φ ) > 0 ∀ φ ∈ IR o n l y help to improve matters by letting solutions decay to equilibrium as

t

^→

+ ∞ . O n t h e o t h e r h a n d w h e n φτ ( φ ) < 0 i s p e r mi t t e d , t h i s ma y produce solutions growing as t

^→

+ ∞ . and is equivalent to making the change of independent variable t

^→

− t and looking at the case of

−

>

φ

φ τ( ) 0, as t

^→

∞ . I n c e r t a i n c a s e s t h i s m a y l e a d t o b l o w - u p o f s o l u t i o n s i n f i n i t e t i m e , a s i n t h e u n d a m p e d n o n l i n e a r

s t r i n g ( L a x [ 5 ] ) . H e r e w e c o n s i d e r t h r e e c a s e s , g e n e r a l l y g r o u p e d t o g e t h e r f o r e a s e o f p r e s e n t a t i o n , i n w h i c h b l o w - u p d o e s n o t o c c u r i . e . f o r w h i c h s o l u t i o n s e x i s t f o r a l l t i m e t > 0 .

H y p o t h e s i s H 1 ) L e t τ ( . ) b e l o c a l l y L i p s c h i t z - c o n t i n u o u s , i . e . IR ∀ φ , ψ ∈ s u c h t h a t f o r s o m e R > 0

, R , R ψ <

<

φ t h e r e e x i s t s a c o n s t a n t ( R ) > 0 , Γ

∞

∞ +

Γ (R)

^→

as R

^→

w i t h τ( φ ) − τ(ψ) ≤ Γ (R) φ − ψ . )

i ) 1

H φτ ( φ > ) 0 , ∀ φ ∈ IR , )

ii ) 1

H τ ( φ ) ≡ 0 , )

iii ) 1

H , some 0 .

) 2

( φ = − αφ α >

φτ

I t w i l l b e s e e n l a t e r t h a t t h e a p p r o p r i a t e 'e n e r g y ' e s t i ma t e b e c o me s l e s s u s e f u l a s w e p r o c e e d f r o m H i ) t h r o u g h H i i ) t o H i i i ) . F u r t h e r hypotheses required for (.), W(.) are σ

H2) Let σ (.) be locally Lipschitfz -continuous (without loss of generality we may take the same Lipschitz constant Γ (R) as in H1), and set σ ( 0 ) ≡ 0 .

H3) Assume W ( φ ) ≥ 0 , ∀ φ ∈ IR . We provide some notation.

L

^P

( R I ) d e n o t e s t h e s p a c e o f m e a s u r a b l e r e a l - v a l u e d f u n c t i o n s o n R I for which

, p

1 , dx

) x ( f

||

f

||

^p

p 1

p

^≡ (

^∞

_∫ ) ^{< ∞ ≤ < ∞}

∞

−

or

|| f || ∞ ≡ ess

_x

∈ IR sup | f ( x ) | < ∞ , when p = ∞ .

The Sobolev spaces m , p ( IR ), 1 p , W m , ( IR ) , m IN ,

W o ≤ < ∞ ∞ ∈

c o n s i s t o f t h o s e f u n c t i o n s i n L

^P

( I R ) a l l o f w h o s e g e n e r a l i z e d d e r i v a t i v e s u p t o a n d i n c l u d i n g o r d e r m b e l o n g t o p ( IR )

L .

We define norms on m , ( IR )

W , ) IR o m , p (

W ∞

by , p

1

|| , dx

f

|| d

||

f

||

^p

1 p p m

0

j j

j p

,

m

⁼ ( ∑ ) ^{≤ < ∞}

=

or

, dx ||

f

|| d f ||

||

^m

0

j j

j ,

m ∞

⁼ ∑

= ∞

respectively.

Each of the spaces m , ( IR )

W , ) IR o m , p (

W ∞

is a Banach space under the

given norm.

let Y be a Banach space and let A be a bounded or semibounded set in IR . Then the Banach space C

^k

( A ; Y ) , k ∈ NU { 0 } is the class of k-times

continuously differentiable mappings ( t ) : A Y , 0 j k . t

d u d

j

j ^→

≤ ≤

) Y

; A (

C

^k

has the norm

||

Y

) t t ( d

u

|| d

| u

|

_j

k j 0 Y j

,

k

∑

tsupA

= ∈

≡

Similarly, we define the Banach space L

¹

(A ; Y) to be the class of measurable mappings u ( t ) : A → Y such that for u ∈ L

¹

( A ; Y )

∞

<

≡ ∫

A Y

Y ,

1

|| u ( t ) || dt

||

u

||

Now we want to consider local existence of solutions to equation 4.

w i t h τ g i v e n b y 5 . T h e d o ma i n o f x w i l l b e t a k e n t o b e t h e r e a l l i n e over which the following Cauchy data are given -

. IR x , ) 0 , x ( u ) x ( u , ) 0 , x ( u ) x (

u

_O

≡

₁

≡

_t

∈ 6.

We present the proof in two stages. The Theorem applies to the space )

IR ( W ) IR ( W

x ∈

^1,∞ _{I O}^1,2 W

hich is the broadest to which our method applies. The possibility of working in a larger space W

_O^1,p

( IR ) , p < ∞ , using other techniques such as a monotonicity approach is not considered here, due partly to the restrictions thereby imposed on σ but also because there appear to be difficulties in the conservative case ( τ = 0 ) . The

Corollary that follows the Theorem is concerned with restricting the i n i t i a l d a t a t o s p e c i a l c l a s s e s o f f u n c t i o n s w h i c h d e c a y a t v a r i o u s r a t e s a s | x |

→

∞ a n d s h o w i n g t h a t t h e s o l u t i o n t o 4 . , 5 . , 6 . b e h a v e s l i k e w i s e ( t h i s i s i n d e p e n d e n t o f t h e c h o i c e o f s u b s i d i a r y H 1 i ) , H 1 i i ) o r H 1 i i i ) .

Theorem 1 Let hypotheses H1) and H2) be satisfied, and suppose that )

IR ( W ) IR ( W ) x ( u , ) x (

u

_{0 1}

∈

^1,∞

I

₀^1,2

. Then there exists a solution

u (x, t) to 4., 5. and 6 . belonging to C

¹

([ 0 , τ [; W

^1,∞

( IR ) I W

₀^1,2

( IR )

defined on a maximal interval [ 0 , τ [ , τ ≤ ∞ . If τ < ∞ , then ,

t as )

t (., u )

t (.,

u

_1,∞

+

_{t 1,∞}

→ ∞ → τ

R e m a r k 1 T h e p r o o f o f t h e T h e o r e m i s d i v i d e d i n t o t w o s t e p s , o n e o f w h i c h w e p o s t p o n e u n t i l l a t e r . H e r e w e s h o w t h a t e q u a t i o n 4 . is formally equivalent to an integro-differential equation which we solve by contraction mapping. The Theorem will be proved when it is shown that the solution found in this way solves 4. in a weak sense.

Henceforth we consider this to be true.

Proof of Theorem 1 We write 4., 5., as

( 1 − ∂ 2 x ) u tt = ∂ x ( σ + τ ) . 7.

and we assume u ( ±∞ , t ) = ∀ t > 0 . Formally operating on 7. with 2 ) ¹

1 x

( − ∂ − we obtain

∫ ξ

∞

∞

−

τ + ξ σ

∂ ξ

= G ( x , ) ( ) d )

t , x tt ( u

∞ ∫

∞

− ξ ξ σ + τ ξ

−

= G ( x , ) ( ) d 8.

where

x , 2 e ) 1 , x (

G − − ξ

=

ξ 9.

and, on integrating twice with respect to time, 8. becomes . )

( ) , ( ) ( )

1 ( )

0 ( )

,

( x t u x t u x

0

t η G ξ x ξ σ τ d ξ d η

u

^t

∞ ∫ +

∞

− −

− +

= ∫ ^10.

The derivative of 10. with respect to x is given by

η ξ τ + σ ξ ξ

∫ ∫ ∞

∞

− − η

− +

= t G x ( x , ) ( ) d d

0

) t ( )

x 1 ( ' u t ) x 0 ( ' u ) t , x x ( u

η τ +

∫ − η σ

− t ( ) d

0 ) t

( 11.

We try to find a fixed point in W

^1,∞

( IR ) I W

₀^1,2

( IR ) for the operator equation

η

∫ ∞ ∫ ξ ξ σ + τ ξ

∞

− η

−

− +

= t d d

0

) ( ) , x ( G ) t ( )

x 1 ( u t ) x 0 ( u ) t , x ( ) Au

( 12.

u s i n g a s t a n d a r d a p p l i c a t i o n o f t h e c o n t r a c t i o n m a p p i n g p r i n c i p l e ( s e e e . g . [ 1 ] ) , w h i c h t h e n i m p l i e s t h e e x i s t e n c e o f a u n i q u e s o l u t i o n for 10. To show that the fixed point exists we need to demonstrate that A m a p s t h e b a l l

: ) IR ( W ) IR ( W

; ] T , 0 [ ( C ) t , x ( u { ) R (

B ≡ ∈

^{1 1,∞}

I

₀^1,2

} W R

, W u

, u

:

1, 1 1,2

1 ∞

+ ≤ 13.

into itself for T sufficiently small, and that A is a contraction, i.e.

that for some R,T>0, 0 < < 1 and all u, v θ ∈ B(R), x , , v 1 x u

, Av 1 Au x and

, u 1 x ,

Au 1 ≤ − ≤ θ − 14.

where 1 denotes the norm inside 13. , x

To satisfy 14. it may in fact be shown ([8]) that because of the

s i m i l a r i t y i n t h e s t e p s o f t h e c a l c u l a t i o n s i t i s s u f f i c i e n t t o v e r i f y that

x D x v D u

) x Av x (

) Au ( D and u x x D ) Au

( ≤ − ≤ θ − 15.

where

16.

2 ) ) t t (., )

t t (., ) 2

t (., )

t (., ( ] T , 0 [ t D sup

(.,.) + φ

φ ∞ + φ

∞ + φ

∈

≡ φ

is the norm associated with C

¹

([ 0 , T ]; L

^∞

( IR ) I L

²

( IR )) . Writing

2

E

(.) (.)

(.) ≡ ψ + ψ

ψ

_∞

17.

and noticing that ( Au )

_x

is given by the right side of 11., we find that +

+ +

≤

_{0 E 1 E}

x D

u ' ( 1 T ) u '

)

Au

(

| D t

0

d d ) x (

G ) t (

| ∫ ∫ ∞

∞

−

η ξ τ + ξ σ η

− +

∫ − η σ + τ η + t

0 ( t ) ( ) d | D

|

∫ − η σ + τ η +

+ +

≤ t

0 ( t ) ( ) d | D

| E C 1 |' u

| ) T 1 E ( 0 |' u

|

w h e r e G > 0 d e p e n d s o n l y o n t h e G r e e n s f u n c t i o n . T h u s b y 1 3 . , H 1 , a n d H 2 . ,

t , 0

D d x | u

| ) T 1 ( ) R ( E C

1 |' u

| ) T 1 E ( 0 |' u D | x | ) Au (

| ≤ + + + Γ + ∫ η

that is

, ) T 1 ( T ) R ( E CR 1 |' u

| ) T 1 E ( 0 |' u D | x | ) Au (

| ≤ + + + Γ + 18.

where the inequalities follow from the definition of the norms and properties of the Bochner integral (Yosida [11]). Hence 18. shows that provided there exists some R > 0, T > 0 such that there holds

R T T R E CR

u E T

u 0 |' + ( 1 + ) | 1 |' + Γ ( ) ( 1 + ) ≤

| 19.

then AB ( R ) ⊂ B ( R )

I n a s i mi l a r w a y , A c a n b e s h o w n t o b e a c o n t r a c t i o n p r o v i d e d 0

T ,

R > e x i s t s u c h t h a t f o r s o me f i n i t e c o n s t a n t C ' > 0 .

1 )

T 1 ( T ) R ( '

C Γ + ≤ θ < 20.

Conditions 19. and 20. ma y be simultaneously satisfied by choosing large enough R and small enough T > 0, which thereby proves the

e x i s t e n c e o f a u n i q u e f i x e d p o i n t f o r 1 2 . a n d h e n c e t h e e x i s t e n c e o f a unique solution u(x, t) to the integral equation 10., with

. )) IR ( W ) IR ( W

; ] T , 0 ([

C (

u

^{1 1,∞}

I

₀^1,2

T h e l a s t p a r t o f t h e T h e o r e m i s p r o v e d b y a r o u t i n e c o n t i n u a t i o n a r g u m e n t a s f o u n d i n R e e d ( [ 7 ] ) , r e p l a c i n g t h e i n t e r v a l o f e x i s t e n c e [0,T] by [0, [, where is the supremum of the T over which existence holds. τ τ For the Corollary to this Theorem we are interested in finding whether if the initial data approach zero at a certain rate as x → ±∞ then the solution to the problem does so also. More clearly, by way of an

e x a m p l e , w e m i g h t a s k t h a t g i v e n e α | x | (| u 0 ( x ) | + | u 1 ( x ) |) → 0 as | x | → ∞ , a some positive number, does ( x , t ) |) 0 , t 0 as

u t

|

| ) t , x ( u

| (|

x

e α | + → >

?

| x

| → ∞ T o t r e a t t h i s q u e s t i o n m o r e f u l l y w e d e f i n e a g e n e r a l c l a s s o f f u n c t i o n , t o g e t h e r w i t h a s s o c i a t e d B a n a c h s p a c e s .

Let satisfy the conditions ψ = ψ ( x ) , 0

| , x e | ) x (

1

2

<

1

α α ≤

<

ψ

≤ 21.

, IR y , x ) y ( ) x ( ) y x

( + ≤ ψ ψ ∀ ∈

ψ 22.

) continuous on R 23.

x ψ ( and

k ) 0 ( ' ) 0 ( '

k < ψ − ≤ ψ + <

− for some k>0. 24.

Further let J = J(x) > 0 , let 1 , 2 ( IR )) W 0

) IR , ( W 1

; ] T , 0 1 ([

C

Z = ∞ I and

} )) R I ( L ) IR ( L

; ) T , 0 ([

C Ju and Ju : Z u { ) J (

Z = ∈

_x

∈

^{1 ∞}

I

²

25.

with corresponding norm (see 14.)

x D x D

, 1 )

J (

z

| u | | Ju | | Ju |

u ≡ + + 26.

Note that when J(x) = 1 ∀ x ∈ IR , Z ( J ) reduces to Z since the norms are then equivalent.

We now prove

Corollary 1. Let ψ ( x ) satisfy 21. - 24. Provided ψ u

₀

, ψ u

₀

,' ψ u

₁

and ' belong to , there exists a unique solution u(x, t) to u

₁

ψ L

²

( IR ) I L

^∞

( IR )

4. under the conditions of Theorem 1, with u ( x , t ) ∈ Z ( ψ ) defined over a maximal interval of existence [ 0 , τ [ . If τ < ∞ , then

. t

as

u

_Z(ψ)

→ ∞ → τ

⁻

Proof The procedure of Theorem 1 can be repeated with the ball B(R) now replaced by

} R u

: ) ( Z u { ) R (

B

_Z

≡ ∈ ψ

_Z(ψ)

≤ 27.

I t i s s u f f i c i e n t t o s h o w t h a t i f u ∈ Z ( ψ ) t h e n Au ∈ Z ( ψ ) s i n c e t h e fixed point argument then completes the proof as before. We therefore establish estimates corresponding to those of Theorem 1 by considering t h e r e p r e s e n t a t i v e t e r m

) x ( 1 ' u ) x ( t ) x ( 0 ' u ) x ( ) t , x x ( ) Au )(

x

( = ψ + ψ

ψ

η ξ τ +

∫ σ

∞

∞

− ξ ξ

∫ − η ψ

− t ( x ) G x ( x , ) ( ) d d 0

) t (

t . 0

d ) )(

x ( ) t

∫ ( − η ψ σ + τ η

− 28.

We note that since ψ u

_x

∈ B

_Z

( R ) then )) IR ( L ) IR ( L

; T , 0 ( L )

( σ + τ ∈

^{1 2 ∞}

ψ I for appropriate T > 0. This can be

seen using hypotheses H1., H2. :-

η η

τ + η σ

∫ − η ψ (.)( (., ) (., )) 2 d t

0

) t (

∫ − Γ +

≤ t

x d x u

u R t

0 ( η ) || ψ (.) ( )(| (., η ) | | η (., η ) |) || 2 η

for {|| || 2 || || 2 } .

] , 0 [

sup u u R

t t

xt

x

+ ≤

∈

Thus

∫

^t

^{− η ψ σ η + τ η η ≤ Γ}

0

2

2

d 2 T ( R ) R .

)) (., )

(., ( (.) ) t

( 29.

A similar result may likewise be established for ( t ) (.) ( (., ) (., )) d ,

t

∫

0

^{− η ψ σ η + τ η}

^∞

^η

a n d s o t h e l a s t t e r m o f 2 8 . i s b o u n d e d i n C

¹

([ 0 , T ]; L

²

( IR ) I L

^∞

( IR )) for 0 ≤ t ≤ T . We need to obtain the same type of estimates as 29. for the s e c o n d l a s t t e r m a l s o . T h e s e a r e o b t a i n e d a s f o l l o w s . S i n c e

) IR ( L ) IR ( L )

( σ + τ ∈

^{2 ∞}

ψ I f o r a l m o s t a l l t , i t i s o n l y n e c e s s a r y t o s h o w t h e r e e x i s t s a f i n i t e c o n s t a n t C > 0 s u c h t h a t

|| (.) G

_x

(., ) f ( ) d ||

_∞

c || (.) f (.) ||

_∞

, 30.

∞

∞

−

ψ

<

ξ ξ ξ

ψ ∫

^ξ

and

, 31.

||

(.) f (.)

||

c

||

d ) ( f ) (., G (.)

|| ψ ∫

^∞ _x

ξ ξ ξ

₂

< ψ

₂

∞

− ξ

for all f(x) with ψ f ∈ L

²

( IR ) I L

^∞

( IR ) .

We denote by K and M, || ψ (.) f (.) ||

_∞

and || ψ (.) f (.) ||

₂

respectively.

Hence, for almost all .

) (

| k ) ( f

|, ξ ≤ ψ ξ ξ

Thus

∞ ∫

∞

− ξ

ξ ψ ξ ψ

−

≤ −

∞ ∫

∞

−

ξ ξ ξ

−

ψ − d

) (

) x

| ( x e | 2 d k

| ) ( f

| | x e | ) x

2

(

1

∞ ∫

∞

−

ξ ξ

− ξ ψ

−

≤ e − | x | ( x ) d 2

k ,

by 22.,

∫ α − − ξ ξ

≤ e ( 1 ) | x | d 2

k , by 21.,

ψ

∞

≤ C || (.) f (.) ||

where C = C ( α ) < ∞ for α < 1 , and so certainly for 0 ≤ α ≤

₂¹

. Hence 30.

follows at once on taking the essential supremum of the first and last

parts of the inequality.

Next we have,

∞

∫

∞

−

ξ ξ ξ

ψ

_ξ ²

||

2

d

| ) ( f ) (., G

| (.)

||

x

( x ) e | f ( ) | d dx

2

2 |x |

2

∫

1

∫

∞

∞

−

∞

∞

−

⎥ ⎦

⎢ ⎤

⎣

⎡ ψ ξ

ψ

≤

^{− −ξ}

∫ ∫ ∫

^∞

∞

−

∞

∞

−

∞

∞

−

ξ ξ ξ

ψ

≤

₄^{1 2}

( x ) e

^−|x−ξ|

d e

^−|x−ξ|

f

²

( ) d dx

≤

₂^{1 ∞}

∫ ∫ ψ

²

( x ) e

^{|x |}

f

²

( ξ ) dx d ξ ≡ I ,

∞

−

∞

∞

−

ξ

−

−

w h e r e w e u s e d t h e C a u c h y S c h w a r t z i n e q u a l i t y f o r t h e s e c o n d l a s t line, and in the last line interchanged the order of integration by applying Fubini's Theorem. On noting that by 22. ,

∫

∫

^∞

∞

−

∞

∞

−

ξ

− ξ ψ ξ ξ ψ

≤ ( ) f ( ) ( x ) e

^{− −ξ}

dx d

I

₂^{1 2 2 2 |x |}

∫

∫

^∞

∞

−

∞

∞

−

−

≤

¹₂

ψ

²

( ξ ) f

²

( ξ ) e

^(2α−^{1|)x ξ |}

dx d ξ ,

||

(.) f (.)

||

C

²

2

2

ψ

≤

we obtain 31., where C = C ( α ) < ∞ for 0 ≤ α <

₂¹

.

I t i s n o w s t r a i g h t f o r w a r d t o c o m p l e t e t h e p r o o f o f t h e C o r o l l a r y u s i n g a g a i n t h e f i r s t p a r t a n d t h e n c o n t i n u i n g a s o u t l i n e d i n t h e p r o o f o f t h e T h e o r e m . F u l l e r d e t a i l s a r e i n [ 8 ] , Y o s i d a [ 1 1 ] , E l c r a t a n d M a c l e a n [ 3 ] .

We make the further remark that the above procedure may also be applied

in some other equations - here for example, when T = T ( x , t , u

_x

, u

_xt

, u , u

_t

)

- but in this case the proof becomes slightly more elaborate involving an

intermediate space Y ( ψ ) of function pairs [u, v],

: )}

R I ( W ) R I ( W

; ] T , 0 {([

] v , u [ { ) (

Y ψ ≡ ∈

^1,∞

I

₀^{1,2 2}

: [ ψ u , ψ v ], [ ψ u

_x

, ψ u

_x

] ∈ { C ( [ 0 , T ] ; L

^∞

( IR ) I L

²

( I R )}

²

} which reduces to Z ( ψ ) once it is shown that v = u

_t

.

To finally complete the proof of Theorem 1 it is only necessary to verify that the solution to the integral equation 10. solves the differential

equation 4. (5., 6.). As is evident, we have so far considered solutions which can generally only be interpreted in a weak sense for 4. Thus we have the following definitions and a Lemma to Theorem 1.

Definition Let ϕ = φ ( x , t ), ψ = ψ ( x , t ) ∈ L

²

( ] 0 , T [ × I R ), 0 ≤ t

₁

< t

₂

≤ T

and < φ ψ > ≡ ∫ ∫

² −^∞∞

φ ψ 32.

1

t

t

( x , t ) ( x , t ) dx dt (.,.)

(.,.),

denote the inner product on L

²

( ] 0 , T [ × IR ) . Let f =(x), g = g(x) ∈ L

²

( IR )

and

^∞

∫ 33.

∞

−

≡ f ( x ) g ( x ) dx (.)

g (.), f (

denote the inner product on L

²

(R)

Lemma 1 The unique solution u(x, t) to equation 10. satisfies ))

IR ( W ) R I ( W

; [ , 0 [ ( C ) t , x (

u ∈

²

τ

^1,∞

I

₀^1,2

and for every φ ( x , t ) ∈ C

¹

([ 0 , T ] ; W

₀^1,1

( IR )) there holds < u

_t

, ϕ

_t

> + < u

_xt

, φ

_xt

> − < σ + τ , φ

_x

>

. T ,

| x ) , u (

| ) u

(

²

1 2

1

t t t

t xt

t,

φ + φ ∀ < τ

= 34.

Proof The first part may be seen by differentiating the right side of 10. twice with respect to t and noticing that all the terms are continuous in t by H1., H2. and Theorem 1.

The second part follows on substituting u

t

and u

x t

from 10. into the first

two terms of 34. and integrating by parts (see [8]) for φ ( x , t ) ∈ C

^∞₀

(] 0 , T [ × IR ) ,

then using density to include all φ ( x , t ) .

Remark 2 Results on regularity may now be found when hypotheses H1.

and H2. are strengthened and initial data are made smoother but since these are straightforward reapplications of the contraction mapping p r i n c i p l e o r s i mp l e ' b o o t s t r a p ' a r g u me n t s f o r t i me d e p e n d e n c e , w e avoid stating them and again refer to [8] for details.

Remark 3 It is possible to prove by a continuation argument that the i n t e r v a l o f e x i s t e n c e [ 0 , τ [ o f t h e r e g u l a r s o l u t i o n s i n R e m a r k 2 i s the same as that in Theorem 1 under the same hypotheses on the initial data and (.) σ and (.) τ , and the same may hold for solutions corresponding to data as given in Corollary 1. Therefore it is only necessary to obtain global existence (i.e. τ = ∞ ) of solutions in W

^1,∞

( I R ) I W

₀^1,2

( I R ) to infer global existence in any of the other spaces alluded to above under the appropriate conditions.

Lemma 1 leads immediately to a result concerning continuous dependence of solutions on their initial data :-

Lemma 2 Suppose u(x, t) and u

mn

(x, t) are solutions of 5., 6, corresponding t o i n i t i a l d a t a 7 . , a n d u

0 m

( x ) , u

1 n

( x ) r e s p e c t i v e l y , w h e r e { u

0 m

} , { u

1 n

} are bounded sequences in W

^1,∞

( IR ) such that

, ) IR ( W in (.) u (.)

u

_0m

→

_{0 0}^1,2

35.

, ) IR ( W in (.) u (.)

u

_1n

→

_{1 0}^1,2

36.

as m , n → ∞ .

Then for some τ > 0 ,

. )) R I ( W ) R I ( W

; [ , 0 ([

C ) t , x ( u ), t , x (

u

_mn

∈

²

τ

₀^1,2

I

^1,∞

Further, for all t ∈ [ 0 , T ], T < τ , as m , n → ∞ ,

) R I ( W in ) t (., u ) t (.,

u

_mn

→

₀^1,2

37.

, ) R I ( W in ) t (., u ) t (.,

u

_mnt

→

_{t 0}^1,2

38.

, ) R I ( W in star weak ) t (., u )

t (.,

u

_mn

⎯ ⎯→

^*

−

^1,∞

39.

, ) R I ( W in star weak ) t (., u )

t (.,

u

_mnt

⎯ ⎯→

^* _t

−

^1,∞

40.

Proof Every bounded sequence in W

^1,∞

( I R ) contains a subsequence which converges weak-* to a member of W

^1,∞

( I R ) (e.g. [11]). If a

sequence converges to a member in the norm topology of W

₀^1,2

( I R ) and is bounded in W

^1,∞

( I R ) then we may show that the entire sequence

converges weak-* to that element in W

^1,∞

( I R ) ([2], P.76).

Thus 35. and 36. imply

) R I ( W in

* weak (.) u (.)

u

_0m

⎯ ⎯→

^* ₀

−

^1,∞

and

. ) R I ( W in

* weak (.) u (.)

u

_1n

⎯ ⎯→

^* ₁

−

^1,∞

S i m i l a r l y 3 9 . a n d 4 0 . f o l l o w w h e n 3 7 . a n d 3 8 . a r e p r o v e d . T h e s e latter are obtained by setting

W

mn

(x,t) = u

mn

(x,t) - u(x,t). 41.

Then by Lemma 1 with <,> defined in L

²

(]0,T[ x R),

>

φ τ

− σ

− τ + σ

<

−

>

φ

<

+

>

φ

< W , W ,

_{mn mn}

, x

xt mnxt mnt t

T

T mnxt x

mnt

W

W , ) |

₀

( , ) |

₀

( φ + φ

= 42.

where the meaning of σ

_mn

, τ

_mn

is evident. We may substitute

mnt

= W

φ in 42. to obtain

. 0 W

, 2

||

) t (., W

||

mnt ^T0 mn mn mnxt

2 2 ,

1

+ < σ + τ − σ − τ > = 43.

Letting

|| u

_0m

− u

₀

||

₁²_,2

= δ

_0m

, 44.

_n ² _1n

45.

2 , 1 1

1

u ||

u

|| − = δ

and using H1., H2., 43. shows

η η

+ η

+ δ

≤ ^c ∫ ^{[ || W (., ) || || W}

_η

^{(., ) || ] d}

||

) T (., W

||

²

2 , 1 T

0

2 2 , 1 1

2 2 ,

1 n mn mn

mnt

47.

where (see the proof of Theorem 1). C = C ( Γ ( R )) B u t ^W

^mn

^{( x , t )} = ^W

^mn

^{( x , 0 )} + ² ∫

0^t

^W

^mn

^{( x ,} η ^{) W}

^mn_η

^{( x ,} η ^{) d} η

2 2

and so

^{|| W (., T ) ||} ≤ δ + ∫ ^{[|| W (.,} η ^{) ||} + ^{|| W}

_η

^(., η ^{) ||}

1²,2

^{] d} η 48.

2 2 , 1 T

0 0 2

2 ,

1 m mn mn

mn

Adding 47. and 48. finally gives an inequality to which Gronwall's Lemma may be applied, and so

, ) T ) 1 C ((

exp ) (

||

) T (., W

||

||

) T (., W

||

_{m n}

mn t

mn 0 1

2 2 , 1 2

2 ,

1

+ ≤ δ + δ + 49.

giving the desired results, since δ

_0m

, δ

_1n

→ 0 as m , n → ∞ .

Before turning to the final question of global existence, we investigate a simple property concerning the propagation of initial discontinuities in the first derivatives of the given data.

W e d e f i n e t h e ' j u m p ' [ φ (.,.)]( x ) in φ ( x ., ) a t a p o i n t x b y t h e r e l a t i o n .

,.) x ( ,.)

x ( )

x (.,.)](

[ φ = φ

⁺

− φ

⁻

50.

Lemma 3 Let u(x, t) be the solution of 4. -6. and suppose IR ,

where )

\ IR ( C ) x ( u ),

\ IR ( C ) x (

u

₀

∈

¹

χ

_{0m 1}

∈

¹

χ

_1n

χ

_0m

χ

_1n

⊂ a r e

arbitrary sets of m , n ∈ N U { 0 } points at which u

₀

' ( x ) , respectively u

₁

' ( x ) has a discontinuity. Then the only points at which discontin-

uities in u

x

(x, t), u

x t

(x, t) may occur for t>0 are contained in χ

_0m

U χ

_1n

. Proof By Theorem 1, the solution u(x, t) for 4. -6. satisfies

η ξ τ

+ σ

ξ η

−

− +

= ∫ ∫

^∞ _{ξ ξ ξη}

∞

−

d d )) u ( ) u ( )(

, x ( G ) t ( )

x ( ' tu ) x ( ' u ) t , x ( u

t

0 1

0 x

x

− ∫

0^t

^{( t} − η ^{) (} σ ^{( u}

^x

⁾ + τ ^{( u}

^x_η

^{)) d} η 51.

for almost every x. It is easy to show that the map ξ

ξ φ ξ

→

φ ^{( x )} ∫

^∞

^G

_ξ^X

^{( x , ) ( ) d takes L}

^P

(IR ) into C

B

(IR ) 1 ≤ p ≤ ∞ , and so the third term in 51.is continuous. Thus

) x ( ] (.) ' u [ t ) x ( ] (.) ' u [ ) x ( ] ) t (., u

[

_x

=

₀

+

₁

. dn )) x ( ] )) n (., u ( [ ) x ( ] )) n (., u ( [ ( ) n t (

t

0

x

∫ ^{− σ}

x

^{+ τ}

^η

^52.

Similarly,

. dn ) ) x ( ] )) n (., u ( [ ) x ( ] )) n (., u ( [ ( )

x ( ] (.) ' u [ ) x ( ] ) t (., u [

t

0

1 x xn

xt

^{= −} ∫ ^{σ + τ 53.}

By hypotheses HI., H2., adding 52., 53. implies that for t ∈ [ 0 , T ] ⊂ [ 0 , τ [ ,

| ) x ( ] ) t (., u [

|

| ) x ( ] ) t (., u [

|

_x

+

_xt

≤ | [ u

₀

' (

₋

⋅ )] ( x ) | + ( 1 + T ) | [ u ' ( ⋅ ) ] ( x ) |

∫ ⁺

Γ +

+

^t

0

)

| ) x ( ] ) n (., u [

|

| ) x ( ) n (., u [ (|

) R ( ) T 1

(

_{x xn}

≤ ( | [ u

₀

' ( ⋅ ) ] ( x ) | + ( 1 + T ) [ u

₁

' ( ⋅ ) ] ( x ) | ) exp ( T ( 1 + T ) Γ ( R )) 54.

where the last inequality is a consequence of Gr ŏnwall's lemma.

Hence for x ∈ χ

_om

U χ

_1n

the right side is zero, and the result follows immediately.

Finally we show that under mild additional hypotheses there appear solutions to 4. - 6. which exist globally in time.

Theorem 2 In addition to the conditions of Theorem 1 let hypotheses H3.

and H1 i), ii) or iii) hold. Then the solution u(x,t) of 4, - 6.

belongs to the class C

²

( [ 0 , τ [ ; W

^1,

∞ ( R ) ∩ W

₀^1,2

( I R ) ) for every finite τ > 0 .

P r o o f We ma y u s e L e mma 1 t o d e r i v e a n ' e n e r g y

^'

e s t i ma t e f o r u ( x , t )

o n r e p l a c i n g φ ( x , t ) b y U

_t

( x , t ) i n t h e e x p r e s s i o n 3 4 . T h i s d e l i v e r s

∫ ∫ ∫

∫

^∞

∞

−

∞

∞

−

∞

∞

−

τ +

+

+

^t

0 2

2

21

( u

_t

U

_xt

) dx W ( u

_x

) dx u

_xt

( u

_xt

) dx dt

, E dx ) ' u ( W dx

) 2 ' u u

(

²_{1 1 0 0}

21

+ + ≡

= ∫ ∫

^∞

∞

−

∞

∞

−

55.

where we have let t

₁

→ 0 and t

₂

= t < τ in 34.

In cases Hl. i), ii) we therefore have the a priori bounds

∫

∞

∞

<

+

_{x 0}

2 2 1 2 t

1

|| u || , ( t ) W ( u ) dx ( t ) E 55 i)

and

∫

∞

∞

−

=

→

_{x 0}

2 2 1 2 t

1

|| u || , ( t ) w ( u ) dx ( t ) E 55 ii)

In cases Hl. iii) we rewrite 55. and use Grönwall's lemma.

∫ ∫

∞

∞

−

∞

∞

−

+

α

=

+

^t

0 2 2 0 2

2 1 t

21

|| u || , ( t ) W ( u

x

) dx ( t ) E ∫ u

xt

dx dt ^. implies

xt

|| ( t ) E e

t

u

||

² ₀

2

≤

α

and, in turn, therefore

∫

∞

∞

+

α

≤

+ W ( u ) dx ( t ) E ( 1 e ) )

t (

||

u

||

_{t 1}²_{,2 x 0 2}^{1 t}

2

1

. 55 iii)

The bounds 55. are in themselves insufficient except in a special case when σ is uniformly Lipschitz continuous. Here we must supplement them by pointwise bounds on u

x

, u

x t

valid almost everywhere. To do this most conveniently, we add two further hypotheses which may however be relaxed, or disposed of entirely when the rod is finite.

H4. Let 0 ≤ K < ∞ be such that o, W are assumed to satisfy, for | φ | ≥ K ,

) ( W

| ) (

|

|

| φ ≤ σ φ ≤ φ .

H5. (case Hl. i)) Let P ∈ N , 0 ≤ K < ∞ be such that for | Φ | ≥ K , some

, )

( φ = αφ

^2p−1

τ α > 0 .

Now we differentiate 10. twice with respect to t and once with respect to x, giving

ξ ξ

∫ τ

∞

∞

− ξ σ ξ +

−

= τ

+ σ

+ ( u x ) ( u xt ) G x ( ( u ) ( u t )) d

u ttx . 56.

Thus in case H1. i), using 55.,

K t | u

|

|, u

| d

|

| x (|

G ttx |

u

| ⎪⎭ ξ ξ <

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ∞ ∫

∞

− ξ σ τ ξ

≤ τ + σ +

K t | u

|

|, u

| d

|

|

| 0 x (|

G ⎪⎭ ξ ξ ≥

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ∞ ∫

∞

−

ξ τ ξ +

+

ξ ξ ξ

Γ

≤ ∫

^∞

∞

−

d ) , x x ( G ) K ( 2

− ξ

∫ ξ

∞

∞

− ∞ ∫

∞

− α + ξ ξ

+ W ( u ) d | u t | 2 p 1 d

∞ ∫

∞

− − ξ

σ ξ + +

Γ

≤ 2 ( K ) E 0 | U t | 2 p 1 d 57.

N e x t w e mu l t i p l y b o t h s i d e s o f 5 7 . b y | u

x t

| a n d i n t e g r a t e w i t h r e s p e c t to time to obtain

∫ ^τ

+

+

^t

0 2

21

U

_xt

( x , t ) W ( U

_x

( x , t )) u

_xn

dn ( x )

∫ ∫

^∞

∞

−

−

ξ

σ + + Γ +

+

≤

^t _{η ξη}

0

1 p 2 0

0 2

1

21

u ' ( x ) W ( u ' ( x )) | u

_x

( 2 ( K ) E | u | d ) dn ( x ) . 58.

In particular therefore, for t ∈ [0,T],

∫

^{t η}

^{≤ + Γ +} ∫

^η

0

t

0 0 1

2

dn ( x ) E ( x ) ( 2 ( K ) E ) | u | dn ( x )

u

_x^p _x

∫ ∫

^∞

∞

−

−

ξ σ

+

^t _{η ξη}

0

1

2

d dn ( x )

| u

|

| u

|

_x ^p

w h e r e E

₁

( x ) ≡

₂¹

U

₁

'

²

( x ) + W ( u

₀

' ( x ) ) So by Hölder's inequality and 55.,

∫ ∫

⎭ ⎬

⎫

⎩ ⎨ + ⎧

Γ +

≤

_η

η

t

0

2 1 t

0

2 0

1

2 p p

p x

x

dn ( x ) E ( x ) ( 2 ( K ) E ) T | u | dn ( x ) u

p p p p

x

2 1 1 t 2

0 2 t 1

0

2

dn ( x ) | u | d dn

| u

|

∞ −

∞

−

⎭ ⎬ ⎫

⎩ ⎨

⎧ ξ

⎭ ⎬

⎫

⎩ ⎨ σ ⎧

+ ∫

^η

∫ ∫

^ξη

t p x p

2 1

0

2 0

1

( x ) ( 2 ( K ) E ) T | u | dn ( x )

E ⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨ + ⎧ Γ +

≤ ∫

^η

p p

t p

x

2 1 1

0 2

1

0

2

dn ( x ) E

| u

|

−

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨ σ ⎧

+ ∫

^η

^59.

It follows that the term on the left side of 59. is uniformly bounded for almost every x and each t ∈ [0,T]. By 58. and H4., t is immediate that also u

x

(x,t) and u

x t

(x,t) remain uniformly bounded, i.e. we have that for some J, 0 < j < ∞ ,

∫ ^τ

^∞

^{≤ ∀ ∈}

∞

∞

t

0 xn xt

x

(. , t ) || , || u (. , t ) || , || u dn || J , t [ 0 , T ] . u

|| _60.

Using these estimates in 10. shows

∞

<

∞

≤

∞

, || u (. , t ) || f ( t )

||

) t , (.

u

||

_{1, t 1,}

61.

where f(t) is uniformly bounded on every finite interval [0,T], Similarly, in case H1. ii) when τ = 0, 56. leads to

∫

∞

∞

−

ξ +

Γ

≤ σ

+ | ( K ) W ( u

_ξ

) d u

|

_ttx

62.

E 0 ) K

( +

Γ

≤

and 58. becomes

) ) t , x ( u ( W ) t , x (

u

_xt² _x

21

+

^{≤ + Γ +} ∫

^t

⁺

^η

0

2 21

0

1

( x ) ( ( K ) E ) ( ( 1 u ) dn ( x ) )

E

_x

63.

Therefore, by Grönwall's inequality, for t ∈ [0,T],

{ 2 E ( x ) [ ( K ) E ] T } exp [ ( K ) E ) t ] )

t , x (

u

_xt²

≤

₁

+ Γ +

₀

Γ +

₀

64.

and we obtain, as before,

∞

<

∞

≤

∞

, || u (. , ) || g ( t )

||

) t , (.

u

||

_{1, t 1,}

65.

where g(t) grows at most exponentially in time.

In case H1. iii) when , 0

2 α >

− αφ

=

τ we have

| u |

2 d | t ) u 2 G (

|

| u

|

_ttx

+ σ ≤ ∫

^∞ _x

σ − σ ξ ξ + σ

^xt

∞

− ξ

2 2

|| u (. , t ) ||

||

) ,.

x (

|| G 2 d ) u ( W )

K

( +

_ξ

ξ + σ

_{ξx ξt}

Γ

≤ ∫

^∞

∞

−

^{+ σ} ₂ ^{| u}

^xt

^|

| . u 2 | E e

E 2 ) K

( +

₀

+ α

₀^{21 2t}

+ α

_xt

Γ

≤

^α

66

w h e r e w e u s e d t h e i n e q u a l i t y p r e c e d i n g 5 5 . i i i ) . M u l t i p l y i n g b y

| u

x t

| a n d i n t e g r a t i n g ,

∫ ^{Γ ⎜} _⎝ ^{⎛ + + σ ⎟} _⎠ ^⎞

^η

+

≤

+

_x ^{t σ} _x

xt

0

2 2 1 0 0

1 2

21

u ( x , t ) W ( U ) E ( x ) ( K ) E 2 E e t | u ( x , n ) | dn

2 u ( x , n ) dn

t xn

0

∫

2

+ σ

⎟⎟ ⎠

⎞

⎜⎜ ⎝

⎛ Γ + +

+

≤

^σ

T

e E T E T ) K ( )

x (

E

_{1 2}¹ _{0 0}^{21 2}

( ^{( ( K ) E E e T} )

^t

^{u ( x , n ) dn}

0 2 2

1 0 , 0

12

2 xn

2 ^σ

∫

+

σ

α + +

Γ

+

₋

67.

from which, again by Grtönwall's lemma

( )

⎭ ⎬

⎫

⎩ ⎨

⎧ ⎟

⎠

⎜ ⎞

⎝

⎛ Γ + + σ + σ +

+ Γ +

≤

^σ

E e

^σ

t

E 2 ) K ( T exp

e E T E T ) K ( ) x ( E 2 ) t , x (

u

_xt² _{1 0 0}^{21 2} _{0 0}^{21 2 T}

68.

and we have

|| u (. , t ) ||

₁

,

_∞

, || u

_t

(. , t ) ||

_1,∞

≤ h ( t ) < ∞ 69.

where h(t) is exponentially bounded on every finite interval [0,T].

We have therefore found in each case that an a priori bound for 2

, w 1 ,

| u

| , w 1 ,

| u

|

₁

∞+

₁

( s e e T h e o r e m 1 ) e x i s t s o n e v e r y f i n i t e i n t e r v a l of the form [0,T] . A standard continuation argument therefore proves

t h a t t h e s e s o l u t i o n s e x i s t for a l l t i me t ∈ [ 0 , ∞ ] (see, for example,

Reed [7], or [8]).

REFERENCES

1. T. B. Benjamin, J.L. Bona and J.J. Mahony, Model Equations for long waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc.

Lond. 272, 47-78, 1220.

2. R.W. Carroll, "Abstract Methods in Partial Differential Equations", Harper and Row, New York (1969).

3. A.R. Elcrat and H.A. Maclean, Weighted Wirtinger and Poincaré inequalities on unbounded domains, Indiana University Math. J., 29, 3 (1980).

4. W. Jaunzemis, "Continuum Mechanics", New York Macmillan (1967).

5. P.D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964).

6. A.E.H. Love, "Theory of Elasticity" (4th Ed.) §278, Cambridge University Press (1927).

7. M. Reed, "Abstract Nonlinear Wave Equations", Springer-Verlag, Berlin (1976).

8. R. Saxton, Ph.D. Thesis, Heriot-Watt University (1981).

9. R. Saxton, in preparation

10. R. Saxton, to appear in the Proceedings of the Conference on

Ordinary and Partial Differential Equations, Dundee 1982, Springer-Verlag, Berlin.

11. K. Yosida, "Functional Analysis", 3rd ed., Springer-Verlag, Berlin.

ACKNOWLEDGEMENTS

This research was carried out with the assistance of a British SERC

research studentship and SERC grant GR/B/78083.