asymptotic behaviour of the solution of a semilinear parabolic equation

Id.onatslaelte liir Mh. Math. 94, 299--311 (1982) Nalhematik

�9 by Springer-Verlag 1982

Asymptotic Behaviour of the Solution of a Semilinear Parabolic Equation

By

Abdeli lah Gmira and Laurent Veron, Tours

(Received 11 January 1982)

Abstract. We study the asymptotic behaviour as t tends to + ~ of the solution of (Ou/&)- Lu +/~(u)--0, ula ~ = 0 where L is a second order self-adjoint elliptic operator and # a maximal monotone graph of ~. If tfl (r)I/I r l 2 e L 1 ( _ 1, 1) and 21 is the first eigenvalue of L we prove that e )* 'u(., t) converges uniformly on f2 to some element of Ker (L + ,tl/) and that the limit is nonzero if [/~ (r) [/J r l is nondecreasing. We give also some properties of the limit (monotonicity, continuity, range).

O. Introduct ion

This paper is concerned with the s tudy of the asympto t ic behaviour as t tends to ~ of the solut ion of the fol lowing evolut ion equa t ion

O u / O t - L u + f l ( u ) ~ O in O x (0, + oo), u (x, 0 = 0 on a ~ x (0, + oo), (0.1) u (x, 0) = u0 (x) in O,

(? ~? . where L = 2 - - ( a i ; - - ~ is a s t rongly elliptic self-adjoint differential

operator , fl a maximal m o n o t o n e opera to r of N vanishing at 0 and D a bounde d open subset of N N. Set ).1 the first eigenvalue o f L with Dirichlet b o u n d a r y data . When/3 = 0 it is well known tha t e x* tu (., t) converges un i fo rmly in ~ to the project ion of u0 onto the first eigenspace of L. Such a l inear result is no t true with a nonzero/3 and there exist var ious hypotheses which insure the convergence of e~ltu(., t); for example if/3 satisfies

1

j" (1/] /3 (s) l) ds < + oo, (0.2) - 1

o r

21

lim infl/3 (r)i/Irl > O. r--+0

Monatshefte fiir Mathematik, Bd. 94/4

(0.3)

0026- -9255 /82 /0094 /0299 /$02 .60

300 A. GMIRA and L VERON

However, under those two hypotheses, the ~ t is 0. In this paper we prove that if ]fll is convex and satisfies

1

S (]fi(s)]/s2) ds < + ~ , (0.4) --1

e~:u (., t) is still convergent in C(~) , that the limit belongs to the first eigenspace of L and is non zero if u0 > 0 in t2. If we set

lim ea'~u (., t) = s162 (uo)ho(.), (0.5) t---~+ ~

where h0 is the generator of K e r ( L + 2t/) such that

0 ~< ho ~ 1 = Max {h0 (x)I x ~ O } , (0.6)

we prove that ~ is Lipschitz continuous from L2(~2) into ~ and we estimate the range of ~ , R (5~ For example if fl (r) = r I r I p- 1 with p > 1, we have

( - ~I/~-',~I/u-~)) c R(~) c (0.7)

\( _ ,~ ]/~,_ ~) tl h0 (.)lJ ,v ,~ ]/e -~) II h0 (.)]1 L, ) C

11 h0 (.)II ~,' ~ : The plan of the article is the following: 1 Preliminaries, 2 L ~ estimates for the heat equation, 3 The convergence theorem, 4 Some properties of ~.

Acknowledgements. This article was written while the second author was visiting Nor thwestern University as a part of an emphasis year on partial differential equations.

1. Prdiminaries Throughout this paper z9 is a bounded open subset of ~N with a C 1

boundary 8Q and L is a seff-adjoint second order elliptic operator in divergence form

L v = ~ - - I a i j - - t (1.1) . . ~ X i ~, (~X]/1 ' l J

such that

i id i

for some ~ > 0 and any (~1-.-@) in [ ~ and x in O. We shall also suppose that aij~ C 1 (t?) for any (i,j).

Asymptotic Behaviour of the Solution of a Semilinear Parabolic Equation 301

Set 21 </~2 < ~3 < ' ' " < /~n < - -o the sequence of the eigenvalues H oo of L and { i}i=l the corresponding eigenspaces. We have the

following hilbertian direct sum (see [3]). 0 0

L2(o) = | H;. (1.3) i=1

Moreover H1 is one-dimensional and admits a generator h0 such that

0 <~ ho(x) ~< 1 = M a x { h 0 ( x ) l x ~ } , (1.4) (see [4]).

Set fl a maximal m o n o t o n e operator of ~ and/~0 (.) the projection of 0 on {/3 (.)}. For any a ~ ~ we set/fla the solution of the following differential equation

{d~oa/dt q- )~l~a -It- fl(~a) ~0 on (0, +oo) , (1.5)

Wa (0) = a .

~0a can be compu ted by inversion in setting a

t = ~ (1/(fl(s) +21s))ds, (1.6) ro (t)

on the suppor t of ~o~. We make on/~ the following assumpt ion

l

(I/~(s)l/s2)ds < + ~ . (1.7) -1

It is natural to compare ~, with a e -~~ 'which satisfies (1.5) with/~ = 0 and we have

Lemma 1.1. Suppose fl satisfies (1.7). Then for any a > 0 and t >>. 0 we have the following

ae-X ' texp {-i21S2fl(s)+ sfl(s) ds} <~ Wa(t) <~ ae -~'~ (1.8)

Proof F r o m the uniqueness ~, is a nonnegat ive nonincreasing function. We set q~ (t) = ae -~t. As we have

a

t = f (1//~ 1 S) d s ( 1 . 9 ) ~(0

we deduce Wa (t) ~< q~ (t) for any t 1> 0. If I is the suppor t of ~a, I is a nonempty interval and for any t in int (/) we have (1.6), which implies

21"

302

so we get

A. GMIRA and L. VERON

ds ~ ~ (~) t = ~ ~ ~ s2 ds (1.10)

~o(,) ~s ~o() ,~ + ~sl~(s) '

1 a 1 ~ # ( s ) or t = -- Log f

~ ~ ( t ) ~ ~;~o ~ s2 + s~(s) ,

{ i #(s) ds}. (1.11) ea'tw~(t) = aexp - ,) 2 1 s 2 31- st(s) 1

From (1.7) S(fl(s)/(als2+ s~(s)))ds < + o0, so we deduce 0

~fa (t) >/ a e_Zlt exp { _ i fl (s) ds}; (1.12) o "hs2+s~(s )

so 1 = [0, + m) and we have (1.8). From the relation (1.11) we see that t ~ e al tW~ (t) is nonincreasing.

If we define l : N ~ N by

l(a) = lim e~twa(t), (1.13) t~q-co

we have the following

Lemma 1.2. Under the hypothesis (1.7) l is a contractive map and the range of 1 is (K- , K +) where

I+;1 K + = exp 21 ,tl s +/~ (s)

K_ = _ e x p { i fl(s) _~ ,hs 2 + st~(s)

i fl (s) ak} (1.14) ~l s2 + s ~ (s) '

ds - :~ l_ ~q s + ~ (s

Proof For the first assertion we set vo(t) = ea'ty~a(t) and we have

dva(t)/dt + ealtfl(e-Z~tv,) ---- 0,

a.e. on (0, + ~ ) . We deduce from the monotonicity of fl

which implies that contraction.

For the second assertion we have from (1.11), as

(1.16)

I d v a - d v b ) ( V a - vb) <<, O, (1.17)

t ~ I v a ( t ) - vb(t)l is nonincreasing and l is a

lim Wa (t) = 0, t~q-oo


f i l(a) = aexp - 21s 2+s f l ( s )

If a > 0 ,

l ( a ) = e x p s - 21s 2+s f l ( s )

M a k i n g a ~ + c o we g e t K + . I f a < 0 ,

l(a) = _ e x p { _ ~1~_+ i fl(s) ds} a S 2 l S 2 " q - S f l ( S ) '

and we get K- .

Remark 1.1. The lemmas 1.1 and 1.2 are still valid if 1.7 is replaced by the optimal condition

1 Ifl(s)] ds < + co. (1.19) s 2 + s (s) - I

Such a condition is not used in the sequel. When f l ( r )= rlrl p-l, wi thp > 1, we get

(K- , K +) = ( - 2 ]/(P- 0, ). I/(p- 1~). (1.20)

2. L ~ Estimates for the Heat Equation

It is natural to associate to (0.1) the linear heat equation

~co/Ot + L co = 0 in s9 x (0, + co), co (x, t) = 0 in 0~ x (0, + Go), (2.1) co (x, 0) = coo (x) in .c2.

We have the following regularizing effect

Proposition 2.1. Set co the semigroup solution of(2.1). Then there exists a constant C = C (L, .(2) such that

IJco(.,0JlL~ ~< (1/tJho(.)Jl• 0 (1 + ct-N/4)e-~'tllcoO(.)llr2 (2.2)

holds for any t > O.

Proof Set (S(t))t>~o the semigroup of contractions of L2(X?) generated by - L with Dirichlet boundary data. The domain of its

+co generator is H 2 (.Q) n H01 (~9) (see [2]). We set H ' = @ / / l and S' (t) the

i=2

304 A. GMIRA and L. VERON

rest r ic t ion of S(t) to H ' . I f we wri te coo ---- co! + co' wi th coleH1 a n d co' ~ O ' , we have

co(., t) = S(t)coo(.) = e-~"col (.) + S ' ( t ) co ' ( . ) , and

(2.3)

I le-~cox (.) IlL= = (1/]]ho(.)HLOe-~"l]col(.)[IL~. (2.4)

o p e r a t o r - - L - 221 is m o n o t o n e on H ' (with the L2(f2) As the s t ruc ture) we have

fl S' (v) q~ lfr 2 ~< e-Z2~ ][q~llL2, (2.5)

for any ~ in H ' and any ~ ~> 0 (see [1]). F r o m [9] there exists C = C(L, O) such tha t

I1S(tT)~ IlL ~ ~ Ct7 -N/4 I]/fl IlL 2 , (2 .6 )

for any cr > 0 a nd w in L 2 (12). I f we take 9~ = co'(.), r = (21/22) t in (2.5) and VJ = S ' ((21/22) t)co', a = (1 - 21/22) t in (2.6), we get

IlS'(t)co'(.)]lLoo <~ C((1 - - 21/22) t ) - N / 4 e - a ' t l l c o ' ( . ) l l L 2 . (2.7)

As [1 col (.)IlL= ~< II coo (.)IlL= and I[ co' (-)IIL~ ~ If coO (-)ILL=, we get (2.2) f rom (2.2) and (2.7).

Corol lary 2.1. Under the assumption of the proposition 2.1 there exists a constant K = K(L, 12) such that

]]09(., t) I[L~O ~ g e -21t [[co0(.) IlL ~ , (2 .8 )

for any t >t O.

Proof As the res t r ic t ion of S (t) to L ~ (9) is a con t r ac t i on (see [8]) we have for any t /> 0

Ilco(., 011L~ ~< Ilco0(.)[[z~. (2.9)

M o r e o v e r f rom (2.2) we get for t ~> 1

Ilco(-, t) l[L~ ~< (11211/2/llho(.)llL ~) (1 + C)e-Xltllco0(.)llr~o. (2.10)

C o m b i n i n g (2.9) a n d (2.10) we d e d u c e (2.8) wi th K- - - -Max{e a', IOI 1/2 (1 + C)/]l h0 (.) l[ L2} �9

3. The Convergence Theorem

W e set D ~ ) c~ L p (12) = {z 6 L p (12)[ z (X) e D (fl) a .e . in 12}. T h e m a i n resul t of this sec t ion is the fo l lowing


Theorem 3.1. Suppose 13 satisfies (I .7) and uo~D(fl) n L2([2). I f u (x, t) is the semigroup solution of(O. t), there exists a real o~ such that e;~tu(x, t) converges to o:ho(x) uniformly on 0 as t tends to +c~.

I f we define ~ by ~ = c~ (u0), the following estimate

15f (Uo) - ~ (Vo)l <~ (1/jlholFfOHuo(.) - v0(.) ILL2, (3.1)

holds for any u0, Vo in D ~ ) n L 2 (S2). Before p rov ing the t h e o r e m we need two es t imates which ex tend

s o m e prev ious resul ts of [6] a n d [9].

L e m m a 3.1. Suppose t3 is a maximal monotone operator of ~ and u (t, x) andv (t, x) are the semigroup solutions of(O. 1) with initial data Uo

and Vo in D ~ ) n L p (O) (17 = 2 or ~ ) . We then have

I{u (., t) - v(., t) I{f~ ~< (1/l{ho(.)l{L2)(t+ct-U/4)e-~ttl Uo(.) -- Vo(.)lfL~, (3.2)

llu(., t) - v(., t)[lf~ ~< Ke-~'~!tuo(.) - v0(.)!lL~ o, (3.3)

where C and K are defined in (2.2) and (2.8).

Proof Set

b(x, t) = L u ( x , t) - gu(x, t) /~t~fl(u(x, t)) a. e. in f2 • (0, + ~ )

a nd b' (x, t) = L v (x, t) - ~v (x, t)/St E13 (v (x, t)).

W e def ine h by

h ( x , t ) = {(0 b ( x ' t ) - b ' ( x , t ) ) / ( u ( x , t ) - v(x, t)) if u ( x , t ) ~ v(x , t ) , if u (x, t) = v (x, t).

T h e func t ion h is nonnega t ive . W e set ~ (x, t) = u (x, 0 - v (x, t) and + oo;H2(o))n oo(0,+ ([1]), so b(x,O-

- b' (x, t) vanishes a. e. on {(x, t) I ~ (x, t) = 0} and we have

@ ( x , t ) / & - L q ~ ( x , t ) + h ( x , t ) ~ ( x , O = O , (3.4)

a. e. in .c2 • (0, + c~). N o w set ~o + the s o l u t o n of (2. i) wi th initial da ta (u0 (.) - v0 (.)) + ; co + is n o n n e g a t i v e and we have

8 c o + ( x , O / ? t - L c o + ( x , t ) + h ( x , t ) c o + ( x , t ) > f O , (3.5)

a. e. in S2 • (0, + oo), w ~ c h impl ies wi th (3.4) a n d (I .2)

l-(d/dt)~ ~ I(~- ~,+)+12dx+ ~ j'[v(~v - co+)+12dx<..O. (3.6) .O' -Q


Hence q~(x, t)~< co + (x, t). In the same way 9~ is minor ized on ~2 x (0, + ~ ) by the solut ion m - of (2.1) with initial da ta - (u0(.) - - v 0 ( . ) ) - . Wi th those two est imates, (2.2) and (2.8) we get (3.2) and (3.3).

P r o o f o f theorem 3.1. As ~ satisfies (1.7), 0 e ~ ( 0 ) and we have

Ilu(., t)llLoo ~< (1/llho(.)llLO (l + ct-U/4)e ~"'lluo(.)llL~, (3.7) so we shall suppose wi thout any restr ic t ion tha t u0 ~ L o~ (~2). I f we set w(x , t) = e ~ t u ( x , t), w is b o u n d e d f rom (3.3) and satisfies

{ Ow/~t - L w - 21 w + e~'t fl ( e - ~ t w)--O a.e . in D • (0, + ~ ) ,

w = 0 on as x (0, + ~ ) . (3.8)

We split w (t, x) into ~ (t) h0 (x) + w' (x, t), with w' (., t) ~ H ' for any t >~ 0 and for the sake of simplicity we shall suppose tha t # is single valued.

Step 1. ~ (t) admits a f in i t e l imit when t ~ + ~ . We mult ip ly the equa t ion (3.8) by ho/11 h0 (.)I[ 2~ and in tegra te over

t2. As - ~ L who dx - 21 ~ who dx = 0, we get

d~ ~(t) +

~2

1

for any t > 0.

i l h o ( , ) l l 2 2 !e'htfl(e-'htw)hodx=O, (3.9)

Set C + = Max IIw+ (., t)llr~, have t>~0

But

C - = - M a x I[ w - (., t)[Iz~, so we t>~0

-]-oo

~< I~1 j" 0

If eXlt t3(e-~"w)hodx[dt <~ o g~

+oo

ea' t f l (e-~" tC+)dt+ I~1 j e*'t I # (e -~ "C- ) ld t . 0

+ oo C + c i # (u) du < + oo and e ~ ' t f l ( e - ~ ' t C + ) d t - ~ U 2

0

+~ '1 c - } Ifl(U)ldu< + j" e ~' #(e-'~"C-)[ dt . . . . . . . . 0 21 C - / /2 "

Hence d~/dt ~ L l(O, + oo) and lim ~ (t) exists. t ~ + c o


Step 2. w' (., t) converges to 0 in L oo (,c2) as t --* + co. With the nota t ions of the propos i t ion 2.1, we set h( . , t)---

= P r o j ~ , ( - e J~t ~(e-~ ' tw)) and we have

Ow'/•t - L w' - 2 1 w ' = h, (3.10)

on ~9 x (0, + oo). So w' is given by

t

w'(., t) = e~"tS'(t)w'( . ,O) + ~ e~ir - s )h ( . , s )ds . (3.11) o

F r o m the p r o o f of the propos i t ion 2.1 and [9] we have for any ~ e H ' and any e > 0

HS'(t)cpIIL~ <~ C(et)-U/4 e-(1-~)~2tltq)llL2 (3.12)

which implies, as in the corol lary 2.1,

IJ S ' (t) ~ II r ~ ~< K~ e -(1 - 0 ~,jj q~ rl L~. ( 3 . 1 3 )

for any q~ e L ~176 (L2)r~ H ' (in fact we can take e = 0 in 3.13). As a consequence lira IIe ~'~ S ' (t) w' (., 0) fl c ~~ = 0. On the other hand we have t~ +

t

II ~ e~l(t-~) S ' (t - s)h ( . ,s)ds llL~ <~ 0

K~\I( + ]}ho(.)l}LZJ o <~

/ I.ell/ 4- K~21 + iih~.)~L=/ ~ e t f l (ea"C-)Jds

/ as IIh(.,s)}lL~ ~< ~1 + IIh0(-)llY lle"'~tS(e-a"w("s)llL~"

We set ff = (1 - e)22 - 21; ff is positive for e small enough and we write t

~ e -~('-~) e~"s fl (e -~"" C +) ds = 0

A t

= I e -~('-~)eX'~fl (e -~"~ C +) ds + [. e -~('- ~)e~'t3 (e -~'" C +) ds. 0 A

+oo

F r o m step 1, ~ eZl ' tg (e-~ 'C+)ds < + oo, so for any ~ > 0 we 0 + c o


can take A such that we have: S e~,s fl (e -~,s C +) ds < 7- On the other hand A

A A

S e-,,(t-S) e~,~s fl (e-~s C+)d s .= e-~,t ~ e(,~,+~,)~ fi (e- ,~ C+)ds, 0 0

so we have

t A

~e-"(t-~ ')e~ ~ (e-,~s C +) ds < ~7 + e-~,t ~ e(~+~,)~ fi (e-;~ C +) ds 0 0

which implies t

lim sup~e -~(t-~) eX~fi (e-a'~C+)ds < ~ for any ~7 > 0. t--* + ~ 0

In the same way t

t ~ + o o 0

II w' (., t) II L~ = 0, w h i c h e n d s the proof of the c o n v e r g e n c e and lim t ~ + ~

theorem. As for (3.1), it is an immediate consequence of (3.2) and

H(Ae (u0) - ~e(v0))h0(.) l lL~ ~< (1/llh0(.)lJL0 Ilu0(.) - v0(.)llr~. (3.14)

4. Some Properties of A ~

In this section we investigate some propert ies of ~ , in particular, in t roducing some stronger assumpt ions on/~, we give estimates on the range of A a, R (A~

As a consequence of the m a x i m u m principle, of the constant sign of ho and of theorem 3.1, we have

Proposition 4.1. Under the hypothesis (1.7) A a is a nondecreasing, Lipschitz continuous function with constant 1/II h0 (.)IlL2 from L 2 (!"2) into •.

Proposition 4.2. Suppose fl satisfies (1.7) and [fl(r)l/lr[ is nondecreasing, then ~ is nonzero. In particular R ( ~ ) D (K- , K+).

We first need the following result

Lemma 4.1. Suppose g e C1 (~), g > 0 in f2 and g = 0 on ~0 with Og/8~ < O. Then there exists c > 0 such that


cho(x) <~ g(x) <~ (1/c)ho(x), (4.1)

for any xtF2.

Proof As h0~ C 1 (~) and satisfies - Lho ----- ,tl h0 ~< 0, h0 > 0 in X2 and ~?hdOv < 0 on 0D (see [7]). F r o m the cont inui ty of h0 and g, for any x e ~?Y2 there exist ex > 0 and rx > 0 such tha t

cxho(y) <~ g(Y) ~ (1/c~)ho(y), (4.2)

for any y e~9 c~ B (x, r0 . As 0.0 is c ompa c t there exist x~, . . . , xq in q q

~?f2 such tha t ~?X? C U ~ (xi, r~). I f we set ~9' = .0 c~ ( U ~ (xi, r~)) i = l i = 1

then D - sg' is strictly inc luded in .(2 and g and ho remain strictly

posit ive on X2 - f2' so there exists c ' > 0 such that

c'ho(x) <~ g(x) <<. (1/c')ho(x), (4.3)

for any x e s9 - D'. I f we take c = Min {c', Min %} we get 4.1. i = 1 . . . . ,q

Proof o f proposition 4.2. Suppose u0 ~ C 1 (D), u0 > 0 in D, u0 = 0 on OD and 3Uo/O~, < 0. Set c > 0 such that c h0 (x) .%< Uo (x). We claim tha t u (x, t) >~ ~c ( t) ho (x) in ~9 x (0, + oo). We define w (x, t) = w ( t) ho (x) ; w satisfies

Ow/~t - L w + ~ (w) = (4.4)

+ t

As fl (r)/r is increasing on R + and 0 ~< ho ~< 1, we get

Ow/Ot - L w + g (w) % O, (4.5)

in O x (0, + oo). By the m a x i m u m principle w (x, t) ~< u (x, t), so

5o(u0) ~> lira eXlt~oc(t) > 0, (4.6) t - ~ + o o

f rom l emma 1.1. M a k i n g c ~ sup D (/~), we get R ( 5 ~ 3 (0, K+). In the same way R (5O) 3 ( K - , 0).

Remark 4.1. I f we take u0 s L 2 (Q) (3 D (fl) such that u0 ) 0 a. e. and Uo dx > 0, then, for any t > 0, u (., t) is a strictly positive funct ion in

S2

Q, vanishing on ~[2. M o r e o v e r u (., t) s C 1 (sg) and Ou/c~v < 0 on &9, (see [17]); so 5O (u0) > 0.


Propos i t ion 4.3. Suppose [fi[ is convex and satisfies (1.7); then

( K - , K +) C R(~LP) C ( K - ~lho(.)L, II ho(.) llr,~

2 , K+ (4.7) II ho (.)II L 2

P r o o f If I/3l is convex and satisfies (1.7) then 0 =/3 (0) and ]/3 (r)I/I r[ is nondecreasing, so R (5r 3 (K- , K+).

We set hi (x) = ho (x)/ll h0 (.) ILL1, and for any u0 e D (~) c~ L 2 (D), u0~>0 a.e. in ~9, we set y ( t ) = S h l ( x ) u ( x , t ) d x ; y is nonnegative and satisfies .o

dy (t)/dt + 27 y (t) + ~ hi (x)/3 (u (x, t) dx = O. 0

As u >t 0 and ~ hi (x) dx = 1, we deduce from the convexity oflfl] that f2_

~ hl (x)/3(u(x, t))dx ~ /3(~ hl (x)u(x, t)dx) = fl(y(t)) D Q

S O

dy (t)/dt + 2xy (t) + fl (y (t)) ~ 0. (4.8)

If we set c = y (0) --- ~ hi (x) uo (x) dx we deduce from the maximum ~2

principle that y (t) ~< ~e (t) for any t ) 0. Multiplying by e ~'~ and going to the limit as t --, + 0% we have

5f(u0) IIho(.)ll~ ~ lim e~"t~c(t) < K +. (4.9) I lho( . ) lTL,

In the same way ~ ( U o ) > ( l l h o ( . ) l l L ~ / l l h o ( . ) l 1 2 2 ) K - for any u0e eD(~) c~ L2(~9), u0 ~< 0 a.e. in ~2.

Remark 4.2. If 1/3[ is convex the hypothesis (1.19) is necessary in order to have R ( ~ ) ~- 0.

Example 4.1. Consider the following equation (for p > 1) _ _ ~2u Ou (x, t) - ~ (x, t) + u p ----- 0 in (0, zO x (0, + ~ ) , Ot

u(0, t) = u0z, t) = 0 in(0, +oo) , (4.10)

u (x, 0) = u0 (x) in (0, ~r);

then ho (x) = sin x, 27 = 1. F rom theorem 3.1 et u (x, t) converges uniformly on [0, ~z] to ~ (u0)sin x and we have

(-- 1, 1) C R(~r C (-- 4/~,4/~). (4.11)

Asymptotic Behaviour of the Solution of a Semilinear Parabolic Equation 3 t 1

References

[1] BREZIS, H.: Op6rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Amsterdam: North-Holland. 1977.

[2] BREZIS, H. : Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In: Contribution to Nonlinear Functional Analysis. Proc. Syrup. Univ. Wisconsin, Madison 1971, pp. 101--156. New York: Academic Press. 197l.

[3] FOLLAND, G. B. : Introduction to Partial Differential Equations. Princeton: University Press. 1976.

[4] KRASNOSELSKII, M. A. : Positive Solutions of Operators Equations. Gronin- gen: Noordhoff. 1964.

[5] LADYZHENSKAYA, O. A., URAL'TSEVA, N. N. : Linear and Quasilinear Ellip- tic Equations. New York: Academic Press. 1968.

[6] MASSEY, F. J. : Semilinear parabolic equation with L l initial data. Indiana U. Math. J. 26, 399--411 (1977).

[7] PROTTER, M. H., WEINBERGER, H. F. : Maximum Principles in Differential Equations. Englewood Cliffs, N. J. : Prentice Hall. 1967.

[8] STEIN, E. M. : Topics in Harmonic Analysis. Princeton: University Press. 1970.

[9] VERON, L.: Effets r6gularisants de semi-troupes non lin6aires dans des espaces de Banach. Ann. Fac. Sci. Toulouse I, 171--200 (1979).

L. VI~RON D6partement de Math6matiques Facult6 des Sciences Parc de Grandmont F-37200 Tours, France

asymptotic behaviour of the solution of a semilinear parabolic equation

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