4lrfqo David R. Harnrin Technical Information Services, ANL. evolution problem with nonlinear boundary condition* HONGFEIZHANG

J < 1

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icat ions Services

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David R. Harnrin

Date

4lrfqo

Technical Information Services, ANL

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Accordingly, the U. S Government retains a nonexclusive. royalty-free licenre to publish or reproduce the published form of this contnbution, or atlow others to do sa, for U. S. Government purposes. I

On

the

sputtering of metals and insulators: A nonlinear evolution problem with nonlinear boundary condition*

HONGFEI ZHANG

Department of Mathematics The University of Texas at Austin

Austin, Texas 78712

In this paper we consider the following nonlinear evolution problem:

{

⁼

- P ( W )

in O X R + on d R x ~ f

where 0 is a bounded domain in

IR” ^{(N 2}

1) with smooth boundary

I? dR,

n denotes the outward normal at ⁵ E dO. The function 9 is smooth such that cp(0) = 0, $(s)

>

^{0 for s}

>

^{0 and cp‘(0) = 0. p(s) is a} nonnegative Lipschitz continuous function on with P(0) = 0 and P(s)

>

0 for s

>

0. uo E Lm(0) is a nonnegative function (for precise assumptions, we refer to Section 1).

Problem

( P )

arises in the study of thermal evaporation of atoms and molecules from locally heated surface regions (spikes) invoked as one of several mechanisms of ion-bombardment-induced particle emission (sputtering). Let

R

denote the locally heated region and let u be the temperature. Then in the case of particle-induced evaporation, the Stefan-Boltzman law of heat loss by radiation is replaced by some activation law describing the loss of heat by evaporation.

In

S i p u n d and Claussen 1121, Sigmund and Symonske [13] the following model, when the spike is approxi-

*This work was supported by the Applied Mathematical Sciences Subprogram of the Office ^i.

qj-

^{, ~}

sh.

Energy Research, US. Department of Energy, under contract W-31-109-Eng-38.

This report was .prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness,

or

usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER

Portions of this document may be illegible

in electronic image products. Images are

produced from the best available original

document.

mated as an ideal gas, is proposed,

ai2

X

IR+

{E=-

^u-2e-l/u

Notice in this example that the function

p(.)

in

( P )

is Lipschitz but non-monotone.

The equation in ( P ) is the so-called degenerate diffusion problem, which has been extensively studied in recent years. See Aronson [l] and Peletier [ll] for references. However, when dealing with the nonlinear flux boundary condition,

p(.)

is usually assumed to be monotone, see e.g. Brezis and Strauss [5] and Benilan, Crandall and Sacks [3]. The purpose of this paper is to provide a general theory for problem ( P ) under a different assumption on

p(-),

i.e., Lipschitz continuity instead of monotonicity. The main idea of the proof used here is to choose an appropriate test function from the corresponding linearized dual space of the solution. The similar idea has been used by many authors, e.g., Aronson, Crandall and Peletier [2], Bertsch and Hilhorst [4] and Friedman [8]. We shall follow the proof of Bertsch and Hilhorst [4].

The paper is organized as follows. We begin by stating the precise assumptions on the functions involved in ( P ) and by defining a weak solution. Then, in Section 2 we prove the existence of the solution by the method of parabolic regularization.

The uniqueness is proved in Section 3. Finally, we study the large time behaviour of the solution in Section 4.

$1. Preliminaries.

We begin with the following hypothesis on y ,

p,

^uo.

H1:

H2:

H3:

y E C 3 ( ~ + )

n c’(E+) ,

^{~ ( 0 ) =} ~ ’ ( 0 ) =

o ,

^Y’(S)

> o

^{for s}

^> o P

^E

Co’l(E+) ,

^{p(0) = 0 and p(s)}

>

0 for s

>

0

uo E LO”(S1)

,

^uo

2

^{0 a.e. in} Q

.

Definition 1.1. A function u : [O, co) ^--f

L1(R)

is a weak solution of

( P )

if

it

satisfies

i)

^{u E C([O,}

T); ^L1(R)) n

^{LCO(Q~) for any}

T >

^{0 and}

QT

ii) y(u) E

V2 ( Q T )

for any

T >

0, where

V2 ( QT)

is the Banach space equipped

R

x (0,

T)

with norm

for any $ E C1(gT) and

T >

0.

It is well known that any function in

V ~ ( Q T )

admits a trace in L2(ST) where

ST

=

dR

x (O,?'], see, e.g., Ladyzenskaja

et.

al [lo, p.781. Therefore the integrals in iii) are well defined.

A weak subsolution (supersolution, resp.) of

( P )

is defined to be a function u :

[O, ^{00) 4}

L1(R)

which satisfies i), ii) and iii) with equality replaced by

5 (2,

resp.).

82. Existence

Consider the following approximating problem ( PE) ut = &&(U) in QT

a(?&>

^- _{- P E}

_{( 4 4 )}

_on

_aS-2

_{x (0,}

_T)

d n

i

^{u(zJ O) =} U O & ( z ) in

R

( P d

where ^{Y E} E

c"(R+),

cp,(O) = 0, y:(s)

2

C(E)

>

^{0 for any s}

2

0, y E and p:

converge uniformly to 43 and q' respectively on compact subsets of

IR'

^as

e 3 0;

where

A

^E C"(Rs)

n C

^{0 1 ' (}

R+

), pE(0) = 0, ^&(s)

>

0 for all s -

>

0, ,Be converges uniformly to ,8 on

'IR+

^and

5

C for some C

>

0.

The construction of y e and tion of uOE, we refer to _[4].

is standard and it is omitted here. For the construc-

Then, for each e

>

0, the classical theory of parabolic equation asserts that there exists a unique solution ^uE E C2J(QT) of problem

(Pc).

Next we derive some a-priori estimates on u,. We begin with the following comparison principle.

Lemma 2.1. Let u and v E C'J'(&T) be the two solutions of problem (Pe) cor- responding to the initial functions uo and vo respectively. Then u(t)

5

^v(t) if

uo Lvo.

Proof. Let z = u

-

^{v. Then z satisfies}

+

^{b,z = 0}

. { ^on dR x (0,T)

where

1

a, =

1

^(p:(6ue

⁺

⁽¹

^{- Q,)}

^dB

1

b& =

1

P:(6(P&(uc)

+

⁽¹

^-

6 ) 9 € ( V L ) ) ^*

The result of the lemma follows from the comparison principle for the linear prob-

lem

. I

Lemma 2.2. Let u, E C2l1(QT) be a solution ofP,. Then, 0

5

u,

5

I I u ~ ( l p in

Proof. Since 0 is a solution of

(P,),

therefore u,

2

0 by Lemma 2.1.

By the strong maximum principle, ue cannot have a positive maximum on

XI

x

>

0 at this point. But = - P e ( v C ( U e ) )

L

^0,

(O,T), because otherwise

which is a contradiction. Therefore the maximum of u, must be achieved on

t

= 0,

2. e.

UE

L

I I ~ O & l l P

L

^II~OllL..

- 1

Lemma 2.3. The problem

(Pc)

^{has a} unique solution: ub E

C2+a(gT)

for each

Q E (071).

Proof. See Theorem 7.4 [14,

LSU]. I

Lemma 2.4. Let 0

5

^r

< t 5 ^2’.

independent of E, such that

Then there exists a constant

C(T) >

^0,

Jn(gradv&(4)2

5

C(2’) ^*

W )

^{= I’ve(r) d r}

J, @ € ( 4 t > - J,

@€(U&-> =

i’ ^J,

v & ( 4 % % ( u z )

=

J,‘ J,, ^%(%)A

( P E ( 4 ) -

J,’ J,

^(grad

v & ( 4 ) 2

J,’ ^J,

(gradv€(4)2

5

C ( T )

-

Proof. Multiplying equation by pE(ue) and integrating by parts. Set

Then,

Therefore, by the boundedness of u,,

I

We shall use the following compactness result, which is due to DiBenedetto [7].

Lemma 2.5. For every r

>

0 there exists wr(-) :

R+

^{+ Et’,} ~ ~ ( 0 ) = 0, continuous and non-decreasing such that

IUe(zl,tl)-U~(~z,t2)1

5 ^v((zi,ti)

^E ~ X [ T , T ] , i ^{= 1,2.}

In addition, if uo(z) is continuous in

a,

^then ( z , t ) ^--+ u e ( z 7 t ) is continuous with modulus of continuity wo(-).

I

Now we use the compactness result to prove existence theorem.

Theorem 2.6. There exists a weak solution u of problem

(P).

Proof. From above estimates and compactness result, we conclude that there exists a u E

L o o ( Q ~ ) n

^C(0 x

(0,T))

and a subsequence of { u e ) , which is denoted again by u,, such that

i) u, + u uniformly on all sets of the form

a

^x

^[.,TI

^{with r}

^>

^{0 (by}

Lemma 2.5)

ii) u, + u strongly in L2( Q T ) and in

L 2 ( S ~ )

(This follows from i) and uniform boundedness of u, in

L"(R).)

iii) p,(uE) + p(u) weakly in L2(0,T;H1(Q)) (This is the consequence of Lemma 2.4. One identifies limit to be ~ ( u ) by the fact that (pc(uE) -+ cp(u) strongly in L2( QT) and the dominated convergence theorem.)

iv) ,BE(yE(u,)) ⁺ @(p(u>) strongly in

L 2 ( S ~ )

where

ST

⁼

^dR

^x (O,T].

One easily checks that u E C((0, T ) ;

L1(Q))

^{and u} satisfies ii) and the integral identity iii) in the definition of weak solution since U, satisfies a similar identity.

It remains to show that Ilu(t)llLlcn> is continuous at zero. For this we consider a smooth approximation vo of uo, ;.e., for any 6

>

0, choose vo smooth to be such that llvo

- UOII ⁵

S/3. ^{Let D} be a solution of problem ( P ) with initial function v g

obtained _as a limit of solution of problem (P,). Then I l v ( t ) l l L ~ ( ~ ) is continuous at zero. (This is the consequence of the second part of Lemma 2.5.) Therefore, there exists a

to >

⁰ such that for all

t 5

t o ,

For the estimation of Ilu(t)-v(t)ll~~ (a), we have the following L1-contraction results:

Lemma 2.7. V

t >

0,

Applying Lemma 2.7, we conclude that

V

6

>

0, 3

to >

0 such

that

for

all t 5

^{t o}

Proof of Lemma 2.7. Let u,,

v,

be the solutions of approximating problem

(P,)

corresponding to the smooth initial data U O , and vo, such that

since both U, and v, satisfy the integral identity in iii) of the definition of weak solution. Integrating by parts and subtracting on both sides, we obtain, for

T >

^0,

where

if U , = v,

.

Both a(u,, v,) and b(u,, v,) are smooth, a(u,, v,)

2

^{C ( E )} and Ib(u,, vE)I

5

^C(e)

for some positive constants c(E), C(E).

Let $ be the smooth solution of the following problem

$t

+

a(u,,v,)A$ = 0 in

R

x

(0,T) - +

^{b(uE,ve)$ = O on}

^aR

^x

^(O,T)

in 52.

{

$(x,T)

²

= ^{X ( X )} where

X

E C,o”(s1) is such that 0

5

X

5

1.

Then $(x,

t )

E

C2J(&T) ,

$(x,

t ) 2

0 and

where

(-)+

= max(., 0).

Let

i?

be

1 if ^uC(z,

T) > ^{v,(z, T)}

0 otherwise

-

X ( x ) = and choose

X

= X, ³

obtain

in L2(s1) as n + 00. Then, letting n ⁴ co in (2.2), we

Identically we can prove that

Therefore,

93. Uniqueness

To

show uniqueness we use an idea due to Kalashinikov [9], see also Bertsch and Hilhorst _[4], to compare an arbitrary generalized solution of problem

( P )

with a solution obtained as the limit of a sequence of classical solutions of the approxi- mating problem

( Pe).

Following Bertsch and Hilhorst _[4], we approximate solutions

n 1

+,O) ^{= .o(x)}

+ ^-

ⁱⁿ

^s1

which in turn is approximated by

U t =

A@(.)

in QT

dn n

+,O) = U O r ( 5 )

+ ^-

1

1 ^--@(.I

^d

⁺ P c ( d ( 4 )

ⁿ =

-

^{Ae-M on 6% x in}

⁵²

^(0,T)

(P.n 1

where

Pe

is smooth,

IIP:

^*

$11~- I

^{A for some} constant A

>

0 and converges to

P

uniformly on

E+

^{as E 3 0.}

Where U O , is such that

uoE E Cm(R) ^? jluo,

-

U O ~ ~ L Z ( Q ) ^-+ O as E ^-+ O and uoE satisfies the compatibility condition

n 1 n

~ - 1 ( u o .

+ A> ^{% + (@}

^(uOe

⁺ ->

^{= 4e-A' on dS1}

where

X

is determined from Lemma 3.1 below and @(s) is smooth and is defined as n an

i f s

5 e

if

8 <

^s

< e ⁺

¹

8 s ) = smooth

{ ^{Cp(6+1) dS)}

i f s ? i ? + 1 with

6

= ~p-'(2y(21/uoll,p)) from lemma 3.1 below.

Lemma 3.1. The solution un, E C21'(g) of ^problem

(Pen)

satisfies, for n large enough, that

-e-xt 1

5

u,,(x,

t ) 5 E

^{EE 'p-l} ( ~ ~ ( Z I J U O llL-

>)

^{for any (x?}

t )

^{E QT} n

where the constant Proof.

does not depend on n and

t.

Let L(u) = ut

-

AP(u) and choose h(z,t) = Then

in

s1

^{x (O,T)} x ^-At

~ ( h ) = --e ⁿ

6 o

^{= L(un,)}

By comparison principle

u,,(z,t)

2

^;e 1 ^{-At for all}

( q t )

E

QT .

In order to find an upper bound we introduce the following function w(z,t) =

3-e-xte(z)

'

P(une) where t(z) E

C2(n) n C'@)

^such

^that

1s

5

2

, -31

_dn

_r ² s

^{for some}

s ^>

^{0 .}

At the maximum point (20, to) E Q x (0, T) of w2, we have wwt

2

0

,

PIwAw

5

0 Substitute into equation (3.1), we obtain

wgradw = 0

%.e.,

Since

E 2

1 and A< is bounded, this is impossible if we choose X large enough.

Therefore, w2 cannot have maximum in St x (0,T). On the lateral boundary point where maximum of w2 occurs, we have

d W

d n

w - 2 0

z.e.,

%.e.,

A A

--

_an

at ^w 5 -

_n

-

e-"P(@(tine))

5 -

_n By the choice oft,

w l -

_n6 A

Therefore, for (2,

t )

E

gT

I w b ,

t>l I

^{,ax{ IIw(zct 0)llP 9}

a}

A

5 -41

^cp(UO€)

^{- t(;c)} (ILrn7 3

- <

IIp(tioe)ll~- (if n is large enough).

Consequently

IF(Une(2,

t)) I _I

⁽³

-

e - x ' r ( 4 )

II~(u0)llL-

L 2llV(~O€>IIP

L

2V(211~OIlL~)

2. e.,

I~n€(zC,f)l

5 e ^I

Now by the method of Section 2, one can obtain further a-priori estimates on tinr of problem

(P,,)

to show that there exists a subsequence of (unE} which converges to the solution u, of (P,) and then that a subsequence of u,, which is denoted by u,,

converges to the solution of problem ( P ) as n -+ 03. We will use this approximation to prove the following uniqueness theorem:

Theorem 3.2. Let u be a solution of ( P ) obtained as the limit ofthe solution un of (P,) and let ii be any subsolution of (P). Then

I

Similarly for any supersolu tion

u

Subtracting on both sides yields

where

and

P ( c ~ ( d ) -

P(4un>>

which is bounded by a positive constant L independent of n. Now define two smooth sequences Anj and B,I, such that

Bn =

Y(U) -

^{4 U n ) >}

and

Let $ n j k be the solution of the following problem

where X(z) E C,"(s2) is such that 0 estimates.

X(z)

5

1. Then $njk satisfies the following Lemma 3.3.

If we let first j -+ ^oc), using ii) of of lemma 3.3, then let

k

^{-+ CQ,} and then let n ^{--+ 00,} we arrive at

As

in

the proof of lemma 2.7, we conclude that

The proof of (3.2) is similar.

Proof of lemma 3.3.

i)

^$njk

2

0 is the consequence

of

maximum principle. For the upper bound, as in the proof of lemma 3.1 let w(z,

t )

= e-M(T-t)t(z)$(z,

t).

Then w(z,

t)

satisfies

Choose ((2) = 1

+

^{cuh(z) where}

Ah(z) = -1 in

R,

in

dR

{

^{h(z) = 0}

and a is large enough such that

[-(at/an)/(]l, 2 L.

[-(dh(z)/dn)]l,

2

⁶

>

0). Next we choose

(This is possible since

Then, similarly as in the proof of lemma 3.1, we conclude that the maximum of w2 can only be attained at

t

=

T,

ie.,

2. e.,

For the proof of ii), multiplying equation by A$ and integrating by parts we obtain

from which we onclude that

ii) follows from A,j

2

~ ( n ) .

I

84. Large time behavior We define the w-limit set

ut = { q E L1(S2) : there exists a sequence

t,

^{+ 00 such that} u(t,) + q in L'(R) as

t,

^{+ co}}

Then by lemma 2.5: the set {u(t; ^ug) :

t 2

1) is precompact in C(Q and therefore w is nonempty and w E C(Q.

Claim. w = (0).

Therefore we have the following result by Proposition 2.1 of Dafermos 161.

Theorem 4.1. Under the hypothesis

H1-H3

the solution of problem

( P )

satisfies u(t;uo) ⁺

o

ⁱⁿ

^C(i7).

any q E

E ,

we have

for any $ E

C2J(Q).

Now set ?I, = 1, (4.1) becomes

z.e.,

Substituting this into (4.1), we obtain

z. ^e.,

y(q)A+ = 0 for any .II, E C2p1(Q)

.

I' ^J,

Therefore q = 0 and

E

= (0).

Next we define the functional V :

L'(f2)

-+ [0, co)

u E L1(sl)

Since zero is a solution of problem

(P)

it follows from Theorem 3.2

that

Therefore

V

is a Lyapunov functional for problem ( P ) and V is continuous since u E C([O, CO); Ll(f2)). An application of Proposition 2.2 from Dafermos [6], that V is constant on ut, ie., for any q E w

~ ( u ( t ; q)) = ~ ( q ) for an

t 2

^O

%.e.,

J,

^{u(t; 4 ) =}

J,

If we choose

II,

= 1 in (4.1), using (4.3); ^then

from which we conclude, as above, that

q rO ⁱⁿ

i2

x (0,t) Thus w =

E

= (0).

The proof of claim is completed and thus theorem 4.1 is proved.

(4.3)

I

Acknowledgement

The author is indebted to Dr.

H.

Kaper for bringing the problem to his attention and for his hospitality extended to the author during his stay at Argonne National Laboratory. Also the author would like to thank Professor S. Kamin for a number of stimulating discussions and Professor

R.

Showalter for bringing the paper of Benilan, Crandall and Sacks to his attention.

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3. Ph. Benilan, M.G. Crandall and P. Sacks, Some L1 existence and dependence results for semilinear elliptic equations under nonlinear bounda y conditions, Appl. Math.

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25 (1973), 565-590.

Evolution Equations,” M.G. Crandall, ed., Academic Press, 1978.

Indiana Univ. Math. J., 32 (1983), 83-118.

133 (1968), 51-87.

17-27.