Minneapolis, U.S.A.(~) ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF PARABOLIC EQUATIONS OF ANY ORDER

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF PARABOLIC EQUATIONS OF ANY ORDER

B Y

A V N E R F R I E D M A N Minneapolis, U.S.A.(~)

Introduction

F o r s o m e p a r a b o l i c d i f f e r e n t i a l e q u a t i o n s i t is k n o w n t h a t a n y s o l u t i o n in a cy- l i n d r i c a l d o m a i n w i t h a x i s t > 0, t e n d s to a l i m i t as t - * ~ p r o v i d e d t h e b o u n d a r y v a l u e s a n d t h e coefficients of t h e e q u a t i o n t e n d t o a l i m i t as t - ~ co. F u r t h e r m o r e , t h e l i m i t of t h e s o l u t i o n is k n o w n t o be t h e s o l u t i o n of t h e l i m i t e q u a t i o n . F o r s e c o n d o r d e r p a r a b o l i c e q u a t i o n s , t h i s has b e e n p r o v e d b y t h e a u t h o r [5] for t h e first m i x e d b o u n d a r y v a l u e p r o b l e m , t h a t is, w h e n t h e s o l u t i o n u is p r e s c r i b e d on t h e l a t e r a l b o u n d a r y of t h e c y l i n d e r . E x t e n s i o n t o e q u a t i o n s w i t h a n o n h o m o g e n e o u s t e r m w h i c h is " s l i g h t l y " n o n l i n e a r in u, is also g i v e n in [5]. I n [6] i t was p r o v e d t h a t if b o t h t h e coefficients of t h e p a r a b o l i c e q u a t i o n a n d t h e b o u n d a r y v a l u e s a d m i t an a s y m p t o t i c e x p a n s i o n i n t -1 (t-->o~), t h e n t h e s a m e is t r u e of t h e solution. A s y m p t o t i c conver- gence for s o l u t i o n s of s e c o n d o r d e r p a r a b o l i c e q u a t i o n s s a t i s f y i n g a n o n l i n e a r b o u n d a r y c o n d i t i o n (generalized N e w t o n ' s l a w of cooling) was e s t a b l i s h e d b y t h e a u t h o r in [7].

The p r e s e n t p a p e r consists of t w o p a r t s . I n P a r t I we consider s e c o n d o r d e r p a r a b o l i c e q u a t i o n s a n d e s t a b l i s h t h e a s y m p t o t i c b e h a v i o r of solutions, b o t h for t h e f i r s t a n d t h e s e c o n d ( a n d e v e n m o r e general) m i x e d b o u n d a r y v a l u e p r o b l e m s . T h e n o n h o m o g e n e o u s t e r m is a n o n l i n e a r p e r t u r b a t i o n . T h e d o m a i n s are " a l m o s t c y l i n d r i c a l , "

i.e., t h e cross sections t = const, t e n d t o a l i m i t as t - * ~ . F o r t h e first m i x e d b o u n d a r y v a l u e p r o b l e m , t h e p r e s e n t t r e a t m e n t is n o t o n l y a n i m p r o v e m e n t of t h e a n a l o g o u s r e s u l t s of [5], b u t i t is also a m u c h m o r e s i m p l i f i e d t r e a t m e n t . T h u s for i n s t a n c e , we do n o t m a k e here a n y use of e x i s t e n c e t h e o r e m s for p a r a b o l i c e q u a t i o n s . W e (1) lOart I was prepared under Contract lqonr-222 (62) (NR 041 214) with the University of Cali- fornia in Berkeley. Part I I was prepared under Contract Nonr-710 (16) (NR 044 004) between the Office of Naval Research and the University of Minnesota.

1 - 61173055. A c t a mathematica. 106. Imprimd le 26 sepbembre 1961.

2 AVNEI~ FRIEDMA:N

use however the Schauder existence theory for elliptic equations [17] and, for the second mixed b o u n d a r y problem, recent results of Agmon, Douglis and Hirenberg [1].

I n P a r t I I we consider general nonhomogeneous parabolic equations of a n y order in an "almost cylindrical" domain, and solutions having prescribed Diriehlet data on the lateral boundary. We first prove t h a t if both the coefficients of the equa- tion and the boundary values tend to a limit as t-> ~ , then the solution u (x, t ) c o n - verges in the L 2 norm to a solution of the limit elliptic equation. The special case of homogeneous equations in a cylindrical domain with zero boundary values was proved b y Vishik [20]. In our derivation of the L 2 convergence, we make essential use of some results of the paper of Agmon, et al. [1], already mentioned above. Having derived the L 2 convergence, we use it to get a uni/orm convergence. Here we make use of the fundamental solutions for parabolic equations [4] [19] and also (for cylin- drical domain--where stronger results are derived) of Green's function considered b y P. Rosenbloom [16]. Finally, we derive asymptotic expansions in t I for the solutions.

Part I. Second order parabolic equations

I n this part we consider the asymptotic behavior of solutions of second order parabolic equations satisfying either the first or the second (and even more general) b o u n d a r y conditions. I n 5 1 we state the main results about uniform convergence (as t ~ ) of solutions of the second mixed b o u n d a r y value problems (Theorems 1, 2).

Theorem 1 is proved in 5 2 and Theorem 2 is proved in 55 3, 4. I n w 5 we discuss the asymptotic expansion in t -1 of solutions, as t-->c~. The results of w 1-5 are extended in w 6 to solutions of the first mixed boundary value problem. Finally, in 5 7 we consider the behavior of solutions satisfying a generalized second boundary value condition.

1. Statement of results for the second boundary value problem

Let D be a domain in the ( n + l ) - d i m e n s i o n a l space of real variables (x,t)

= @1 .. . . . x~, t) bounded b y a bounded domain B on t = 0 and a surface S in the half space t > 0. We denote b y B~ the intersection D N {t = 3) and assume t h a t for every T > 0 B~ is bounded and nonempty. We further denote b y D~ (Dr162 = D) the domain D N { 0 < t < 3 ) and b y S~ the set S f i { 0 < t < 3 } . The b o u n d a r y of a domain G i s d e - noted b y ~ G, the closure of a set G is denoted b y G, and the complement in a set G 2 of a set G 1 is denoted b y G 2 - G1. Later on we shall assume t h a t there exists a

A S Y M P T O T I C B E H A V I O R O F S O L U T I O N S O F P A R A B O L I C E Q U A T I O N S

b o u n d e d d o m a i n C in t h e x-space such that, as

t--->oo, Bt-->C

in a certain sense.

F o r simplicity we assume t h r o u g h o u t this p a p e r t h a t C a n d S are each composed of one surface, b u t all t h e results can easily be e x t e n d e d to the case t h a t C a n d S are each composed of a finite n u m b e r of surfaces.

D E F I N I T I O N . W e s a y t h a t

w (y, t)---~z (x)

u n i f o r m l y in

(y, t) ED, x E C

as

y->x,

t-->oo a n d also write

lim w (y, t) = z (x),

y--~X t--) Oo

if for a n y s > 0 there exist 6 > 0 , t 0 > 0 depending on s such t h a t

I w ( y , t ) - z ( x ) i < s

whenever (y,

t) E D,

x E C, ] y - x I < 8, t > to, A similar definition can be given for func- tions defined only on S.

Consider the equations

x t ~2u ~ ~ u ~ u

Lu~--~.j=l ~

a i , ( ,

) ~ § ~ l b ~ ( x , t ) ~ x + C ( x , t ) u - - ~ = / ( x , t ) + k ( x , t , u )

f o r ( x , t ) E D , (1.1) c ~2 v

L~ ~ a " ( x ) ~ § r

for

xEC,

(1.2)

where

u = u

(x,

t), v = v (x),

a n d the b o u n d a r y conditions

~ (x, t)

T ~- g (x, t, u (x, t)) = h (x, t)

for

(x, t) E S,

(1.3)

dv (x)

d T +g(x) v(x)=h(x)

for

x~_cqC.

(1.4)

Here

(?u~O(X,T t)

limy_~ ~.j=l ~

aij (x, t)

cos [v (x,

t), xj] cgUo(yY' t)

(1.5)

yEy

for all r a y s y issuing f r o m

(x, t)

a n d pointing into the interior of

Bt.

W e call ~

u/~ T

the

transversal

(or

conormal)

derivative of u. I n (1.5), v (x, t) is the o u t w a r d l y directed n o r m a l to

~Bt

at the p o i n t (x, t). Similarly we define

dv (x)= lira ~ a~j (x)

cos [v (x), xj] ~v (y)

d T ~ " ~,j=t ~ xt

Yelp

(1.6)

as the transversal derivative of v (x), where t h e r a y s ~, s t a r t at x a n d p o i n t into t h e

4 iV~ER FI~IEDMAN

interior of C. I n order to avoid confusion later on, we h a v e d e n o t e d t h e transversal d e r i v a t i v e on S b y 8 / 8 T a n d on 8 C b y d i d T .

D E F I N I T I O N . Given a b o u n d e d d o m a i n G in the x-space, its b o u n d a r y 8 G is said to he of class C z+~ (m integer, 0 < f l < l ) if to each point y of G there corre- sponds a sphere V (in the x-space) h a v i n g y for its center a n d such t h a t V N 8 G can be represented, for some i, in t h e f o r m

xi=yJ (x 1, ..., xi 1, X~+l, ..., Xn), (1.7) where ~v possesses m H61der continuous (exponent fi) x-derivatives. If t h e functions

~v are only assumed to be m times c o n t i n u o u s l y differentiable, t h e n 8 G is said t o belong to the class C m.

F o r a n y function w = w (x) in G we i n t r o d u c e t h e norms:

Iwlg=l.u.b. Iw(x)l, Iwl2=lwlo § (w),

x E G

where 8 / 8 x denotes a n y partial derivative with respect to the xj a n d 0 ~ < ~ < 1, a n d H i (w) = 1.u.b I w (x) - ~(Y) I

x.y~G I x - y

W h e n there is no confusion, we o m i t the superscript G f r o m the n o r m sign.

W h e n we write, for functions z (x) defined on 8 G, t h e n o r m ]z] ~ we m e a n t h e following. A finite covering of 8 G is given and, hence, in each such p o r t i o n z be- comes a f u n c t i o n of n - 1 variables. W e t h e n t a k e I zl~ a to be the sum of the d- n o r m s of z in these portions. W e shall clearly assume t h e n t h a t 8 G is of class C e w i t h e>~ d. L e t t h e a b o v e finite covering be composed of portions 8 Gj of 8 G a n d let F = ~ v r be the representation (1.7) for 8 Gj. W e t h e n define

1

:Finally, we d e n o t e b y I G[ t h e diameter of G.

We shall need, later on, various assumptions on L , Lo, /, k, 9, h a n d D. F o r t h e sake of clarity we list m o s t of t h e m now.

(A) T h e coefficients of L are continuous in /) a n d are b o u n d e d b y a positive c o n s t a n t M , a n d L is u n i f o r m l y parabolic in /), t h a t is, there exists a positive con- s t a n t M ' such t h a t , for all (x, t) in /) a n d for all real vectors $ = (~x . . . . , ~=),

A S Y M P T O T I C B E H A V I O R OF S O L U T I O N S OF P A R A B O L I C E Q U A T I O N S 5

g , j - 1 ~-1

(A0) T h e following limits exist, u n i f o r m l y in (x, t ) e 2) a n d y E C:

lim a~j (x, t) = a~j (y), lim b~ (x, t) = b~ (y), lira c (x, t) = c (y).

X - - > y X - - > y X - ~ y

t ---> oO t ---> oO t - ~ oO

T h e functions a~j (y), b~ (y), c (y) are ]-I51der continuous (exponent ~) in C.

(B) / ( x , t) is a continuous f u n c t i o n in 2~.

(B0) As x->y, t---> oo, / (x, t)--->] (y) u n i f o r m l y w i t h respect to (x, t) e D a n d y e C, a n d / ( y ) is HSlder continuous (exponent ~) in C.

(C) k (x, t, u) is continuous for (x, t, ) e / ) , - c~ < u < oo, a n d

Ik(z,t, )l<z01u], (1.9)

where /t o is a sufficiently small c o n s t a n t (depending on M , M', t h e co-norms of t h e coefficients of i a n d 1.u.b. ([Bt[ + [~Bt[1); see (n)).

t

(Co) As x-->y,t-->~, k ( x , t , u ) - - > k ( y , u ) u n i f o r m l y with respect to (x,t) CD, y E C a n d u in b o u n d e d intervals. T h e f u n c t i o n k (x, u) is }tSlder continuous in (x, u) f o r x E C a n d u in b o u n d e d intervals, a n d ~ k (x, u)/~ u is continuous for x E C a n d u in b o u n d e d interval, a n d

t k (x, u)[ ~<

/ % (1.10)

t u I

where /z 0 is a sufficiently small c o n s t a n t (depending on t h e s a m e quantities as t h e

#0 in (1.9) and, in addition, on b o u n d s on ]/[, g, I hi a n d ]~ C]e+~).

(D) F o r e v e r y t > 0 , r U t is of class C 1 a n d 1.u.b. ( I B t ] + [ a B t l l ) < ~ .

t

(Do) ~ C is of class C 2+~ a n d to e v e r y p o i n t x on ~ C t h e r e corresponds one a n d only one p o i n t (xt, t) on each Bt ( t > 0 ) such t h a t (i) xt-->x as t-->c~, u n i f o r m l y w i t h respect to x on ~ C, a n d (ii) as t--> ~ , t h e direction cosines of the n o r m a l v (xt, t) t o

~Bt t e n d to t h e direction cosines of the n o r m a l v ( x ) t o ~ C a t x, u n i f o r m l y w i t h respect t o x on ~C.

Remarks. (a) B y (A0), (Do) it follows t h a t if z (x) is c o n t i n u o u s l y differentiable in a n e i g h b o r h o o d of ~ C, t h e n as t--> ~ ~ z (xt)/~ T-->dz ( x ) / d T u n i f o r m l y w i t h respee~

to x on t C. (b) If D is a cylindrical domain, t h e n (Do) reduces to t h e a s s u m p t i o n t h a t tB~:--~C is of class C 2+~.

(E) h (x, t) is a continuous f u n c t i o n for (x, t) on S.

6 A V N E I ~ F I ~ I E D M A ~

(Eo) As t - - - ~ , h (xt, t)---~h (x) u n i f o r m l y in x E ~ C.

(F) g (x, t, u) is continuous for (x, t) E ~q, - ~ < u < c~, a n d there exists a positive c o n s t a n t #1 such t h a t

g (x, t, u)

U

- - ~ > t t l for (x,t) ES, - - ~ < u < ~ , u # 0 . (1.11) (Note, b y t a k i n g u~O, u-->O t h a t g (x, t, 0 ) ~ 0 . )

(F0) As t-->~, g (xt, t,u)--~g (x)u u n i f o r m l y with respect t o x on ~ C a n d u in b o u n d e d intervals. (Note, b y (F), t h a t g (x) ~>/~1 > 0.)

(G1) a~j (x) belong to C 1+~ in some outside n e i g h b o r h o o d of ~ C.

(G~) ~ C is of class C ~+~.

D E F I N I T I O N . We say t h a t u (x, t) is a solution in D of t h e s y s t e m (1.1), (1.3) if (i) u is continuous in D, (ii) the derivatives ~ u/O x~, ~2 u/O x~ ~ xj, ~ u / ~ t are con- tinuous in D a n d (1.1), (1.3) are satisfied. We s a y t h a t v (x) is a solution in C of the s y s t e m (1.2), (1.4) if (i) v is continuous in C, (ii) t h e derivatives ~ v/O x~, ~ v/~ x~ ~ x~

are continuous in C a n d (1.2), (1.4) are satisfied.

We can n o w state t h e m a i n results on the u n i f o r m convergence of solutions of (1.1), (1.3) as t---~oo.

TItEOREM 1. Assume that (A)-(F) hold and, in addition, that

lira h (x, t) = 0, lim / (x, t) = 0, lim sup c (x, t) ~< 0 (1.12)

t--> r t-->~r t-->ao

uni/ormly with respect to (x, t) E S, (x, t) E D and (x, t) E JD respectively. I / u (x, t) is a solu- tion in D o/the system (1.1), (1.3), then, uni/ormly in (x, t) E .D,

lim u (x, t) = 0. (1.13)

t - - ~

THEOREM 2. Assume that (A)-(F), (Ao)-(F0) (G1), (G2)hold and thatc(x)<~O. I / u (x, t) is a solution in D o/the system (1.1), (1.3), then

lim u (x, t) = v (y) (1.14)

X--> Y t-->OO

uni/ormly with respect to (x, t) in D and y in C, and v (y) is the unique solution in C o/the system (1.2), (1.4).

I n a p r e l i m i n a r y r e p o r t [9] we have p r o v e d T h e o r e m 1 as s t a t e d above, a n d Theo- r e m 2 w i t h o u t assuming (G1), (G2).

A S Y M P T O T I C B E H A V I O R O F S O L U T I O N S O F P A R A B O L I C E Q U A T I O N S

I n t h e proof of T h e o r e m 2 t h e r e a p p e a r s a decisive l e m m a ( L e m m a 3, below) whose p r o o f i n v o l v e d tedious p o t e n t i a l theoretic calculations. T h e p r e s e n t proof avoids these calculations b y s i m p l y using a recent result of [1]. H o w e v e r , we h a v e to as- sume (G1) , (G2)

I n the course of t h e proof of T h e o r e m 2 it will be shown t h a t if h (x), g (x) belong to C 1+~ t h e n v (x) belongs to C 2+~ in C, a n d thus it satisfies (1.4) in t h e classi- cal sense.

:From t h e proofs of T h e o r e m s 1, 2 it will b e c o m e clear t h a t t h e y r e m a i n t r u e ff

~/~ T is replaced b y a n y other oblique d e r i v a t i v e ~/~ T provided, as t--~ oo, a / ~ T - - > d / d T (at the corresponding points).

2. P r o o f o f Theorem 1 W e introduce t h e f u n c t i o n

cf (x) = e ke - e ~~1, (2.1)

where R is a positive n u m b e r satisfying 2 x I ~< R for all (x, t ) = @1 . . . x=, t) in /), a n d 4 is a positive constant. 4 a n d R will be d e t e r m i n e d later, cf (x) satisfies

( L qJ) (x, t) = - a l l (x, t) 2 ~ e axl - b 1 (x, t) 4 e ~ ' § c (x, t) (e ~R - e ~xl) for (x, t) E D ,

~ q ~ ( x ) ~ - g ( x , t , ~ ( x ) ) > ~ - 4 e ~x~Oxl T ~ +/s~ (e ~R - e ~z<) for (x, t) e S.

Using (A), we m a y choose 4 sufficiently large such t h a t

(L ~v) (x, t) < - 2 e ~z~ + c (x, t) (e ~R - e ~ ' ) for (x, t) E D. (2.2) H a v i n g fixed 4, we choose R so large t h a t

~ (x) ? g (x, t, ~o (x)) ~>/z 2 > 0 for (x, t) E S. (2.3) 9 T

N o t e t h a t t h e c o n s t a n t s 4, R , / z 2 are i n d e p e n d e n t of (x, t). B y (1.12) it follows t h a t there exists a sufficiently large n u m b e r 5 such t h a t

c ( x , t ) ( e ~ S - e ~ Z ' ) < e ~ for all ( x , t ) E D - D ; . (2.4) S u b s t i t u t i n g (2.4) into (2.2), we get

(L ~v) (x, t) < - 2 (5 for (x, t) E D - D ; , (2.5)

8

where 2 d = g.l.b, e ~xl > 0.

(X,t)r

AVNEI~ FI~IEDMAN

F o r later purposes we define d o = g.l.b. ~ (x), d l = 1.u.b. ~v (x).

( x , t ) e D ( x , t ) e D (2.6)

The function ~ (x) will n o w be used to construct a comparison function which will majorize t h e solution u (x, t). Consider t h e function

~f (x, t) = 2 e ~ 2 ~ + s ~ (x) + A (p (x) e_~(t_o) ' (2.7)

#2 do

where e, A , ). are a n y positive numbers, a n d a ~> &. Using t h e properties of ~ (x) derived above, we get

v (x, t) < e dl, (2.s)

L ~ (z, t) < - 2 s - 2 d s _ 2 d A e_r(t_~) + Y A dl e_,(t_~) (2.9)

#2 do do "

~2 = d / d 1 , d 2 = d / d I ( 2 . 1 0 )

Defining

a n d using (2.8), we o b t a i n f r o m (2.9) L yJ (x, t) < - s - d2 ~ (x, t) Using (1.11) a n d t h e choice of R, we also get

y~ (x, t)

~ T ~ - g ( x , t , y ~ ( x , t ) ) > s

for (x, t) E D - D o . (2.11)

for (x, t) e S. (2.12)

T h e function yJ (x, t) will n o w be used t o estimate u (x, t).

L e t s be a n a r b i t r a r y positive number. I f we prove t h a t for a sufficiently large n u m b e r Q = Q (e)

l u ( x , t ) l < A o s for (x,t) e D - D ~ , (2.13) where A 0 is a c o n s t a n t i n d e p e n d e n t of s, Q, t h e n t h e proof of T h e o r e m 1 is completed.

Now, b y (1.12) there exists a = a ( s ) > 0 such t h a t

I h (x, t) l < e for (x, t) e S - So, (2.14) I / ( x , t ) l < e for (x,t) e D - D ~ . (2.15) We t a k e a such t h a t also a > 5 (and t h e n (2.11), (2.12) hold). W e n e x t t a k e in

A S Y M P T O T I C B]~HAVIOI~ OF S O L U T I O N S OF PAtCABOLIC E Q U A T I O N S

t h e definition of to a b o v e t h e n u m b e r s a, s t o be the s a m e n u m b e r s as t h e p r e s e n t ones, a n d

A = 1.u.b. I u (x, a)I" (2.16)

W e shall p r o v e t h a t

u ( x , t ) < t o ( x , t ) in D - D , . (2.17) T h e proof is b a s e d on a n a r g u m e n t similar t o t h a t a p p e a r i n g in [21]. W e first note, b y (2.14), (2.15), (2.16) a n d (1.9), t h a t

L u > - e - ~ o ] U I for (x,t) E D - D , , (2.18)

~ u / ~ T + g ( x , t , u ) < r for (x,t) E S - S o , (2.19)

u (x, (r) < to (x, a) for x E B~. (2.20)

Consider n o w t h e set ~ of points t >~ a such t h a t to > u i n / J ~ - D,. B y (2.20), Z is n o n e m p t y . I t is clearly a n open set. I f we p r o v e t h a t ~ is closed, t h e n the proof of (2.17) is completed. Suppose t h e n t h a t t is such t h a t to (x, T)> u (x, 7:) in D r - D o , a n d we h a v e to p r o v e t h a t ~p (x, t) > u (x, t) for x E B t . I f this is n o t t h e case, t h e n the function 4 ( x , t ) ~ t o (x, t ) - u ( x , t) obtains its m i n i m u m zero in t h e set D t - D ~ a t a point (x ~ t) on /~t. W e shall derive a contradiction b y p r o v i n g t h a t (x ~ t) can belong neither to ~Bt nor t o Bt.

I f (x ~ t)E ~ Bt then, noting t h a t ~/~ T is a d e r i v a t i v e along a n o u t w a r d direction t o ~Bt, we get

a (x ~ t) - ~ (y, t) ~ ~ ~ (x ~ t)

o>/ izO_y I

= ~ - T ~ ( ~ , t ) - ~ ~ T '

as y-->x ~ along t h e t r a n s v e r s a l r a y issuing a t t h e p o i n t (x ~ t). Since also g [x, t, u (x, t)]

= g [x, t, to (x, t)] a t x = x ~ we get to (x ~ t)

a ~ ~- g Ix~ t, to (x ~ t)] < - - which contradicts (2.12), (2.19) combined.

I f (x ~ t) E Bt then, a t t h a t point,

u (x ~ t)

~ T

~ = t o , l ~ l = t o , ~ / ~ x ~ = ~ t o / ~ x . Also, since (aij) is a positive m a t r i x ,

~ to

~a~j ~ x ~ x j

~- g Ix ~ t, u (x ~ t)],

~/~ t >I ~ to/~ t.

9 2 u

- - 1> Y a~j ~ at (x ~ t).

(2.21)

(2.22)

10 AV~ER FRIEDMA~

Combining (2.21), (2.22) we conclude that, at the point (x ~ t), L u + e + t t o [ U l < ~ L v +e+~uo~f.

The last inequality however, contradicts (2.11), (2.18) combined, provided

~u0 ~< 52. (2.23)

Hence, assuming /~o to be sufficiently small so t h a t (2.23) is satisfied, we conclude t h a t (x ~ t) cannot belong to Bt. This completes the proof of (2.17).

In a similar way, replacing (2.11), (2.12) b y

L ~ > e + ~ [v~ 1, 0 + g ( x , t , ( f l ) < - e , (2.24) where v~= - V and replacing (2.18), (2.19) by

L u < s + t t o [ u [ , ~-~ + g (x, t, u) > - e, ~u (2.25) we can prove t h a t u>v~ in D - D a . Combining this ineqality with (2.17), and recalling the definition of V in (2.7), we have

e e A

] u (x, t) l < -($ q~ (x) + ~ q~ (x) + ~-o q~ (x) e-v(t-") for (x, t) 6 D - na. (2.26) Taking @ sufficiently large such t h a t A (~1 e-V(e-")/~0 ~< e the proof of (2.13) is completed.

From the above proof the following corollary follows.

COROLLARY 1. I / the assumption (1.12) in Theorem 1 is replaced by

lim sup I h (x, t)[ ~< s, lira sup [/(x, t) l ~ e, lira sup c (x, t) ~ 0 (2.27) uni/ormly in (x, t) 6 .D, and i/the other assumptions o/Theorem 1 remain unchanged, then

lim sup l u (x, t) I ~< A18 (2.28)

t---> Oo

uni/ormly in (x, t) 6 D, where A 1 is a constant independent o/e.

3. Proof of Theorem 2 for smooth h, g

In this paragraph we prove Theorem 2 under the additional assumption t h a t h, g are C 1+~ in some outside neighborhood of ~ C. This assumption will be removed in w 4. We need a few preliminary results.

A S Y M P T O T I C B E H A V I O R O F S O L U T I O N S O F P A R A B O L I C EQIIATIOI~IS 1 1

W e recall t h a t ~ C is of class C 3+~. N o w a t e v e r y p o i n t x ~ of ~ C we d r a w a n o u t w a r d l y directed n o r m a l r (x ~ t o ~ C a n d denote b y ~ (x ~ t h e s e g m e n t on v (x ~ of length (~' ( ~ ' > 0) a n d initial p o i n t x ~ W e o b t a i n a f a m i l y N of s t r a i g h t segments.

I t is e l e m e n t a r y t o see t h a t e v e r y p o i n t x outside C a n d sufficiently close to ~ C lies on one a n d only one n o r m a l s e g m e n t ~ (x ~ p r o v i d e d (~' is sufficiently small, s a y 3' ~< ~.

I n w h a t follows we t a k e ~ ' = ~ .

W e n o w m e a s u r e a n y fixed distance ~, 0 < ~ ~< ~ on each ~ (x), x e ~ C a n d denote t h e set of the end points b y ~ C~. T h e following l e m m a is well known.

L w M M A 1. Each ~C~ is a sur/ace orthogonal to the /amily N , and 1.u.b. 1~C~[2+~

<~ const. < oo.

Using local coordinates x~=/~(s) ( l < ~ i < ~ n , s = (s 1 .. . . . sn-1)) for ~C, we can re- p r e s e n t ~ C~ locally in the f o r m

x~ =/~ (s) + g~ (s) t, (3.1)

where gi (s) is ( - 1 ) ~-1 times the d e t e r m i n a n t of t h e m a t r i x o b t a i n e d f r o m t h e m a t r i x (~/~/~ sj) (i indicates rows, ] indicates columns) b y erasing the i t h row. t is defined b y

t = (~/g (s), (3.2)

where g ( s ) = L= ~ (gi (s))~] ~. (3.3)

W e n e x t need a r e c e n t result of A g m o n et al. [1, C h a p t e r I I ] :

L E M ~ A 2. Consider the system (1.2), (1.4) with k - - O and assume that ~ C is o/class C 2+~, that / and the coe//icients o/ L o are C ~ (C), that a~j are C 1+~ (~ C), and that h, g are C 1+~ (~ C). / / c (x) ~< 0, g (x) > 0, then there exists a unique solution o/ (1.2), (1.4) which is o/class C 2+~ (C), and

< K (I h + l l If), (3.4)

where K depends only on bounds on the quantities

a oc

la,jlL Ib, lf, lelf, Igloos, I1/glo ~176 IcI,

MP, t] l + a ,

W e are n o w going t o consider differential s y s t e m s analogous t o (1.2), (1.4) in each C~, C~ being t h e interior of ~ C~. T h e solution v ~ (x) will be " c l o s e " to b o t h u (x, t ) ( t - - > ~ ) a n d v (x) a p p e a r i n g in t h e f o r m u l a t i o n of T h e o r e m 2. W e p u t C ' = C~, where ~ a p p e a r s in L e m m a 1, a n d write C ' for the closure of C'. W e m a y a s s u m e t h a t t h e a~ (x) are C ~+~ in C ' - C , as follows b y a s s u m p t i o n (G~).

E v e r y function p (x) defined in C or on ~ C can be e x t e n d e d t o C ' - C as follows.

L e t x e C ' - C a n d let x ~ be t h e p o i n t on ~ C such t h a t x lies on O(x~ W e t h e n

1 2 A W I E R F R I E D M A N

define p ( x ) = p (x~ We extend in this manner the functions b~ (x), c (x), /(x),/c (x, u) (u fixed). I t is clear t h a t the extended functions have the same HSlder-continuity properties (in x) as the original functions. We denote the transversal derivatives at the point x on ~ (x ~ b y d / d T ~.

Consider the system

L 0 v ~ (x) = / (x) +/c (x, v) in C~ (3.5)

W e s h M l p r o v e : LEMMA 3.

a s ~ O ,

d v ~ (x)

d T s § g (x) v ~ (x) = h (x) on ~ C~. (3.6)

The system (3.5), (3.6) has (/or 0 <~ (~ ~ ~) a unique solution v ~ (x) and

{ dye(x) dv~(x') }

l.u.b.x~oc ]v~(x)-v~ + d T d T ~ ->0, (3.7) where v ~ v ~ and x' is the point on a Cs which lies on ~ (x).

Proo/. Using the m a x i m u m principle [11] and (1.9) we easily conclude t h a t if solution v s exists, it m u s t be bounded independently of (~, the bound being dependent only on the given functions of the system and on I C I" Hence, without loss of gener- ality we m a y assume t h a t k (x, u), for l ul larger t h a n a certain a priori determined constant, satisfies the regularity assumptions in (C), (Co) with constants independent of u.

We next consider the set ZN of functions w defined in Cs which satisfy Iw I ~ N.

We define a transformation T w as follows. Replace in (3.5) k (x,v) b y k (x,w). T w is the solution of the modified system (3.5), (3.6). B y L e m m a 2 it exists and (using L e m m a 1)

I Twlp. <-K (I § lll § l kl ),

where K is independent of ~. Noting t h a t ] Ic ]~ <~ K 1 +/~o K2 N, where K1, K 2 are con- stants independent of N and ~, we conclude, upon taking N = K (I h ]1+~ + I/I~ + g l ) + 1 and assuming /~0 to be sufficiently small, ITwl2+~<~N. Hence, T w m a p s Zy into a compact subset.

T is also a continuous transformation on Zg. Indeed, if we write the differential systems for T w l , T w 2 a n d subtract one from the other, we find, using L e m m a 2, t h a t

I Twl--Tw212+~<.Ka I It (x, wl)--Ic (x, w~)I~<~K4 Iwl-w2]~.

H a v i n g proved t h a t T is a continuous transformation of a convex and bounded subset ZN

A S Y M P T O T I O B E H A V I O R OF S O L U T I O N S O F P A R A B O L I C E Q U A T I O N S 13 of a Banach space Z~o into a compact subset, we can a p p l y Schauder's fixed point theorem [18] and conclude t h a t there exists a fixed point v = T v .

To complete the proof of L e m m a 3 we have to prove (3.7). The second state- m e n t of (3.7) follows from the inequality I v~ IC+~< N, which, in particular, guarantees the equi-continuity of {v ~} and of their first derivatives in their respective domains C~.

The first s t a t e m e n t follows b y either an appropriate use of the m a x i m u m principle, or b y the comparison argument of w 2.

COROLLARY. Erom the above proo] i/ /ollows that the convergence in (3.7)is uni- /orm with respect to /, h, provided ]/I~, I h]l+~ are bounded by a ]ixed constant.

Proo] o/ Theorem 2. Given a n y positive n u m b e r s, we shall prove t h a t there exists fl > 0, ~ > 0 depending on s such t h a t

] u ( x , t ) - v ( y ) i < A 8 for (x,t) E D - D e , y E C ,

I x - y l < ~ .

(3.8) Here and in the following, A is used to denote a n y constant independent of e. I n [5] we simply defined w = u - v and applied Theorem 1 to w. This method, however, fails in the present case, mainly since D is not necessarily a cylindrical domain. To overcome this difficulty, we shall not t r y to estimate u - v directly. Instead, we shall a p p r o x i m a t e v b y a family of functions v~, (6-+0) and estimate u - v ~ , .

We introduce the functions

h . ( x ) = h ( x ) - s for x E ~ C , and a p p l y L e m m a 3 with / , h replaced

0 < 6 ~<~ there exists a unique solution v~. (x) of the system L 0 v~. (x) = / . (x) + k (x, v~.) for x E C~,

d v~, (x)

- d T ~- § g (x) v~, (x) = h, (x) for x e ~ C~.

/ , ( x ) = / ( x ) + s for x E C (3.9) b y /,, h,. We conclude t h a t for every

(3.10) (3.11) We define v, ( x ) = v~ (x). B y L e m m a 3 and its corollary we also conclude t h a t there exists a /ixed 6 > 0 depending on s, such t h a t

1.u.b. IvY, (x) - v, (x)[< s, (3.12)

X E C

l.u.b, dv~*(~) dv~*(x')l ~

x~oc d T d T ~ ]~<~, (3.13)

14 A V N E g F R I E D M A I ~ "

l.u.b. I g (x) v~, (x) - g (x') v~, (x')l < ~-

xe0c 4 ' (3.14)

1.u.b.

I h ( x ) - h ( x ' ) l < V~"

8 x e O C

(3.15) Consider t h e function

w (x, t) = u (x, t) - v~, (x) for (x, t) E D - Do.

H e r e a is a sufficiently large n u m b e r such t h a t all t h e d o m a i n s B ~ (which are t h e projections of Bt on t = 0 ) lie in a fixed closed set c o n t a i n e d in t h e interior of C~, p r o v i d e d t>~ a (recall t h a t ~ is a fixed n u m b e r ) . The function w (x, t) is t h u s defined in D - D , , a n d it satisfies t h e differential e q u a t i o n

L w = L u - (L - Lo) v~, - L o v~,

= [ / ( x , t) - / , (x)] + [k (x, t, u) - k (x, u)] + [k (x, u) - k (x, v~,)] + (L - Lo) v~,

F (x, t). (3.16)

B y t h e corollary a t t h e end of w 2 we o b t a i n

I u (x, t) ] ~< A for all (x, t) E D - D , , (3.17) p r o v i d e d (r is sufficiently large.

(x, t) E D. W e also h a v e

Hence, k (x, t, u)--->k (x, u) as t - + ~ , u n i f o r m l y in

1.u.b. IvY, ]gt 4 A, (3.18)

t>cr

since U B t is contained in a closed set interior to Co.

Combining these r e m a r k s a n d using (3.9) we o b t a i n

Z w < it(x, t) w, (3.19)

where I/~ (x, t) i ~</~o ((x, t) E D - Do), p r o v i d e d a is sufficiently large. W e t u r n to t h e b o u n d a r y condition. B y (Do),

xt---~x, direction of v (x~, t)---> direction of v (x) (3.20) as t--->~, u n i f o r m l y in x E ~ C . Using t h e definitions (1.5), (1.6) a n d R e m a r k (a) in w 1 (following t h e a s s u m p t i o n (Do)), we get

v~, (x)--->0 t---> c~, (3.21)

g (xt) v , (xt) - g (x) as

ASYMPTOTIC B E H A V I O R OF S O L U T I O N S OF PA1KABOLIC EQUATIOI~S 15 v~, (z~) d v~, (~) - ~ 0 as t - ~

~ T d T ^(3.22)

u n i f o r m l y with respect to x on ~ C. Now, on a Bt we h a v e w (xt, t)

T ~ g (xt) w (x~, t) = [g (xt) u - g (x~, t, u)] + h (xt, t) - ~ @* (xt) a T - g (xt) v , (xt)

r

= [e (x~) u - g (z~, t, u)] + [h (x~, t) - h . (~')] +

[ d v . ( z )

[ d ~ ~ + [g (x') v~, (x') - g (x t) v~, (xt)] ~ I , § I S + I a + 14.

~ f j

(3.23) As t - - > ~ , I 1 - ~ 0 b y (F0) a n d (3.17); I S becomes larger t h a n ~ s , b y (E0), (3.9) a n d (3.15); I a becomes smaller t h a n 89 b y (3.13), (3.22), a n d 14 becomes smaller t h a n 89 b y (3.14), (3.21). The a b o v e s t a t e m e n t s hold u n i f o r m l y w i t h respect to x E~ C. W e conclude t h a t

aw (x,t)

T ~- g (x) w (x, t) > 0 for (x, t) C S - S~, (3.24) p r o v i d e d ~ is sufficiently large.

W i t h the aid of (3.19), (3.24) we proceed to e s t i m a t e w. ~ is n o w a fixed n u m - ber. Consider the function

0 (x, t) = - A o ~ (x) e -~(t ~ (3.25)

where ~ (x) a n d y are defined in w Using t h e properties of 9~ (x) derived in w 2 we conclude t h a t

L 0 > ]c (x, t) lO I for (x, t) e D - D~, (3.26) a 0 (x, t)

~ T

- - + g (x, t, 0) < 0 for (x, t) E S - So, (3.27) p r o v i d e d a > & which we m a y assume. T a k i n g

A0 ^{= ~ 1} 1.u.b. [w (x, o) [ + 1,

x ~ B q

we can use t h e c o m p a r i s o n a r g u m e n t of w t o conclude t h a t

w (x, t) > 0 (x, t) for (x, t) E D - Do. (3.28) T a k i n g Q > a such t h a t A 0 ~ (x)e-'(~-~)~<~ a n d using (3.25), (3.28) we get, using t h e definition of w,

u (x, t) > v~, (x) - s for (x, t) e D - D~. (3.29)

16 AWSWl~ l~I~IE DMA/ff

Since v~, (x) is a continuous function in Co, there exists fi > 0 such t h a t I v , ~ ( x ) - v , ~ ( y ) [ < e if I x - y l < / ~ , y e C , xEC~.

Combining (3.30) with (3.29), (3.12) we get u (x, t) > v. (y) - 3 e

Consider now the function

if y e C , (x,t) E D - D e, I x - y l < f i .

(3.30)

(3.31)

I t satisfies the system of differential inequalities

L o v > - s - / & [ v I for x E C , (3.33) d ~ (x)

d T ~-g(x)v<~e for x E ~ C . (3.34)

Using the comparison argument of w 2 (considering L 0 v as (L 0 - ~/~ t)v) we easily obtain,

(x) ~< A ~ for x E C. (3.35)

Combining (3.35), (3.32) with (3.31) we get

u ( x , t ) > v ( y ) - A e , if (x,t) E D - D ~ , y e C , [ x - y l < ft. (3.36) In a similar way, b y defining h* (x)= h (x)+ e, /* ( x ) = / ( x ) - e we can prove that

u ( x , t ) < v ( y ) + A e , if (x,t) E D - D ~ , y E C , I x - y i < ~ f l . (3.37) Combining (3.37) with (3.36), the proof of (3.8) is completed.

From the above proof we easily derive:

COI~OLLAI%~ 2. I/ the assumptions: / ( x , t ) - - ~ / ( x ) , h ( x , t ) - ~ h ( x ) , g ( x , t , u ) - >

g (x, t) u are replaced by

lim sup l/(x, t) - / (y)] < e, lim sup ]h (x, t) - h (x)] ~< e,

X--~ y X--> y

t ~ t--,~ (3.38)

lim sup

Ig(x,t,u)-g(x)ul<elul

(~>0),

t--> Oo

uni/ormly with respect to (x,t) ED, y E C ; (x,t) ES, y E O C and (x,t) ES, yEOC, u in bounded intervals, respectively, and i/ the assumptions o/ Theorem 2 are otherwise the same, and i/ g ( x ) , h ( x ) are C 1+~ on ~C, then

(x) = v (x) - v. (x). (3.32)

A S Y M P T O T I C B E H A V I O R O F S O L U T I O N S O F P A R A B O L I C E Q U A T I O N S 17 lim sup l u (x, t) - v (Y) I ~< A e (3.39)

X---~y t--->~

uni/ormly in (x, t) 6 D, y 6 C, where A is a constant independent o~ e. I / h (x), g (x), 1/g (x) and /(x) are bounded by K o, then A depends on K o but not on h (x), g (x),/(x).

This corollary will be used in the following section.

4. Proof of Theorem 2 (for general h, g)

I t remains to prove T h e o r e m 2 in case h, g are only a s s u m e d to be continuous.

The essential point is t h e proof of t h e existence of a solution v (x) of (1.2), (1.4).

Once this is proved, the proof of T h e o r e m 2 can be completed as follows.

We have to prove (3.8) for e v e r y e > 0 . We c o n s t r u c t C 1+~ functions ~, /~ in a neighborhood of ~ C which satisfy

1.u.b. [Ig(x)-~(x)l § Ih (x)- ~ (x)l]< ~. (4.1)

x e O c

L e t ~ (x) be t h e solution of (1.2), (1.4) with g, h replaced b y #, )~. B y the results of w 3, ~ (x) exists and, b y Corollary 2,

l u ( x , t ) - ~ ( y ) ] < A s if (x,t) E D - D ~ , y e C , I x - y l < f l (4.2) for @ sufficiently large.

Next, the function w (x) = v (x) - ~ (x) satisfies:

L 0 w (x) = ~ (x) w (I fc] < #0) for x E C, (4.3) d w / d T § ( x ) ] + [ g ( x ) - ~ ( x ) ] ~ H ( x ) for x 6 ~ C . (4.4) B y the m a x i m u m principle we find t h a t ~ is b o u n d e d i n d e p e n d e n t l y of ~ (provided we take, as we certainly m a y , )~, # a n d 1 / # t o be b o u n d e d i n d e p e n d e n t l y of ~). I t thus follows t h a t IH(x) I<~A le, W h e r e A 1 is independent of e.

We can n o w a p p l y to the s y s t e m (4.3), (4.4) either the m a x i m u m principle, or the comparison a r g u m e n t of w 2 (writing L o w = (Lo-O/~t) w). We conclude t h a t

I w (x) I = Iv (x) - ~ (x) I < A2 ~ f o r all 9 e e , (4.5) where A 2 is independent of e. Combining (4.5) with (4.2), the proof of (3.8) is com- pleted. I t thus remains to prove the existence of v (x).

E X I S T E N C E OF V. We recall t h a t the coefficients of L 0 have been e x t e n d e d to C'.

W e n.;w need to introduce a principal /undamental solution of L 0. This is a funda-

2 -- 61173055. A c t a m a t h e m a t l c a . 106. I m p r i m 4 le 26 s e p t e m b r v 1961.

18 AVNEI~ FI%IEDMAI~

m e n t a l solution F (x, t) in the whole n-dimensional space E~ of an elliptic equation which coincides with L 0 u = 0 on C'. F u r t h e r m o r e , it satisfies (uniformly in ~ in b o u n d e d sets)

F ( x , ~ ) - ~ 0 , ~ F ( x , ~ ) *0 as ] x l - - > ~ . (4.6) x~

I n the general case t h a t L 0 is a l r e a d y defined in the whole space En, t h e construc- t i o n of F is fairly complicated. I t was given b y Giraud [10]; see also [13, w 20].

I n our present case, t h e c o n s t r u c t i o n can be simplified a n d we proceed to describe it.

W e first e x t e n d the coefficients of L 0 into t h e whole space E~ in such a m a n n e r t h a t for some R > 0

a~j(x)=6~j, b~(x)=0, c ( x ) = - k 2 < 0 if [ x l > R

(k constant) a n d such t h a t all the coefficients are again HSlder-continuous (exponent ~) in En a n d c (x) ~ 0 in E~. I n w h a t follows we shall consider only t h e case t h a t n > 2.

I n the case n = 2 some of the formulas take a different form, b u t t h e m e t h o d s a n d results are t h e same.

L e t J (t) be t h e Bessel f u n c t i o n which solves t h e e q u a t i o n

a n d which, for t->O, satisfies

d 2 J n - l d J

~- J = O

d t 2 t d t

J ( t ) = K t ~-~ (l + O ( t ) ) , J ' ( t ) = ( 2 - n ) K t 1-n (l + O ( t ) ) , (4.7) where K is a positive constant. F u r t h e r m o r e ,

J (t) = 0 (e-rot), J ' (t) = 0 (e -mr) as t-+ 0% ( 4 . s ) where m is some positive constant. Following the p a r a m e t r i c m e t h o d we proceed t o c o n s t r u c t a f u n d a m e n t a l solution F 1 (x, ~) in En for t h e elliptic operator

L 1 = [L0 - c (x)] - k s, (4.9)

which has for its essential singularity t h e kernel

~n -2

F 0 (x, ~) = i det (a~j (~)) 189 J [k (5 a ~ (~) (x~ - ~) (xj - ~j))89 ( 4 . 1 0 ) Here (a *j) is t h e m a t r i x inverse to (a,j).

ASYMPTOTIC BEHAVI01~ O F SOLUTIONS OF P A R A B O L I C E Q U A T I O N S 1 9

We write F 1 in the form (compare [13, p. 55])

F 1 (x, 8) = P0 (x, 8) + I _ Pl (x, •) K (~, 8) d~. (4.11) Noting that ~ aij (8) 92 F0/9 x~ 8 xj =/c ~ Fo, and assuming t h a t the second term on the right side of (4.11) is of smaller order of singularity compared with the first term (this can very easily be verified a posteriori), the equation L 11~1 = 0 implies t h a t

P~ + 5 b~ (x) ~ F~ (4.12)

K (x, ~) = 5 [a~j (x) - a~j (8)] ~ x~ ~ xj ~ x~"

Note t h a t

I g ( x , ~ ) l <<-

i x _ ~ l n _ ~ e x p Ao ~ = m ( ~ ) , (4.13)

and A 0 is independent of k. We next observe t h a t if we prove that

f IK(x,~)ldx<e <1

(e constant), (4.14)

n

then the solution of (4.11) is given by iteration, t h a t is,

F o

(x, ~1) K(m) (~1, 8) dr],

(4.15) P. (x, ~) = F0 (x, 8) + ~ 0

where

K~

Indeed, using (4.14) and the elementary inequality

fE

1 1 const.

lx- l < (4.16)

provided 0 < fl < n, 0 < y < n, fl + 7 < n, one can prove, by induction, t h a t for j > n and for all xEE~, 8EE~

(x,

~)[+ I _ IK(J)(~' ~)]g~ < const, e j, (4.17)

I

n

where the constant is independent of j. Furthermore, noting by (4.8), (4.10), (4.12), t h a t (4.6) is satisfied for F replaced b y F 0 and for F replaced by each term

f F0 ^{(x, ~)} K(m) (~1, 8) d~,

D,

we conclude, upon using (4.17), t h a t (4.6) is satisfied also with F replaced by F 1.

I t remains to prove (4.14).

~ 0 AVl~ EI:r :FRIEDMAN

Noting t h a t in (4.13) m (/c)-,oo as lc-->oo, it follows t h a t if k is sufficiently large then (4.14) is satisfied.

We proceed to construct F. We write it in the form

r(x,e)=rl(x,~)+ f F(x,~)y(~)r~(~,~)d~,

(4.18)

1,71<R

where Y(~])={; (~)+k2 if t ~ ] K R } i f , ~ I > R " (4.19)

In the bounded domain (~, V), ] ~ I < R, ]V ] < R we can apply the Fredholm theory.

I t follows t h a t if (for any fixed x) a unique solution F

(x, ~)

of (4.18)does not exist, then there exists a nontrivial function w (~) which satisfies the equation

w ( ~ ) = y ( ~ ) f I

Fl(~,V)w(v)d~=_y($)~(~)

(4.20)

,~L<R

for I}I~<R, where ~ ( } ) is an abbreviated notation for the integral. However, we can then define ~ ( } ) for all } E E n (in terms of the i n t e g r a l ) a n d it satisfies the equation

Lo~=(L,+y)~=O

in En.

B y (4.6) with F replaced by F 1, ~ ( } ) - * 0 as I}1-*oo. Applying the maximum prin- ciple [11] we conclude t h a t ~ 0 , which is a contradiction.

We have thus proved t h a t for every x e E,~ there exists a unique solution F (x, }) of (4.18) for [}[~<R. We can now use the right side of (4.18) to define F (x,_~)also for I } l > R .

In order to study the behavior of F ( x , } ) as x-->~ and as ]xl-+oo we first mul- tiply b o t h sides of (4.18) b y F~ (x', x) and integrate with respect to

x, I xl ~ R.

N e x t we multiply the resulting equation by 1~1

(x", x')

and integrate with respect to x'. Proceeding in this manner n - 2 additional times, we obtain n + 1 integral equations: the first one determines F, the second equation determines ] F~F, etc. The last equation (with variables x ('), }) determines ] ... ~ F 1 ... F 1 F (n integrations) and the nonhomogeneous term is continuous in x (n), and tends to zero as I x(~)l-->oo. Therefore, the same can be proved for the solution ] ... I F , . . . F 1F (n integrations). We now turn to the ( n - 1 ) t h equation, ( n - 2 ) t h equation, etc. In this manner we conclude t h a t

F (x, ~) = F 0 (x, ~) + F'

(x, ~),

^(4.21)

where F' satisfies (4.6) with F replaced by F', and F' has a smaller order of singu- larity than F 0. Thus,

A S Y M P T O T I C B E H A V I O I % O F S O L U T I O N S O F P A I % A B O L I C E Q U A T I O N S 21

A ~ F ' A ~2 F ' A

Ir'l<l~-~l~-~-~' ~x ~ l z - $ l ~ ~-~' ~ < l x - ~ l ~-~'

^(4.22)

where A is a constant. We have thus completed the construction of the principal fundamental solution F.

We now return to the proof of the existence of v (x). We consider the space ZN of functions w(x) on C with norm I w ] ~ < N for some e > 0 . We define ~ = T w as follows :

ffJ(x)= fo c r(x,~)/~(~)d2- fc r(x,~)[/(~)+kff],w(~))]d~,

(4.23) where # (x) is defined for x 6 ~ C as the solution of

( [d r (x, ~) ]

= (x) + f [ a " (x' ") , (x,,)] [/(,)+

k(~, w (~?))] d~--- ]~ (x). (4.24) Here d ~ is the surface area element on 8 C. B y the properties of F [15] [13, 28-30]

it follows t h a t if # (x) is continuous on 8 C then ~ (x) is a solution of the system

L 0 ~ = / (x) + k (x, w) in C (4.25)

d ~ (x)

d T + g (x) w (x) = h (x) on ~ C (4.26)

in the sense defined in w 1. Hence it remains to prove the following two statements:

(a) # (x) exists as a unique solution of (4.24), (b) @=

T w

has a fixed point.

Proo/ o/

(a). Since the kernel of (4.24) is integrable, it is sufficient (by Fred- holm's theory) to show t h a t if

8 9 cL d ~ {- g(x) F ( x ' ~ ) # ( $ ) d 2 = 0 , (4.27) then #----0. Consider the function

z (x) = f F (x, ~)/~ (~) d ~. (4.28)

J 0 c

22 A V ~ E I ~ F R I E D M A : N

B y (4.27), d z (x)/d T + g (x) z (x) = 0 on ~ C. Using t h e m a x i m u m principle a n d the p o s i t i v i t y of g (x) we easily conclude t h a t z ~ 0 in C. W e n e x t consider z (x) in E n - C.

I n this d o m a i n it satisfies Loz = 0 a n d it vanishes on ~ C. Since b y ( 4 . 6 ) i t also t e n d s t o zero as ]x] --~ ~ , t h e m a x i m u m principle yields z ~ 0 in E n - C. A p p l y i n g the j u m p relation for t h e t r a n s v e r s a l d e r i v a t i v e s of simple layers (a simple layer is a function of t h e f o r m (4.28) with a n y f u n c t i o n /~) we get # ( x ) ~ 0 .

Proo/ o/ (b). B y a c o m p a r i s o n a r g u m e n t similar to t h a t given in w 2, we find t h a t v (x), if existing, is a priori bounded. H e n c e we m a y change the definition of k (x, u) f o r large u w i t h o u t restricting the g e n e r a l i t y of t h e proof. W e thus m a y a s s u m e t h a t k (X, U) ~< K1 for all x E C, - ~ < u < + , (4.29)

I k (x, ~)1 < K1, /

where K 1 is a constant. Solving (4.24) we t h e n find t h a t

1.u.b. ]# (x) l < K 2 1.u.b. ] ]~ (x) l, (4.30)

x e O C x e O C

where Ke is i n d e p e n d e n t of h. Using (4.29) a n d t h e definition of h we conclude t h a t

1.u.b. I/~ (x) l<K3,

(4.31)

x e O c

where K a is i n d e p e n d e n t of b o t h N a n d t h e p a r t i c u l a r w of ZN.

Using [15, T h e o r e m 8] we f u r t h e r get ] @ I ~ < K 4 , where K 4 is also i n d e p e n d e n t of b o t h N a n d w in ZN. Hence, if we t a k e N = K 4, t h e n T m a p s ZN into itself.

T (ZN) is c o m p a c t , since b y [15, T h e o r e m 8] we h a v e I@ ]8 ~< K5 for a n y fi < 1, a n d it is enough to t a k e fl > s.

T h e c o n t i n u i t y of T on ZN is easily p r o v e d using (4.24) a n d (4.23). W e can t h u s a p p l y S c h a u d e r ' s fixed p o i n t t h e o r e m [18] a n d conclude the existence of a fixed p o i n t for T. H a v i n g c o m p l e t e d the p r o o f of (b), t h e proof of T h e o r e m 2 is completed.

Remark 1. T h e a b o v e proof of the existence of v (x) does n o t m a k e use of t h e a s s u m p t i o n s (G1) , (G2). F u r t h e r m o r e , ~ C need o n l y to b e C 1+~.

Remark 2. Corollary 2 a t t h e end of w 3 holds also u n d e r the w e a k e r a s s u m p t i o n t h a t g (x) a n d h (x) are o n l y continuous on a C.

Remark 3. I f g (x, t, u) is m o n o t o n e decreasing in u, t h e n existence of a solution for the s y s t e m (1.1), (1.3) was p r o v e d in [7].

A S Y M P T O T I C B E H A V I O R O F S O L U T I O I ~ S O F P A R A B O L I C : E Q U A T I O N S 23

Remark

4. If a~jEC 2+~ (C), b~ E C x+~ (C), t h e n we can write

Lou

in a variational form and use the (1 + ~) estimates of Agmon

et al.

[1] instead of the (2 + ~) estimates.

I t is t h e n sufficient to assume in the above proof of T h e o r e m 2 t h a t ~C belongs to C 2+~.

5. Asymptotic expansion of solutions We shall need the following assumptions:

(D*) D is a cylinder and ~ B (or ~C) is of class C ~+~.

(C*) k (x, t, u) = 0.

(Am) F o r (x, t) in /),

aij (x, t) = ~ a~j (x) t -~ + t '~ o (1),

.t=0

b~ (x, t) = ~ b~ (x) t - ~ + t - ~ o (1),

A=0

c(x, t) = ~ c~'(x) t-x+t

too(l),

2=0

where o (1)--~0 as t--~ ~ , uniformly in x E/]; the functions a -~-~J, b~.9 c x belong to C ~ (/~) and a 9.~J also belong to C 1+~ (OB).

(Bm) F o r (x, t) in D,

/ ( x , t ) = ~ / X ( x ) t

x+t-mo(1),

where o (1)-->0 as

t--+c~,

uniformly with respect to

x EB,

and the f belong to C ~ (/~).

(Era) F o r (x, t) E ~B,

h (x, t ) = ~

h~(x) t-~§

where o ( 1 ) - + 0 as t - + ~ , uniformly in

xEgB,

a n d the h A belong to

CI+~(~B).

(Fro) g (x, t, u) ~- g (x, t) u and for x E b B ,

g (x, t)= ~ g~(x) t-~+t-m o (1),

2 = 0

where o(1)-->0 as t-->c~ uniformly in

xE~B,

and the g.1 belong to

CI+~(gB).

We introduce the operators

2~: A V N E R F R I E D M A : ~

~2 V n ~ V

L A V---- i,/'=1 ~ a~ ( x ) ~ § ~1= b~ (x) ~x~ + c (x) v, (5.1)

dv(X)dT ~ *.y=~ a~ (x) c~ (~(x)' xs) ~V~(xX) (5.2)

W e c a n n o w state:

THEOREM 3. Let the assumptions (A), (B), (C*), (D*), (E), (F) and (Am), (Bin), (E,~), (Fro) be satis/ied /or some non-negative integer m and let e ~ (x) <~0. I[ u (x, t) is a solution o/ the system (1.1), (1.3) then

u (x, t) = ~ u 4 (x) t -4 + t -mo (1), (5.3)

4=0

where o (1)->0 as t-->~, uni/ormly in x EB, and the u 4 (x) are determined successively by the [ollowing system:

4

L ou ~ ( x ) = / 4 ( x ) - ( z - 1)u 4-1 (x)-- ~ L#u 4-"(x) (xeB) (5.4)

,u=l

d u 4 (x) ~ gO (x) u 4 d T (x) = h 4 ( x ) - ~ g~(x) u4-~(x) - ~ 4 4 u4-~(x) ( x E ~ B ) . (5.5)

g = l ,a=l w..t

It is understood that lor 4 = 0 the right sides o I (5.4), (5.5) are replaced by ]o (x) and h ~ (x) respectively.

6. The first mixed boundary value problem

I n this chapter we shall prove analogs of T h e o r e m 1.2 to the case of the first mixed b o u n d a r y value problem. T h e b o u n d a r y conditions (1.3), (1.4) are replaced b y

u (x, t) = h (x, t) for (x, t) E S, (6.1)

v (x) = h (x) for x e ~ C. (6.2)

T h e assumptions (D), (Do) are replaced b y the weaker assumptions:

(D') 1.u.b. I Btl < ~ ,

t

(Do) ~C is of class C 2+~ and to e v e r y x on ~C there corresponds one and o n l y one point (xt, t) on each ~ Bt such t h a t x r * x as t---> ~ , uniformly in x E ~ C.

THEOlCEM 4. Let the assumptions (A)-(C), (D'), (E) be saris/led and assume that l i m h ( x , t ) = 0 , l i m / ( x , t ) = 0 , l i m s u p c ( x , t ) < 0 (6.3)

t - ~ t--~oo t-*oo

A S Y M P T O T I C B E H A V I O R O F S O L U T I O N S O F P A R A B O L I C E Q U A T I O : N S 25 uni]ormly with respect to (x, t ) ~ S , (x, t ) E D and (x, t ) E D respectively. 11 u (x, t) is a solution in D o I the system (1.1), (6.1), then u ( x , t)->O as t-->c~, uni/ormly with respect to (x, t) in D.

T h e proof is similar to t h a t of T h e o r e m 1, a n d e m p l o y s the s a m e function ~ (x) a n d a c o m p a r i s o n a r g u m e n t similar to t h a t used in w 2. Details are omitted.

TH]~OI~:nM 5. Let the assumptions (A)-(C), (D'), (E) and (Ao)-(Co), (Do), (E o) be satislied and let c ( x ) ~ O . I / u ( x , t) is a solution in D o/ the system (1.1), (6.1)then

lira u (x, t) = v (y) (6.4)

X--~y t-->~

uni/ormly with respect to (x, t)E D, y E C, and v (y) is the unique solution in C o / the system (1.2), (6.2).

Proo I. W e first p r o v e the t h e o r e m in t h e case t h a t h (x) is a polynomial. T h e proof is t h e n similar t o the proof in w 3, e x c e p t t h a t i n s t e a d of using L e m m a 2 we use S c h a u d e r ' s (2 + ~ ) e s t i m a t e s [17] (see also [3], [13]). T h e existence of v (x)follows b y using these e s t i m a t e s a n d S e h a u d e r ' s fixed p o i n t t h e o r e m , as in w 3. T h e f a m i l y v ~ of a p p r o x i m a t i n g functions is c o n s t r u c t e d as follows:

L e t C~ be a sequence of d o m a i n s which t e n d to C (as (~-->0) f r o m t h e outside, a n d which satisfy:

1.u.b. [~ C~ ]2§ < ~ . (6.5)

W e can construct t h e C~ in such a m a n n e r t h a t t h e r e exists a one-to-one correspond- ence x*-*x ~ f r o m ~ C o n t o ~ C~ such t h a t x~-~x as (~--~0, u n i f o r m l y in x E~ C.

W e n e x t t a k e C' to be a n y fixed d o m a i n containing C, a n d e x t e n d the coef- ficients of t h e s y s t e m (1.2), (6.2) t o C' in such a m a n n e r t h a t t h e y r e m a i n HSlder- continuous (exponent a). This can be done even with preserving the HSlder coefficients (see [12]).

I n each C~ we solve the p r o b l e m

L o v ~ = / (x) § k (x, v ~) in C~, (6.6)

v ~ (x) = h (x) on a C~. (6.7)

B y t h e Schauder e s t i m a t e s (and on using (6.5)) we get

v 2+~ ~ const, i n d e p e n d e n t of (~. C~ (6.s)

2 6 AVNER FRIEDMAN

F r o m this i n e q u a l i t y we get a l e m m a analogous to L e m m a 3, a n d we t h e n complete the proof b y the m e t h o d of w 3. F u r t h e r m o r e , Corollary 2 can also be extended to t h e present case.

I n the general case t h a t h (x) is n o t a polynomial, b u t only a continuous func- tion, we construct, for a n y given s > 0, a polynomial h (x) such t h a t

l -hlo~

^(6.9)

The existence of v (x) is p r o v e d b y a p p r o x i m a t i n g h b y s m o o t h functions hm a n d finding, b y using interior (2 § a) estimates [17, 13, 3], t h a t t h e corresponding solu- tions Vm converge to a solution in the interior of C, whereas, b y using the m a x i m u m principle, we find t h a t the convergence is uniform in C. Hence lim vm is the desired solution v.

:By t h e m a x i m u m principle we have

I ~ - v Ig ~< A e, (6.10)

where A is i n d e p e n d e n t of ~, a n d ~ is the solution of (1.2), (6.2) when h is re- placed b y ~.

The proof of T h e o r e m 5 can n o w be completed (similarly to w 4) b y a p p l y i n g to ~, u a corollary analogous to Corollary 2, a n d b y using (6.10).

Remark 1. If a~j E C 2+~ (C), b~ E C 1+~ (C), t h e n we can write L 0 is a variational form a n d use the ~-estimates of A g m o n et al. [1] instead of the (2 + ~) estimates. I t is t h e n sufficient to assume t h a t ~ C in T h e o r e m 5, is only C%

Remark 2. I n [6] we have p r o v e d an analogue of T h e o r e m 3 for the first mixed b o u n d a r y value problem.

7. Generalized s e c o n d b o u n d a r y v a l u e p r o b l e m

I n this section we discuss the extension of Theorems 1-3 to the case where in- stead of (1.3) we h a v e

~ u ( x ' t ) + g ( x , t, u ) = h ( x , t) on S, (7.1)

~T

where ~ u / ~ ~ = fi (x, t) ~ u / ~ t + ~ u / ~ T. I t will be assumed t h a t

(G) fi (x, t) is continuous on S a n d 0 ~< fi (x, t) ~< eonst. < oz.

Theorem 1 remains true i] we replace (1.3) by (7.1) and assume that (G) holds.

A S Y M P T O T I C B E I - I A V I O R OF S O L U T I O N S OF P A R A B O L I C E Q U A T I O N S 27 To prove this statement we proceed along the proof of w 3 with appropriate modifications. Thus, in the definition of ~ (x, t) we take y smaller than t h a t in (2.10), depending on 1.u.b. ft. We thus derive (2.11) and

~

(x, t)

Or b g ( x , t, ~f(x, t))> s. (7.2)

If we prove t h a t the function w (x, t) = ~# (x, t) - u (x, t) is positive in D - D~, then the proof is easily completed.

The proof can be given similarly to that of w 2, noting that ~ / ~ is a derivative in an outward-upward direction.

We note that the uniqueness of u, for more general quasi-linear equations and with h in (7.1) being a nonlinear function of u, Ou/~x~, was proved in [8].

Theorem 2 can also be extended to the present problem, and also Theorem 3 with the u ~ (x) depending also on the coefficients in the expansion of ~ (x, t).

Part II. Higher order parabolic equations

I n this part we prove t h a t if tile boundary values and the coefficients of a para- bolic equation of a n y order tend to a limit as t--> oo, then the solution also tends to a limit which will be the solution of the limit elliptic equation. The convergence is first proved in the L 2 sense and then it is extended to a uniform convergence.

Naturally, since an appropriate m a x i m u m principle for higher order equations is not known, the regularity assumptions on the differential system will be stronger than in the case of second order equations. The methods are also quite different.

I n w 1 we state some results of Agmon et al. [1], part of which overlap with results announced b y Browder [2]. These are used very substantially in the following.

I n w 2 we formulate the main result on L~ convergence. (The domain is not neces- sarily cylindrical.) The proof is given in w 3. Using the L 2 convergence we proceed in w 4 to establish uniform convergence. We finally discuss in w 5 the question of asymptotic expansion of solutions.

I n what follows, the notation introduced in P a r t I, w 1 will be used freely. All the functions are real.

1, Auxiliary theorems on elliptic equations

Let G be an n-dimensional bounded domain and denote x = (xl . . . x~), i = (~1 .... in),