Technical SummaryS~MRC Report #2422 WEAK SOLUTION CLASSES FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS Hans Engler and Stephan Luckhaus

:

:I

fI

Mathematics Research Center University of Wisconsin-Madison 610 Walnut Street Madison, Wisconsin 53706 September 1982

(Received June 22,

1982)

Approved for public release

DW

FILE CoPY

Sponsored by U. S. Army Research Office P. 0. Box 12213. Research Triangle Park North Carolina 27709

Distribution unlimited

Reproduced From

Best Available Copy-

S...

t,)

UNIVERSITY OF WISCONSIN-MADIS MATHEMATICS

RESEARCH CENTER

WEAK SOLUTION CLASSES FOR PARABOLIC INTEGRO-DIFFERENTIAL

Hans Engler

EQUATIONS

and Stephan Luckhaus

Technical Summary Report #2422 September 1982 ABSTRACT

We study partial integro-differential equations of the type t

atu(*,t) + Au(',t)

(I)

+

f

0 4 t < T

a(t - s)Bu(.,s)ds = f(-,t),

,

0 in

some spatial domain

elliptic

A

operator of second order,

initial of

n C Rn,

being a linear and

and

B

of (I)

t)

together with

We give conditions on the structure

that lead to a priori estimates and show how to get the

existence of weak solutions a.e.

a quasilinear

both in divergence form,

and various boundary conditions. A

B

(u(.,t)

e W 'p()or

u(.,t)

e W 2 ,2 (g)

loc

for

from approximating solutions (that solve finite-dimensional versions or versions with modified coefficients).

on 13 testimates u(-,t)|2 + f G(V u), if solutions, and estimates on t~e L2-product

The main tools are "energy"

Bu - -div (V G(Vx u)), for W (Au,Bu ) 2 for Wloc-SOlutions.

LL

AMS (MOS) Key Words:

Subject Classifications:

35K60,

45K05,

73F15

Partial Integro-Differential Equations, Solutions, Materials with Memcry

Energy Estimates, Weak

Work Unit Number I (Applied Analysis) U

Institut fUr Angewandte Mathematik, W. Germany.

Universitat Heidelberg,

Sponsored by the United States Army under Contract No. Supported by Deutsche Forschungsgemeinschaft.

6900 Heidelberg,

DAAG29-80-C-0041.

o

SIGNIFICANCE AND EXPLANATION -

This paper studies a class of integro-differential equations that arises in some models for heat conduction in materials with memory or for the deformation of visco-elastic membranes.

*

assumptions are given that ensure the existence of weak solutions for these models; i.e., configuration.

*

Some classes of constitutive

stress or heat flux are integrable fields over the reference The models are hybrids between damped nonlinear wave equations

and perturbed heat equations, and mathematical techniques for these different problems are combined to establish existence results.

.

The resjonsibility for the wording and views expressed in this descriptive summary lies with MRC, and not with the authors of this report.

WEAK SOLUTION CLASSES FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS Hans Engler

and 7tephan Luckhaus

Introduction

1.

In this paper we want to consider the integru-differential equation t (I)

u(xt)

in together with an initial

Auxt)

-

- f

alt

0

s)divxg(Vxu(xs))d

- fixt)

2 x (O,T)

condition

(1.0)

u(*,0)

u

0

in

0

and boundary conditions •(1.1)

u I uI

on

(a\r)

x [0,T)

t (1.2)

-V * (V u(x,t) +

r

on Here

9 C 30

The function

is g

f 0

a(t

s)g(Vu(xe))ds)

-

a gradient,

+ EPn is

uOu

are traces of some

certain regularity classes. In

on, r c an, v

is

the outward normal.

a

subject to certain growth conditions,

scaler kernel with some regularity properties The functions

B(u(x't))

x (0,T).

bounded with Lipschitz boundary : 1

-

and

function

a(O) u0

B is 0u

-

i

x [O,T)

Section 2 we prove the existence of distributional

and

u0

are in

the sections below.

solutions, using a version of a

technique that has been used by J.

Clements

Dirichlet boundary conditions.

Section 3 we consider specifically the "isotropic* case

9(C)

g0 (JEJ)

Institut ftr W. Germany.

Z

In

((41)

a

a monotone function. R, f

The precise assumptions are stated in

is

for the case

and prove some results on inner regularity,

Angewandte Mathematik,

a - 1

showing that all

Universitat Heidelberg,

Sponsored by the United States Army under Contract No. Supported by Deutsche Forschungsgemeinschaft.

and constant

terms

6900 Heidelberg.

DAAG29-80-C-0041.

domain

2

Sections

to the boundary.

general cases

is

in Section 4 that the regularity estimates hold up

shown

and M. Crandall/S.-O.

Barbu ((2])

3 and 4 use a device by V.

No claims concerning the uniqueness of the solution are made in the

Nohel ([6]).

Londen/J.

it

with C -boundary,

0

For the case of constant Dirichlet data and a

(0 x [0,T]). L 10c

appearing in (I) are in

for this question and some other remarks see Section 4.

Fquation (I) has a physical interpretation from the theory of heat conduution in Consider a homogeneous rigid heat-conduicting material occupying

materials with memory. 3

Let

Q C R .

some regicn

has been proposed that

it

denote the heat flux,

q

and

history of the temperature and its

and

(K > 0

ate(x,t)

(1.5)

denoting heat sources or sinks)

temperature history

u

"temperature outside

of

corresponds, [9]).

e.g.,

relation

up to

t -

a

conductivity,

a suitable relaxation

+ divxq(x,t) - r(x,t)

0.

The boundary condition (1.1)

corresponds to a fixed

$1 and pe- ect heat conduction through the boundary;

(1.2)

to a radiation law or to local temperature control at the bondary (cf. as a perturbed heat equation.

(I) comes (I)

from the theory of viscoelastic

with

bar composed of a Kelvin solid (cf.

between strain

E

and Picla-Kirchhoff

(1.6)

I

+ K * u(x,t)

e 0 (x)

The one-dimensional version of

of a homogeneous

14•

- s))ds

a(s)g(Vx u(xt

- f 0

This physical model leads us to regard (I)

materials:

•

The constitutive assumptions

then give (I) after rescaling time and prescribing the

Another physical interpretation of

L

[19])

together with the law of energy balance

kernel)

C(r

gradient.

denoting heat capacity resp.

a0 > 0

[17],

(5],

(cf.

e

should depend both on the present value and the

e(x,t) -

(1.4)

L

e

q(x,t) - -a0 VxU(X,t)

(1.3)

the absolute temperature and

u

various general models for heat conduction

In

the internal energy.

q

a linear tensor-valued function, *

-

G(E)

a E I (21],

stress

describes (101),

longitudinal motions

assumine the following

E:

+ L(E)

denctiing the time derivative of

-22

E.

The two-

/

dimensional equation (I)

then arises In a model for the normal displacement

membrane composed of such a material.

however,

the friction coefficient depending on the displacement.

that one would have to take

arrive at (1), (1.6).

of a

correspond.

to a fixed

(1.2) can be interpreted as a friction-type boundary

portion of the edge of the membrane, condition,

The boundary condition (1.1)

u

2

It

should be noted,

to be the linear infinitesimal strain in order to

which somewhat disagrees with taking

G as a general non-linear function in

Nevertheless this leads us to view (1) as a damped non-ý'neat wave equation. It should be noted that the fundamental differences between these two physical

interpretations essentially appear in the asymptotic properties of the kernel forcing term

cf.

f;

(181

a

and the

for a discussion of these problems.

Various authors have discussed the one-dimensional vers.'on of the visco-elastic model problem leading to (M) (hence

and shown existence, uniqueness, and asymptotic

a E 1)

properties of classical solutions ([(], general equation (1) (a

arbitrary,

(71,

[12),

(231).

weak solutions of the more

have been discussed in [201 and as

n - 1)

applications of abstract theorems in [2) and [61.

The n-dimensional case for

homogeneous Dirichlet boundary data has been treated in

a 3 I

and

[4) where distributional solutions

are shown to exist. A few words on the notation that we are going to employ: For

x e ik, jxj

For

f i in

(gradient for n - t.

For

iRi, Vf - Vx f

0CW C 1 ,

(the closure taken in

r(FPX)

1-1

is reserved to Banach space norms.

is the matrix uf (weak)

I - I)i divxf a div f

Banach space);

(fl)

denote the normi

is

is

the divergence operator applied to

the usual Sobolev space (for

C0 (0) is the space of c-functions 0)

with recpect to the

derivatives wherever it

is compact,

also if

k *P-norm.

2

is

X

f 10 +3

not open;

R

is

x e Be

if

X M R

such that

wkP(fl) 0

Dependence on the variables

or

f,

exists

or

supp(f)

the closure of or

t e R is

suppressed where no confusion will arise. By

a * b(t), a e LI(oTI3),

with respect to

b e LI(OTX),

t:

-3-

X

a Banach space, we denote convolution

I

C

t a

The symbol

C,

*

b(t)

f "0

aft

-

o)b(u)ds

when appearinq in proofs, denotes a constant whose value can change from

line to line but which depends only on given properties.

d

I

4--

!I-

4

2.

of the Dirichlet Problem.

Weak Solution,

In this seetion we want to show the existence of solutions of (I)

setl i.e.

should hold on all of

the boundary condition (1.1)

if

30 e (0,T].

r

is the empty

We shall use the

following assumptions & open and bounded.

(Al)

The region

(42)

The function

g t RO +F P

C -function.

There exists a constant

is

iis

A C

is

g(')

given by

1g(c)

(A3)

The kernel

(A4)

The function

u0

i

W ' ([o,TI,R)r x

[O0T)

R

C% 3 0

• (c(, ) + G()

' C

2 1

is in

a

• n

such that

L )C>,

convex and positive, and there exists a

(2.1)

V G(C), G(0)

-

a(O) -

0, G

-

P"

R

G(a) - G(m) + 2

such that for all

being a

(lei

+ 1)

E Ir e

+ 1)

1.

satisfies

I (GO3vau0 (',2)) +÷Iva,-01(,o,)l

l01",3i) 9 0o..

0 2

u 0 (*,0) e W2 '2 (a) u

and

f (AS)

The function

f

is

in

G(VxU A

1

w 'll(0,T]t

(*,0))

< -

2

L 10M).

We are going to prove the following results Theorem Z.lt

Suppose (Al)

and b"mdary conditions

through (AS)

(1.0),

(1.1)

hold.

Then the equation (1) together with initial

has a distributional

solution

ul

i.e.

u

satisfies

2

(Vxu + a

(2.2)

g(Vxu))

V

-

u

3

-

f • *}dxdt -

f

u 0 (-,0)

4

,e2(0,R)

*

-,0)dx

09a

for all test

functions

6 C (I3 x e

[0,T),R)i

and

-5.-

(u

-

u 0 )(.,t)

for a.e.

t.

"[j

I

Pat t vcu(..s)1 2da + sup

(2.3) 0

a

(jatu(o,t)1 2 + G(Vxu(ott))) 4 K < -

(0,T)

K depending only on the data of the problem. We shall use a Galerkin procedure and

Proof.

1. find approximating solutions, 2.

deduce a priori estimates for them,

3.

show that some of their weak clusterpoints solve (1).

Step is

Let

be a sequence of finite-dJUmensional subspaces of

(V )

Lj

WIs (0),

'1 in

1dense

VM C

W"(SOO

C'(i).

00

'

3 'u(-,t)

- v +

f

v e V.

and

0 4 t 4 Tj

uVu is of class

Steop 2 U0 , f,

Let

• (*I0)

um -u

sup [O,TI a

twhich 3 u ,

(0.,T)

of

V,

the

systems

of

be a

g(Vxum + Vx-0)(*,t)) :* Vv

0.

a

Vxv x

By standard theorems on functional

and

I (•(VXu)(.,t)

+

g,

u at [OT]

4

V, for

t.

We show that there exists a censtant

+ u0 .

*

(.2.4) has a unique local solution

such that for all

IatU(.,t)12 ) +

depending only on

C*,

a

I VatU.,t-l2dt

j

" .

0o

shows also that solutions of (2.4) To show (2.5),

-

with respect to

C"

and the properties of

(2.5) "

•

(VxU'(.,t) + a

differential equations (see (131), all

solutions

- fV u (-,t) -3tu % - (',t)) 0..t, By- Vstandard - ! (f~~~"(.*,O t 0x 'tl•

.1-f(f(*,t)

for all

seek

equations

ordinary integro-differential (2.4).

We

exist on

(OTI.

we shall transform (2.4) such that

integrate over

dominate all the rest.

a 3 1, differentiate test with

(0,t),

and show that the "goodO terms (that appear in (2.5))

Let

be the resolvent kernel of

r

a

ii.e.

r t (10T] * R is

defined by t r(t) +

I 0

4.

r(t

+

-s);(s)da

(t)

-0,

0 Ct

T .

-6-

'!

It.

I

.

.

.

.

.*

-

..

..

*

-

,

-.

*

•

•.

4

. .

.

-V

,-

4.

Than

r

is as regular as

"(2.6) "We apply

a * y (2.6)

a,

- a

to (2.4),

and for

on

y,s e L (oTIR)

differentiate the resulting identity,

t.

-s

+ r

on

a

[0,T]

with

"ylt) I- f

v e L2(0,T;V_).

functions

1 * y

iff

[0,T)

g(Vx u(lt)) * Vxv

and note that it

We then choose

v(T) -

is possible to take t-dependent teot u (',),

and integrate from

0

to

The result can be written in the form

(2.7)

1 1 (t)+

2

l (t)

+ 1 3 (t)

1 4 (t) I

with the following notations

,

11 (t)

t

22 +*+r(o).

-I1Iat..t)I 2a

t

2dou+l-,fld fIf 1i'C(.e)1 . ÷5 '1

0a

Ifft%(.t)l2 2 (ti -

t

ff IIV3(• * I2ds

f IVs(..t)12

r(÷) .

00 a

S, :t)

fft

r(0)

ff/

IFC )(.,)d

s

a (3f'(*.) a ÷ r "I

f(*.s)

14(.) ff (

-

rr

ff(-s +(r (.a

s)

0

30(*.)

-

r

2t a

2

3

Vxl(e.e)I 2 ds'

/I/

-C"Sf~

j-

0 a

pt

IVxU"i(et)1 2

j

2 _ 2"( G(m('.t)12 +(',))d rxlu(,)gvu ds =f IV .)-*clI.

~ ~~ (..)1d Irm(as.gCV

I3(t) . I~

00 I 4 (t) 1.

a,•(..,s).(;, a 00

1 fIt24

13

C

I3/oI3 (u',)1 d,

-.

t

3

(1.).(;

0a

IV"us.¶.I)12 o(,u

+'

a

0(..2))*

iIf .,e)ds

32 au0 Os)3aUmsd

t

ff j

*:

(-,a)

00aVxJ~(*,s) u

ft

S+

*( gl

t

• I ~(s)'(f 0

a

3(

r

(*.mllld9

-t

)

,))u(-,)d

dh, o.)12, + .

ff

t 2 d" + Ifx "'.(',9)I

0Oa

-7-

*I

C'f I G'(V u (-,s))ds . 0

C

ee LI(0,TItR);

with

Inserting all

using the properties of

these estimates into (2.7)

u.,

and

(u')m~l*

in

f

and using Gronwall's

We extract a subsequence of the

Step 3s

g,

the last

estimate.

lemma we get (2.5).

again labeled in

the same way,

such

that m

(i)

ua + u

(ii)

Vx u

(iv)

2

strongly in

2

L (0,TiL ())

1

Vxu

weakly in

L 2(0,TiL 2(1)

3(ii)ua + atu

weakly in

L (0,TjW ,

g(V A)

x

a) weakly in

1

2

+ C

with a suitable function the choice (iv)

2

1

1

n

All these limits exist due to suitable imbedding theoremal

C.

sequentially precompact in

(1))

.2(O,TiL (2,10))

g(Vu )

possible since the

is

2

LI([O,T] x 12,30)

are equi-integrable (of.

[9]

and hence weakly

and Lemmas 5.3,

5.4).

Next we want to use thet actually (v)

atU

tu

Suppose this is

L (0,TiL (Q1))

strongly in

trues

what is

needed to complete the proof of the theorem now is " g(V u)

(2.8)

a.e.

on

0 x 10,T]

To show this, we use a version of a eonotoaLcity argument which has first

" Clements ([41). "identity once,

Transform (2.4) and take a test

resulting identity from to

s

(2.9)

function

to

-

I

(T

-

t)

I

(At,,"Ov>dt

•

3 v, v C wl12 (0,T',V

with respect to

+

f

t

Integrating the

and from

T

I at-" u

0 a

+ f 0

using the abbreviations

.t.-

a

differentiate the resulting

0

to

T

with respect

then gives

T

*

0

again by means of (2.6)F

been employed by

(I

,

1

(v)(t)

v dt - T

f

tu (.,o)

v(.,o)

a

+ Ist, 2 (v)(t) + Zm,3(v)(t))dt

f 0

1 4 (v)(t)dt.,

(T

1 1 ,(V)Ct) -

It

fa

-t)

(r

3 u")(*.t) a

*v(*,t)dt

t x atu'C..u)V xvC(ui0de + (T

Im 2 (v)Ct) - f f

+

As

t)

Iý. 33 v)(t)

(T

-

14 (v)Ct)

(T

-t)

V ua (,O))v

(g(Vxu"'C',t)) +

f(3 tfC*,t)

at,~uVatu'~ by

V3tu, g(Vu*)

v

to be in

by

Writing

(2.10)

f

-4t e

u C',t)

m-O

)

-~

by

I k(v)(

V %(*,t)

't

*

,t) (1 4 k 4 3) and

(by density).

atu'

by

The resultin~g

gore precisely, we

and additionally

GCV VC.,t))
0

1

equipped with the

there exist

(2.13)

C(C)

> 0

IzA L (

and

£

K e N

zl

be the

W1,'-closure oE

We claim:

such that for all

1,2M + C(M)

"

|Z:

*

z e W 2(a) ,

K For else we could find an

c > 0

and a sequence

1

in

(zK )K>

2

W , (1)'

ZK w¶,2

1, 1

such

that

(2.14)

1ZKK 1 2

1

Using the compactness of the imbedding limit

zlzl

L 2 (0l)X > C.

By density, 1(k)+: all

* 0

C > 0

We apply (2.15)

seueceinan sequence in it

any

to

2

(0)

+ L (f()

by (2.14)

zK + 0

we extract a subsequence with 1z K 1

C,K 2

From (2.13)

w e L (0,T;W 1

and see that it

in

suffices to show that this In a Cauchy-

2

(0,T;X).

In

fact,

will be enough to show that

shows that for fixed

an equi-intograble set in XK

L (0T;XK)

2 since

w

(w )

-y

is

Dtu

2

weakly in

L (0,TjL2(0)) 2

precompact in any

L (0,T;X•)$

by the claim

Now the differentiated version of

K (13tWm

is

we conclude that for

(1))

1 2 L (0,TiW , (ML))

then follows from standard diagonal sequence arguments. (2.4)

2

2

((1))
0

5.6.

Hence in

(4.2)

in order to take care of boundary terms.

ve simply choose

1 I1

step 2

. IVx

i(

and

fT IV (f0g 0

Ul 2 1) 14

jf. ai,j

'(lvuI)Il3auNI 2

12 + f•;

f

u

u") 1

2

- C

c(IVuN1) • IVuN12

(IV u) x

+ Z V U) 2+

1) • Hn,

can be estimated from below by

f 9j a

is a priori

sup (0,T]

-

va" 2 bounded by (3.4) and Leam Theorem 3.1 fulfill

fe IV2uI (.t)


0,

e

N 7 UN" V1 "x x

f ;(IV xVU¶I)

XX

9

C

lag 5 0,

a small constant.

(see Le.ma 5.1),

r

the expression

X

manipulations similar to those in

f a UN

in (1.1)

- &tv(c(IVxUMI)VxuM) • C2

f'Axu

39

if

0.

In step 2 of the proof of Theorem 3.1,

2

of Section 3,

(N

-I

the mean curvature of

if

go,

general nonlinearities

aa

is

non-neegative.

A possible

class is described in the hypothesis

(BS)

R3

go 1 [0,-)

d C

((gor) + L)

*

r) ) 0 on

There exists

10,) r ) 0

such that for all

Cc ) 0

there exists

0

(0,-).

such that

(4.3) For any

C,

g(C) - go(1(1)

being locally Lipschitz-continuous on

L ) 0

a constant

is given by

"

g s 3"

The function

r

(4.4)

Io(r) •rl

S•

(g0 s) + L) 1f

0

ode + Cc

We then get the Theorem 4.2t H

be bounded,

1 C le

Let

(with respect to the outer normal).

u0

V' 2 1()

t.

not depending on

C 2-sfooth,

30

Let

(35), (A3),

hold with

with boundary conditions

satisfying (3.4) and 2 sup f IVxu% (.,t) (0,T] 0

(4.6)

and (AS)

10

utag0x(0TI u

(A4)

Then the equation (I)

(4.5) has a distributional qolution

with non-negative mean curvature

< "

Sketch of the proof, N ) 0

As in the proof of Theorem 3.1, we define for

*

g90(r) - inf(g0 (r),N), gA(C) - 9("I(4) Then

g0

still

satifies (35).

distributional solutions above,

2

uN

2

um e t7(0,TWM' (A))

for all

N.

-.

(3.11) with

i = 1.

-1

and get unique

a

to the equation (M) to get (3.9), and multiply with

-A u N which is an x

This gives the identity (3.10) and - after rearranging terms

-

We then apply Lemma 4.1 and conclude that

f .-

gm

Still following the lines of the proof of

(take difference quotients),

test function.

•"admissible

replaced by

By an argument similar to the one used

that satisfy (3.14).

Theorem 3.1, we apply the resolvent kernel of

"differentiate formally

g

We solve (1) with

C •

AuuMdiv((gl(gV.U M) + L) • V u

"-24-

0

N

a step towards a simple non-linear version of Sobolevskil-type estinmates

This Lemma is

for linear second-order elliptic operators as stated, e.g.,

in

[3].

We thank Prof. A.

Friedman for pointing this out to us. Obviously the identity of (4.1) still u e W2' 2(f() 6 W1' 2(0),

g

holds under the assumptions

locally Lipschitz and bounded,

r w gt(r) - r

bounded, by a

standard approximation argument. It

in (1.1)

r

1) in order to take care of boundary term!

If

is now possible to use (4.1) to modify the arguments of Section 3,

is empty and

if

0.

uIx[OT]

In step 2 of the proof of Theorem 3.1, the expression

f A UM

M0

is

2

C -smooth and

C e W 'm 0,

uNIag M

.

*2

x

$x xX had to be estimated from below,

)

div(;(Iv UM1 V

we simply choose

1 and get by Le-a I

4.1 and

manipulations similar to those in step 2

fOx A

•N* div(,(IVxU"I)VxuM )

SC.

K > 0

(IVxUMI)

a small constant.

I 2 f (see Lemma 5.1), 5.6.

I

•IVx1u12 +

x

g(I'vuMIII)

auNI2

"*lu

2

(n

1)

H

By a standard trace theorem this can be estimated from below by

i,j

C > 0,

• Oi,j I I(/(uI)u

13

and

g( IVUKI) auM)1 2 _ C

f ;(IVu"I)

I2

IVu

f ;(.IVuHI) 0

1 VuMI

2

is a priori bounded by (3.4) and Lemma

Hence in this special situation the solutions found in Theorem 3.1 fulfill T

(4.2)

fI 0

(g0Vx 30IV ul) + L V u)I

+

sup

IVuI 2 (-,t)


0

64(r) C (5(r)

* L,

*

r 6

1161).

let

L ) 0

and

N.

C(C)

; 0

rL [

C(C)

such that for all

r e R

.

r

Let

(r) -

9

Then for any

,'5S) L a'ds. 6 > 0

there exists some

i6(r)J Cl Proofs

Let

r e 3.

) 0

a

a simple consequence of vell-known trace theorems (cf. Let

1)-dimensLonal

luI 2

6lul IV., l2 + C(C)

2

Lema 5.2:

(a -

(,)

(5.1)

This is

0

DO of class

Z(6) ) 0 *

3r) W

such that for all

r a a

CMI).

then.

r

ISI0J +

f

C

11(0)M *

J

4

1J(0011

B (r)

1i(r) + L * r4

($'(a) * 1ldm

a

44s(m) +

* /ij#s)

* V¢1J(r) + LrI

* € 21r) 4 1(0)Ca(rl

J(l)J

C

46

-S

L *-t

+ C()"

10(r) + LrI

4

hence

li5Cr)I

C

8 2Cr1 W * C1 1€1 * L • LI)

2 C 2C8 (r)

-27-

(c).)

÷ Clc? do

/'

ima S.3s

ý(C)_-G(C) aC)

0 1 1n

Let

It

be differentiable,

let

L ) 0,

+ h (ICl2 + ) eplteade ) 1) be positive, and lot 2C

let

=VG

satisfy any of the following

V

hypothesem. (1)

ftere exists

C)

0

such that for all

IS(M) • U(l)

The function

9

(

0 4 L Thnnfor all

moditidon (i1)

"Pr.of

(I),

Ig(C)l

in

case (ii).

put

Ig(C)

+ 1)

- C. go

is locally Lipschitz

much that for all

C ) 0

0(r) + L ), 0

9

((q0(r)

+ L-)o r)

there exists

let

B(d)

ums

-

i(r)

max

-

I,(C)

*

g0(r) + L. 9

"I 1

;(ICI)

C(d)

r )0 )

C • (b,(r) +'L

0

such that forall

Z()O.

C aIF

nl (d.*m , G-(C) ÷

Then for

,r +L"

(a(.)* a)'

aft d

f

I 0

•++

ro 0

0

and we are back in

and

. 1 • (o(

,

case (I).

-28-

(m)

:1

ads

) .

r0 "ax(IuIICI)

r0

0

*

n.C t 3'

*

ICI - 1l11

r0

""•

C

Then for any

ICI -• ll + L.- ICI •

C ;(ro) •

.4

01

Implies condition (I).

Zn case

*

0

1)

g(j(IC)

-

continuou,. and there ex*Ktz a constant 0-

+

4Ii C g(C)

is gven by

C.n

- • I

I

l

.

Lama 5.4t N

Let

L (0,3t").

for a..

9

be a finite measure space,

For all

u e N

and

9 > 0

M C L (0,10

let there exist

a bounded net and v e M and

C€

such that

x ea

I'(x)l 4 a

Ivlx)t + cc

A is equi-integrable (and hence weakly sequentially compae. in

Then

0

J

liz

k-

lal

" 0

uniformly in

u

L (AR )),

i.e.

e AM.

(Iul-lk}

Proof$

I Iu! {lul,)

I

S(lul)-k}

(lv

• K+ Cc

C + C)

k

MI

where

@supf IlI, yalA

N

Lesa 5.5,Lo

Cv (Le

sup f hal ueN 0

M1

be a sequence in

f

N + C1

L([0,T] x 0,10),

I

such that (in the notation

of Section 2) m

g(w ) .

w weakly in

C

LI (0,T]

x 2,e)

\2

and ego sup f (G(v)

for all

Let

+ L *

CK X

2)

a.

v C LI([0,T] x 12,),

[0,T) f glw)

(G(v) + LV

eon sup

" v(°)

*

2

l)(0.0t)
0,

For

f0

• v)(.,t)dt-

f (g(w)

*(t)

I

+j

0 (,-

9

9*f') •WC

*it)

0

then

LO(0,TsL).

f qw,)

#(t)

0

T0

t

Let

inf(Nsup(-N,v)).

VN)(.t)O

t

(.Ol-t)dt N

The first term can be estimated by T"

L

p- 2

÷

G(p) -G(p)

by (2.1),

N.))

-

+ G(R,1

(G(,)

C.

.1,,

(5.3)

+ 1)

00a

for any

+ 1,

R 3 1, (C)) *

(CC*

0(K 0 p)

also Implies

e (.1) sinc*

is bounded by

oence the expression (5.3)

I

•'

T

[C.

n-))

l-

+ 1)

a

0

.( - V. - A,") ° -' G•O)., as N ,am

~~~~ic

for any R., we. thus, der.ive. from

0

O0 2

(S.2) and the convergence of the second term in (5.2)

lim sup

V

at,l

j eC

which shows the claim.

R

and

N,

Lema *0 5.61

Let

go & (0,-) # i

for any

•v-

#(t) •f(c(Vm)

r

Than for all

C > 0

there exists some # al 4 a

IW g 0 lr) Proofs

First note that

positive for

0.

r

rs# (g 0 (r)

Further for

G(0 r) -

be as in (32).

*

+ Lis

(Go() • L1 )

r2 > r 1

such that for all

) 0

C()

5ds, L, a

g(ea) 0

2

) + C(c)(G00r)

• L1

a

in (32).

ras ) 0 r

2

1.

is monotone and (without loss of generality)

*r

0

-30-

1"

il

r

....

'~7-/ -

(9 (r2 + L1 *r (q0 (l1 ) + L1 1 * r

~

-

2 9otM r) r

"r2

r2c+ d..±gim

4 exp(J

So either •

9

*

r

asg then (gol)

+ LO)

r

(

(goI(r) + LI)

*

* +

(90

0 1(CS

CO

r2

l ÷rI I)

• (g

• 4

• •

÷ L)

(9001

O)+L

I*

I.)

(go('O

82

On the other hand, L 1

2

+ L11

ads )

(g+(r)

(90

r) + L

2

(90(s)

0 r+J (9,()

C

r2

r

(G.or)

2 0a

r1 2

+

hence (WO)

2 * r

+ )1

Hence 0C(golf) * . 1 1 • r • .4

L1 +-Lr

(C + 2) * (G(O)

CC ÷ 2) * € • (Golf) +*

2

r ) + •

Combining this with Ir"

*I

AJ€ r2

46

2 1 W(solj)÷5- )•

2+

*'

for all

2 482 8

given the desired estimatle.l

-31-

/.

*

RZFZR9HCZS

-

*

11)

on the Existence of Solutions to the Equation

G. Andrewa,

Differential Equations 35 (1980), 121

V. Barbut Cuaxa

[31

ua

+ ONu )X*

xx

.

Ann. St. Univ.

*Al. 1.

365-383. A Variational Inequality Approach to the Bellman-Dirichlet

H. Broxis, L. C. Evanst

Equation for Two Elliptic Operator.. (4) J. Clement..

uxx

200-231.

Integro-Oifferential Squatton. in Hilbert Spaces.

19 (1973),

-

Existence Theorest

Arch. Rat. Mach. Anal. 71 (1979), 1-13.

for a Quasilinear Evolution Equation.

SLAM J. AppI.

Math. 26 (1974), 745-752. (51

3. D. Coleman, N. 3. Gurtint conductors.

(61

squipreastnce

and Conatituti~ve Equations for Rigid Beat

2. Angew. Math. Phys. IS (1967),

M. 0. Crandall, U.-O. londen, J. A. Nooel: Differential Equation.

(7) C. M. Dafermoss

199-207. An Abstract Nonlinear Volterra Integro-

J. Math. Anal. Appl. 64 (1976),

701-735.

The Mixed initial-Boundary Value Problem for the Equations of

Nonlinear one-Dimensional Viscoelnaticity.

J7.Differential Equation. j (1969),

71-06.

Schwarta,

161

n. Dunford, J. T.

[91

G. Duvaut, 7. L. Lions:

U.

Linear Operators, 1. New York 1955.

Inequalities in Mechanics and Physics.

Berlin, Heidelberg,

New York 1976. (101

N. N.

Findley, .7. 5. Lai, L. Onarant

Materials. fill

Amsterdam, New York, Oxford 1976.

H. Ga~oewki,, X. Gr5ger, K. Zachariave Operatordifferentialgleichungen.

1121

J. K. Greenberg: 00 Xtt - 3(XX)X)

Nichtlineare Operat..rgloichungen und

Berlin 1974.

O,% the Existence, Uniqueness, and stability of the Equation AX

*x

.J

Math. Anal. Appl. 25 (1969),

(131

7. Hale:

114)

0. A. Ladyzenskaja, N. N. Urallcevas

Functional Differential Equation..

Elliptique.

*

Creep and Relaxation, of Nonlinear Viscoelastic

S75-591.

Berlin, Heidelberg, New fork 1977.

Equation. ens D~rivies Partiellea do Type

Paris 1968.

-32-

1151

J.

Queleries M~thodes do R6solution des Problemes aux Limites

L. Lionse

Paris 1969.

non Lintaires.

[161

J. L. Lions, 1. Mageness Berlin,

(171

[181

R.

C.

MacCamys

An Integro-Differential Equation with Applications in Math.

R.

A Model for One-Dimensionalo

C. MacCamys

Math. 35 (1977),

J. W. ILnziatot 29 (1971),

(20)

M.-L.

M.

J. Serrins

(231

Nonlinear Viscoelasticity.

Quarterly

29-33.

1978.

Rheology.

Berlin,

Heidelberg,

Quarterly Appl. Math.

Proc.

Helsinki Symp. on

New York 1979.

Handbuch der Physik (Ed.v

S. rlugge),

vol. VI.

Heidelberg 1958.

The Problem of Dirichlet for Quasilinear Elliptic Differential Equations

with 14any independent Variables. (1969),

1-19.

On Some Nonlinear Problems of Diffusion.

Berlin, G~ttingen, 1221

(1977),

Heat Flow.

187-204.

Raynals

Reiner:

35

On Heat Conduction in Materials with Memory.

Integral Equations (21]

New YorK 1972.

Quarterly Appl.

Appi.

(191

Heidelberg,

Non-homogeneous Boundary Value Problems and Applications.

Philos. Trans.

Roy.

goo. London,

Ser. A,

264

413-496.

Y. Yamadat

Some Remarks on the Equation

ytt

(Y)y

Yxtx

f. Osaka 3. Math.

17 (1980), 303-323.

• (I

HE:SLs

ccr

-33-

L

SECURITY CLASSIFICATION OF THIS PAGE ("omn Des Entered)

REPORT DOCUMENTATION PAGE

READ INSTRUCTIONS

BEF EORE COMPLETING FORM

2. GOVT ACCESSION NO.

I. REPORT NUMBER

2422

t, o

.'&

RECIPIENT'S CATALOG NUMBER _I

_

s. TYPE OF REPORT A PERIOD COVERED

4. TITLE (and Subltl)

Summary Report - no specific reporting period

WEAK SOLUTION CLASSES FOR PARABOLIC INTEGRO-

DIFFERENTIAL EQUATIONS 7.

S.

I_

6. PERFORMING ORG. R9POR7 NUMBER S.

AUTNOR(a)

CONTRACT OR GRANT NUMBER(e)

DAAG29-80-C-0041

Hans Engler and Stephan Luckhaus III. PERFORMING ORGANIZATION NAME AND ADDRESS

SO. PROGRAM ELEMENT. PROJECT. TASK

Mathematics Research Center, University of Wisconsin 610 Walnut Street

Work Unit Number 1Applied Analysis

AREA & WORK UNIT ":UMBERS

Madison. Wisconsin

53706 ...

....

....... 12. REPORT DATE

. C CONTROLLIG OFPICE NAME AND ADDRESS

U. S. Army Research Office P.O. Box 12211 Research Triangle Park, North Carolina

September 1982 IS. NUMBER o0 PAGES

27709

MONITORING AGENCY NAME S ADDRESS(If differmf tham Cmontrlliln

14.

33 Office)

1I.

SECURITY Ci.ASS. (of tfle report)

UNCLASSIFIED

* IS14

DECL ASSI FICATION/DOWNGRADING SCHEDULE

DISTRIBUTION STATEMENT (of tOf* Report)

*-S.

"Approved for public release; distribution unlimited.

I7. DISTRIBUTION STATEMENT (of the abstract etered in Bloc

. it dlifrae

ho

Report) -.

IS. SUPPLEMENTARY NOTES

IS.

4

KEY WORDS (Continue an reveree side It necessary and imdltl

by block mmbor)

Partial Integro-Differential Equations

Energy Estimates Weak Solutions Materials with Memory 20. ABSTRACT (Continue -n ,revese sid, I necessry and Identify' by block ,,mbo*)

We study partial integro-differential equations of the type t

(I)

atu(,t)

+ Au(',t) + f a(t - s)Bu(-,s)ds - f(',t), 0 < t < T 0 "in some spatial domain fl C eRn, A being a linear and B a quasilinear elliptic operator of second order, both in divergence form, together with initial and various boundary conditions. DD

IJAN73

1473 , EDITION •AN.OF

I NOV 65 IS OBSOLETE

We give conditions on the structure

UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Wien Data Bntered)

'2.

20.

ABSTRACT

cont'd.

lo of

A

and

B

that lead to a priori estimates and show how to get the existence

of weak solutions

u(*,t) e W2, (92) for a.e.

(u(-,t) e W1P(Q~) or

from approximating solutions

(that

solve finite-dimensional

or versions with modified coefficients).

Son

IItu(-,tl 2 2 L2

+ f

I

G(Vu), 2

if

Bu = -div (Au,Bu)

versions of (M)

The main tools are "energy" estimates (V G(V u)),

x

and estimates on the L -product

t)

2

for

for

Wl'

-solutions,

2,2

W2-soluticns.

LL

'I

'1-

"•, " /