Viemam J. Matli (2016) 44:587-601 DOI 10 10D7/SI0013-015-0171-X
Construction of a Control for the Cubic Semilinear Heat Equation
TluMinhNhatVo''"©
Received 28June2014/Accepted: 18June2015/Publishedonline 21 October''015
0 Viemam Academy of Science and Technology (VAST) and Spnnger Science+Business Media Singapore
Abstract In this article, w e c o n s i d e r the null controllability ptxjblem for the cubic semilin- ear heat equation in b o u n d e d d o m a i n s SJ of R " , « > 3 with Dirichiet boundary conditions for small initial data A constructive way to c o m p u t e a control function acting on any nonempty open subset « of £2 is given such that the corresponding solution of the cubic semilinear heat equation can be driven lo zero at a given final t i m e T. F u r t h e r m o r e we p r o - vide a quantitative estimate for die smallness of the size of the initial data with respect to T that ensures the null controllability p r o p e n y .
Keywords Null controllability • Cubic semilinear heal equation - Linear heat equation
Mathematics S u b j e c t Classilication (2010) P r i m a i y 3 5 K 5 8 Secondary 93BOS
1 Introduction and Main Result
Many systems in physics, m e c h a n i c s , or more recently in biology or medical sciences are described by partial differential equations ( P D E s ) It is necessary to control the character-
Mi DM,!^., " " " " " " ^°'- '•'^'""^ ""^ Umversity of Orleans (France). She ihanks die MAPMO departmenl of mathematics of die University of Orleans. The aulhor also wishes lo acknowledge Region Centre for its Hnancial suppon.
S Thi Minh Nhai Vo v(mnhal@hcmpreu edu.vn
Umversite Piuis 13. Sorbonne Pans Cue. LAGA. CNRS UMR 7539. Insuiut Galilee 99 Avenue J.-B Clemeni 93430 Villeianeuse Cedes, France ' '
^ 7 ^ t ^^S!}^T-1-^'"™'^'™ MAPMO, CNRS UMR 7349, Federation Denis Poisson.
bR CNRS 2964, Bal.menl de Malhf maliques. B.P 6759. 45067 Orleans Cede^ 2 France Ho Chi Minh City Umversuy of Natural Science. Ho On Mmh City, Vielnam
^ Springer
588 T.MN, \^y=
istic variables, such as the speed of a fluid or the temperattire of a device, etc. to guaiBoIee that a bridge will not collapse or the temperature is at the desired level for example. In tbe specific words, given a time interval (0, 7"), an initial stale and a final one, we have to find a suitable control such that the solution matches both the initial state at time r = 0 and the final one at time / = T. Let SJ be a bounded connected open set in W(n > 3) with a boundary 9S2 of class C-; w be a nonempty open subset in n . Consider the cubic semilinear heat equation complemented with initial and Dirichiet boundary conditions, which has the following form-
- Ay + yy^ = ! L« in Q x (0, T).
•0 on 9£ix (Q.T). (i) [y(-,0) = / in Q.
where y e {1. - 1 1 . Well-posedness property and blow-up phenomena for the cubic semilin- ear heat equation are now well-known results (see, e.g., [2,4]), It will be said dial (1) is null controllable at dme T if diere exists a control funcdon M such that die corresponding initial boundary problem possesses a solution y which is null at final time 7". The basic discus- sion of tins article is how to construct a control function that leads to die null control!^ility property of system (1).
Our main result is ihe following.
Theorem 1 There exists a constant G > I such that for any T > 0. any y" e HhSij satisfying
iiy'iil
there exists a control function u e L^iw x (0, T)) such that the solution of (I) satisfies yi-. T) = 0. Furthermore, the control can be computed explicitly and the construction of the control is given below.
Remark I
I, Theorem 1 ensures the local null controllability of (1) for any control set w, any small enough initial data y° e H^iQ), at any time T. It is well-known diat die system (1}
witttout control function blows up in finite time for the case y = - 1 . But thanks to an appropriate control function, Theorem 1 affirms that the blow-up phenomena can be prevented for very specific initial data This issue (i.e., the null controllability for semilinear heat equations) has been extensively studied (see, e.g., [1,5-7] and the references dierein) Obviously, the result is not new fixim the point of view of null contiollabilily, but the method completely differs from otiiers.
An important achievement of our result is that we can construct die control function. An outline of the construction is descnbed as follows, firstly, we remind the construction of the control for die Imear heat equation with an estimate of die cost (see, e.g, [13] or [5]). secondly, from die previous result, we do similarly when adding an outside force using the method of Liu et al. in [9]. The solution will be forced lo be null at time T by adding an exponential weight function; lastly, dianks to an appropriate iterative fixed point process and linearization by replacing the outside force by cubic function the desired control is constructed, but the result is only local, i e., the initial condition must be small enough. The preci.se construction of the control function is found in tile praof
of this Theorem 1. "^
Another main achievemeni of our result is lo give a quantitative e.stimate for die small- ness of the size of the initial condition with respect to die control time T. The upper
CimmiUmi nf , Conlrol fa ih, C.bic Stmillnei Hon E,n,
bound of inilia] dam is a function wi± respect to the final control lime r . which obvi- ously increases to a certain value and then keeps to be a constant until T tends lo
Bai:kgrotmd We now review die achievements of contioljabihty for die heat equadons whrch has been intensively studied in the past. Consider the heat equadon in die foilowinv
form: "
J 3,y - Av + at. x)y + flt.x.y)=l | , „ + g in J! x (0. T).
For linear case With / = 0, (2) is null controllable widi no lesuiction on v" r and a, which means Ihe global null conliollability holds. There are at least iw,o ways to approach such result. The first one is due lo Lebeau and Robbiano [8], who connect null controilabd.
ily 10 an inlerpolalr^Dn esdmale for elliptic system. Tie second one is due to Fu™kov and Imanuvilov [7], and is based on a global Carleman inequality which is an estimate widi an enponenlial weight funcdon and on a minimizadon technique to consdiiel die control func- tion For nonlinear case, Fursikov and Imanuvilov |7] also give us the proof of global null controllability when fU.x..s) satisfies the global Lipschitz condition in ., variable wid, .":, u , " . ' " ™ ' ° ' ^ ' ' • • ' " ' " ' ' ^"^ Piinl theorem, and assert die local null con- irollabilily when / ( , . ,-. s) saiisfies the superhnear growth condition m ., by means of die implicit function theorem. In 17], Fursikov and Imanuvilov point out that null conlrollabilitv
«o,ks in case die minal data is .small enough but widioul an explicit formula In addition Amta and Taiaru [1] improve tbe result of Fursikov and Imanuvilov b , providin. sharp esdmale. for die eonlrollabiiity lime in terms of the size of the iniual data A liitlehi dif- ferem fram diis document, m |6]. Femdndez-Cara and Zuazua e.stabhsh the first result in the Itieralure on die null controllability of blowing-up .semilinear heat equation. In detail ftey prove dial the system is null-controllable at any lime provided a dobaily defined and bounded uajeciory exists and the nonlinear lerm /(.v. i. s) is such that | f(,,l| grows slower
* » M log! (1 -I- Isll as III - , oo. Furdiermore. they observe dial it is not possible to obtain a globd eonuollabdily resuli for a cubic nonlinear term More recently, die controllability of a parabolic .system widi a cubic coupling term has been studied by Coron el al in [3]
Another interesting problem is 10 study die case where ihe blow-up phenomena will not occur, for example when ). = I. Our method gives die following result-
Comllary 1 There exists a constam C > I such that for an\ T > 0 anv i" e / -lOl
satisfy-ing '
< max - i"-''tcii+t)2jiey'
™Si™''" °""'^'''>""""•" " 6 /-'(!» X (0. r i ) such that the solution of II) ,vitb y = | The article is organized as follows. In Secuon 2. we deal with die hnear heat equadon.
1 ^ consmictiou of the control f„, die linear beat equadon with outside force ,s descnbed tete. In Section 1 we apply this consuuclion widi a fixed point arenmem in order to prove
the main results. Theorem 1; Corollary I. ucriopro^e
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2 L i n e a r Cases
in this section, we survey the null controliabUity properties for the linear heat equation.
2.1 Basic Linear Case
Now we recall the results about the null controllability and observability for linear heat equation.
Theorem 2 Far any T > 0 and any ;o S Z,^(S2). there exists a control function u g L-iw X (0, T)) such that the solution z of
3,z-Az= ILu in Q x (0. T).
; = 0 on QQx (0, 7").
zi:Q) = zo in S2,
satires ;{•. 7") = 0 in Si. Furthermore, u can be chosen such that the following estimate holds:
\\"h'-«^M0.T))<Cer\\zo\\L2tn) for some positive constant C = CiQ. u>).
The positive constant C is given in the following equivalent dieorem (observability estimate for die heat equation)
Theorems There exists a constant C > 0 such that, for any T > 0. for each ^T e i^(Q).
the associated solution of the system
iJfitH-A(t) ^ 0 in Six (Q.T).
It) = 0 on dQx (0. T).
*(-. T) = $7- in n satisfies
The two above results are quite an old subject which started at least from the works of [8! and [7|, Many improvements are given in [5, 6, 10-13] We turn now to study the nuU controllability problem for linear case, but with an outside force.
2,2 Linear Case with the Outside Force
Consider die linear heat equation with the outside force, which has die following form:
|
9,v - Ay = / -I- 1|^H in n X (0, T),
y = " on 'dU X (0. T). (3) .v(-. 0) = y in S2.
For Ihe moment, we choose / ' e L^i^) and / e L-(Q x (0. T)).
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r
Consmiction of a Control forthe Cubic Semilinear Heat Equa Let ( 7 t ) t > o be die sequence of real p o s i d v e n u m b e r s given by
Tk = i - - ^ . (4)
where a > I . P u t ft ^ l | ( 7 i , 7 i . , | / . We start to d e s c n b e die algorithm to construct the control: w e initiate witii zo = y° and w^i = 0. Define die sequences ( z t h ^ o , {«i-U>o.
(i'*h>o. luit]k^o as follows. Let vt be the soluUon of
I
S.Vk - Avk = fk in n X iTk. Tk+y).Vk=0 on 3 S 2 x f 7 i , 7 t + i ) . (5) Vki:^Tk) = Wk-\i-.Tk) in Q..
Introduce
Zk+\ = U i ( - . r ( + , ) . (6)
a,tp* + Aqu = 0 in S2 X iTt. 7 i + i ) .
»Pi = 0 on a n X ( T , . T i + i ) . tpA('. 7 i + i ) = tp['*' in £2.
H e r e , ( p / * ' 'S die unique m i m m i z e r (see die proof of T h e o r e m 1,1, page 1399. [5]) of die following functional depending on e* > 0: ^ : L-iQ) -* R given by
where C is die constant in T h e o r e m 2 and
|
3,$t -t- Aijii = (1 m n X ITi, n+D,
• 1 = 0 on 3 ! 2 x ( r , , 7 i + , ) ,
• i ( - , 7 i + l ) = • [ ' * ' e Z , ' i S 2 ) Let uii be the solution of
|
3,u', - A i o , = l | . , , „ m S2 X ( r , . F i + i l ,
•"» = » on a n x i r , , 7i+i).
m i l - , 7 i ) = a m J!.
Therefore (see. e g . , {5])
u.,(-, 7-n.i) = e,ip[**
Cel^ 'Tl i.'"*''''"'' + ^ X ' ' ° ' ' ' ' ' ' + ' " ' ' ' ' - '""I'l^
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-^1
TJM.N- Va Finally, put y^ = uj. + u^, then it solves
d,yo - Ayo = /o + 1 UHO in fi X ( T . T, ).
.vo = 0 on aS2 X (T. Tl).
yo(_:Q) = y° in S2
•k+t - Avi+i = /i-+, + 1 U ( + | in n X ( n + i . 7i+2), h i = 0 on dnxiTk+i.Tk+2), i-l(--7(+l) = wi(-.T(_(.|)4-2A+i in n .
Notice tiiat >•;{, Tt+i) = yi+](-. 7(+i), therefore the functions y = Ylk>o 'lin.rt+iiJ'j andH = Lt>ol|[Ti,7i^.i|",t satisfy (3),
Now we are able to slate our result; (recall tiiat a and Sk are needed in (4) and (9) respectively).
Theorem 4 Let C be the constant in Theorem 2 There are k > 0. a > 1 and a sequence (£iU>o of real positive numbers such that for any y" e M^{^) and any f 6 L^(Q X (0, T)) such that feT^ € ^.^(£3 x (0, T)), the above constructed coniml function
-Y^\in n+ii"*
is in L-ia>x(O.T)) and drives the solution of ii) toy i .T) = 0. Furthermore, there exists a positive conslanl K such that the following estimate holds:
Now, we come to the proof of Theorem 4 2.3 Proof of Theorem 4
Our strategy to prove Theorem 4 is as follows: we want to get ILveJ^ |lc([o T] L^<a)] < +°°
for some suitable constant M > 0 in order to deduce tiiat .v(-, 7") *= 0. To do so, smce r = Lt^^o Ilin.7i.niyi; and y^ = vt -\- w* is given by (5H8), we start to esti- mate ||Ui||t-,|7j Ti^ii.t^tQ)) and \\">k\\cnn,Tt-n\.Li(n))- 'n the same time, we also derive an inequality for 11"^'^ llf.J(,„^(o.7-|, for some suitable constant fi > 0 in order to gel u e L^(a> X (0, T)) Finally, we will focus on estimating ||Vv^r^ \\r.,a TI.-J/QI, for some suitable cn-^— f^ - " ic(|i),rj,i. (m
By Ihe classical energy estimate for the heat equation widi outside force, one has frotn (5H8)
ll^'<illciir„,r,|,i.^(Ri) < ^WJhh^-ia^a^Toi-
ll^i+illain^i.Ti^.u^mii ^ ^ll-A+ill/.3if2-(r.^,.r,^j„ + Ni('.7i+i)||t,,jji and
ll""«cnn.r..,u.,o„ < v^l>«lt=,..,r.,r,.,„ + iz,l,.:,ay
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Cmslnictlon of a Conlrol for ibe Cubic Semilinear Heal Equanon By using die following estimates, which are implied by (10):
ll"*(l/.^to..7,.r..,„ < ^/Ce^-^'^^^WzkU,^^, and |)t,,t(., r,+,)|i^:,^, < ^kWzkWc-,, we get (1
ll''^+illa[n_, n,,].i2,«„<Vf||A+,||^,,„,,j...,.r,_,„ + V^lkti[t.,f3, (i
and
ll«'*llcnr.,7,.,,.t:m), < '^cfe^-^^^^WzkWc-^^. + WzkU,^^, (I Since by (6) Vk+\ ( . r(+2) = Zk+l. it implies using (12) that
As a result, for any constant A > 0, we get
= ^ ' ^ llzollt=,fi, + e ' ^ \\zi\\c-mi + E ^ ' ^ ll^*+?lli.=<iii
^ ' ' * ^ ILv°lll!,„, -F . ' ^ s/f | / , l k . , „ „ , „ r, „
+ ' ^ r ' - ' = * ^ l l / w i l i i | „ „ r . , , r , . , „ + X ; e ' ^ V ; a . - , | | ^ i , „ ,
^••'^l.v»llt.,n, + v ^ | ; e ' 4 ^ t l / . | | , i , „ „ , ^ , , _ _ „ + J 3 , T 4 ; T ^ I | „ , ^ , _ ^ ^ 5 ' ' ^ l . v ° I U : , n , + s ^ E f ' * ^ l l / , l l i i , „ . , r , r i . , „
+ i ; ' ^ * ^ v ^ l : i l r t m - (14 Choose
J, = - , - T ^ ^ . , , 5 ,
in order dial e l ^ H ^ y ^ J j ^ e ' = f c . dien (l4)becomes
w l ' ' ' ^ ^ " " ' " ' ' " " ~ ' ^ ' ' ^ " ' ' ° " ' - ' ™ " ^ ^ ^ 2 ] ' ' ' ^ l / i l f i ! l « c r , . r , , „ r (161
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On one hand, for any constant M > 0, we obtain by (12), (13), and (15) 2^e'"-'''-n l|.V([lci|r,,rt.,i|;i.-(fl))
il>0 k>0
i«'^^ll/»lt=ltixir..r„, + E ' ' ^ ^ l l / ' l l i = i < l « r a , r . . i l )
+ J2''^^--Ml\-I:h'is»+T,<'''^ (' + '^"'''"''*^)i'l,h'm1,-20 t>0 ^ I
+5 E'•''^^^'f^ 1:.Iliiiri, + E " ' * ' «=»lli=(!!l
where /i/ = max{flM - Aia- - 1), M.M -\- ^TST,1 which implies under die condition N s A with (16) that
E''^"^li«llni7l,7i.il,£=(S3l, < 3(l+^/CT)e^|Lv<'||^2(J^, t=:0 ^ '
+3-yf(l+VCT)Y.eT^Ut.hH^„tT,.Ti„,y
Therefore
i-*^'^llc(lO,r|.tJm)) - E ^ ""'*'li>''--llc(in.Ji.nl:t2(£i))
<3(i+Vcr)(.A|iy>||,,,„,)
-l-3yf(l+^/cf)||/ef^|| (17)
On Ihe other hand, by the first inequality in (11). one has forany constant S > 0E^^^""*"t'(.x,7-.,n,„, < E^'^^^^^'^^ll^ilk^.n,
< VcTe(''+'^)r^l|.,ll ,
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Qmstnicdon of a Control for the Cubic Semilinear Heai E which implies under die condition B + j j ^ < A widi (16). diat
E^''^'^ll«tllt:,^„(7-,7i^,„ < 2 V C e ^ | | y ' ' [ | t . , j j )
-f2Vc?X;«'^IIAIll=,tl.,r.r,_,„.
Therefore
< 2 V c ^ ^ | | / | | ^ , , J J ^ + 2 ^ / C T | / e ^ | | (18) Il lli-mA(0.n) By takings = M = ^ ^ ^ and A = ^ , we conclude fiom (17) and (18) dial
lye^^M + | | « . ^ T ^ | |
< c ( l + y f ) . ^ ^ | | / j | ^ , , ^ , + , y f ( l + y j ^ ) | | / e £ ^ 7 b | | ,19) 'I llz.-(n«io7-)i for some constant c. We turn now to die case .v" e //,{(Q.). For any constant £) > 0, put P = pil) = e"^! and g = py dien g .satisfies die following system
I
d,g-Ag = p'y + p{\\^u-f) in S2x(0. T), S = 0 on 9J^ X (0. T).gi.l3) = e7y^ in S2.
Applying classical energy estimate, one has
ll^sllcdo.nt^itj), < "^W^y^Wi.HiD + Wp'yWL^in.m.Tv, + ll/'"ll/.^(Q.(0,ni + llp/ll;.^(£2xm.n,- whieh implies, for any /> e (1. 3/2) die existence of A'^ > 0 such tiiat
l'"*lc„.,.,..=,ii„^'-'«'^°"'^.'..-*^»l|-"*^ll«„.,o,,
-i-||„t^|| +1/^^11 _ (20) Take
in order that a>\.pD = ^ ^ and ^ = 3D. Then, it implies by combimng (19) and (20) that
l''-"*lln,u.ru.,a„=^'^(' + ^ ) ' - " « ' - ' " « « » . + 'f(l + n | | A « | ^ , , „ ^ ^ „ , ,
r '22) lor some constant K With k = ^^ —_^^- in order thai D = ,l.C. we have completed die
proof of Theorem 4,
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3 Proof of Main Results
This section focuses on die proof of die main results. Theorem 1 and Corollary I, whfclJ;
ensures diat system (1) is null controllable widi die different conditions of the initial datt.
First, we stan widi the proof of Theorem 1.
3.1 Proof of Theorem 1
The idea of die proof of Theorem 1 is as follows: first, by applying die result m TheoiBm 4, we construct a control sequence u„ e L^ (^ x (0, T)) such tiiat die solution of
3,ym - Aym + yyl_.^ = l\a,u„, in H x (0, T), ym = 0 on 3S2 X (0. T), y„(.,0) = / in S2
satisfies y„(-. T) = 0 in //J (S2); secondly, by proving y„ converges to y and u„, converges to u, we will get tiie deshed result. Now, we start die first step by checking diat die function f = "yym-] satisfies die condition of Theorem 4. Denote D = AC. Firet take yo such that yo(-. 0) = y° and yy^e^^ e L^{Q. x (0, D ) . for example yo = e-rr.eTy'i. Now by mducdon, we will prove diat yy^eT^' e L\^ X (0, T)) for any ni > 1. Indeed, suppose
>'m-i^'^ e L^iQ X (0, T)), by Theorem 4, y„ verifies
Using Sobolev embedding, we obtain
\Wym(:t)e
< c*:r (^(i + y f ) cf |v,»[ii,,„, -I- (I + D I ,= _ h^in-io.Tn) ftom the inducdon assumpdon. Thus, the control u„ constructed in Theorem 4 lends 10 )»(•. T) = 0, which completes die first step. Now, we pass lo die second step by proving
*at (.v»(-. l ) c ' = l is bomided in C(IO, T], ffj(n)) for «,y m > I. From the inequality in Theorem 4 widi D = AC (or simply (22)) and Sobolev embedding, we gel
v,-„(.,,)cA ,
II WLhii)
<ff(l-H^/f).^||VyVc«,+.^(l + n(_^'||v,„^,(.,,,,^||^^^^^^,y if({i + ^)e'f\iVy'^U2,n,+cKVfi\-i-T)( sup \\vy„ ,(. t)eT^\\ V
® Springer
r
Ccfflgmcuon of a Conlrol for tlie Cubic Semilinear Heal Equation which implies
,.Z|ll''-"""^IL^,n, - ''{' + ^)''^tl^y°h:,^, + c>:^l) + T)
X t s u p V v „ _ | ^ f - ' V'eio.rj'i • U-<Q,J
8 ( / f ( l - H y f ) . ^ , | V y ° | | , , , ^ , ) - <
cKVf(l + T)
f^l|Vy°||i:,sj,= sup ivyoe^i S2K (\-\-Vf) e'-^ \\Vy°\\r.„s- dien by induction, w e have for any m > 1
sup | v y „ ^ ^ | | _ S2K(i + Vf)e^\\Vyf>\\ ,
„e7^ is bounded in C l [ 0 . T]: H^iil)) for any m > , . w h e n e v e r l h " l l „ o , o , is Thus. ;
sufficiendy small. N o w w e prove that {y,„erh] and l i . ^ e ' r ^ l are C a u c h y sequences m C l i o , T]. H^lQ)) and L'ia, x (0. T)). respectively, for any m > I. Indeed, put Y„+i = ym-hl -ym sndU„+[ = M „ . n -u„ for any/M > 1 dien Y„,^i is a solution of
I
3,yn,+\ - ATnp+i = -yiyl - v ^ _ , ) -t- 1|,„(,'„+| m £2 x (0, T).•'ni+i = 0 o n a n X (0. D . l ' ™ + i ( . 0 ) = U ,n fi Rrsdy. we will estimate U„.i-\er^ Recall that tiie control function u ^ is constructed by
wherc<p„+| ( - ( p „ t solves
I
3/(9".+ i , i - < p ™ , ( ) - l - A ( ( p , „ + i i - ( p „ i ) = 0 in Q x ( T ^ T i ^ , ) . ( < P " . + U - < p ™ t ) = 0 on 3 i i x ( r t , 7 i - i . | ) .«P«->1 i - <Pm.i)(-. 7 i + | ) e £ . - ( « ) .
Fufdiermore. we also h a \ e constructed the functions u ' „ + , j and w„, t by applying T h e o r e m 4 widi / = - j / y ^ and / = -yy^_^ respectively:
3 , ( „ . _ u - u v t ) - A ( « . „ + , , - u - „ , i
= _ C ^ ' . - t - ' . , ( p „ ^ U - < P ^ . i ) in Q x ( 7 i J i ^ i ) . ' " • " - I * - ' ' ' " , , t ) = 0 o n dQ X ( T i . r , ^ , ) 124)
< u W i . l - w ™ i U - , 7 i ) = ; „ ^ l , l - ; „ i m Q.
I ( u ' ™ ^ n - i i , „ i ) ( - . 7 i 4 - i ) = f*((p„-i-n - q ) „ i ) ( . . 7 i + i ) m n .
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Muldplying die two sides of die first equation in (24) by (,p.+, . - p . .) and integrain.
over L2 x (7i, 7i^[). weget t: i-T,.i ,
-Ce i»i '• j ^ J |(|i,+|.i -i|)„jpdxrf7
= Jjf.,yl.t, - >«.,,»)(•. r i + , ) ( 9 , + ,.J - ,p„,)(., T,y,)dx - j (fm+l A- - Ulm.iK-. Tt,)tip„yi.i, - tp,„^)(., ri)d.r
= « ^ l«P»,+i.. - <?.,.,)(•. r.+,)prf» - X'=^"+i' - J-,.K'P-+u - •p-.iK-, r,)d,.
Therefore, we can write
Ce^^^^ I I | ( p ^ ^ , k-~<P.n.t\^dxdt+Sk f Ktp,„+u- - (Pm,t)( , n + | ) P r f . t
= y (;^+i„t - z™.i)((p™+i.t - ip,„.0('. 7()rf.v
< l l W i . t -z„a!!/::,„,||(<p„,+ ,,A -(p„,()(., 7i.)||L.ijj,.
By die classical observability estimate (see Theorem 3), we deduce tiiat c f'^M f
Ce't-M-^»y / i'p,„+i.k->p,„.i.l-dxdt
< Vz„,-^!.k -Zm.khHii) {Ce'M^'''' I J \<(>m-^\.i-<f^,.k\-d.xdtY Therefore, we gel
Y^'''^+U-u.„.k)e^l
Y^^''•*'''"'^'^('Pm+l.k -f>m.k)l lliL-(rixi7i,r,+
< J2C''':^'''^«ii.y,.i-„„.itti,,„,,^^,^
s ^ E ' - < ° * ' ^ ' - * ^ I U . i . - z . . . | | , ,
(nxin.T^,,))- (25)^ springer
Conanicnon of a Control for die Cjbic Semiiuiear Heal Equanon
Recall dial the constants a and D were given m die proof of Theorem 4 (see (21)) and satisfy D 4- ji:—^^ ~ ipia-t) + '>{a-\) - 7a=TT- Following tiie same compulations as in the proof of Theorem 4 (see (16)). we get
Ye"-' ^'"''+' \\Zm + \ k - ZmkWLhil)
< 2^Y^^'^^'\\yi.k-yU.,\\LH^.ak,n.o^
k>0
^ 2 V f | | ( y ^ - y ^ _ , ) e f t | | , (26) II llz.^|£2x(0,r) ^ '
Therefore, combimng (25) and (26), it holds:
| | f / m + i e ^ | | , < 2 V C T | | ( V ^ - V 3 Aef^W (97) Secondly, following die same computations as in the proof of Theorem 4 (see (22)), we
obtain
|vy„,(..,)cAII <*:,i + r)||(v=-y,;_,).«l| , vi 6 (O, n .
, , (28) But, from \a' - b^\ < 2\a - b\ia H- h)- and Holder inequahty. we have
llj'm - -V,^,-i Ili:,j3, < 4 /" (v„ - y,„^i)^(.v„ + y „ _ , ) V r Jn
^Hx—'fa—f
5c|iVy„-V,._,||I,,„,,V,. + Vv„_,[|ii,„,
<4c|V>-.,,^,,„,(||V,.„|J,,„, + ||Vv,.^i||J,,„^)\
As a result,
ll<'^->--'^*IL..«.,.,r(i''<^J-ri-i>iii.,=,^-<")-
< ( f 4c||vy.,||:,„, (l|Vv„iii=,„, + l | v , : , - , l l i , , „ ) % e A ) '
< ( / ' 4 c | | y y . , c f t f f | | v r , . , . A f , - f l l v v , _ i c A f V d r V
y o " IlL-ini V" llt^isli 11 h:mj I
So by (2.1), il implies
| | ( v ; ; , - v ; ; _ , ) . « | | ,
II lli.-(£is(0.rii
s4s/fc sup ||vy.,pA|| I sup ||vv„cA|| "i
,€lo,ri> ' I ' l r a V e i o r i ' Itsinjy
<4s/fc»p^J|vr„.>*i||^^^^(2^(, + yf),¥llv,,o,|,,„^)' ,2„
© Springer
_. _--^
^ TMJI.\(ji Gathermg (28). (27), and (29) yields
< KO + T)\\ivl - vi_.)eT^\\
<4K{\-hT)^/fc sup ||vy„,eT^| , (^K (l-\-Vf)e'-^\\Vv%,,^,Y (e|o.r|ll llt-(S) \ *> y II > iit'tfl);
and
II Ili-lmxiO-DI
<2Vcf\\iyi~yl_^)e^\\
II llL^t^xiO.T)
- ' - ^ " ' . s , II"'"""*' IL^ii. (''^ ( ' + ^ ) ' " ii'^°"^=<i..)'-
Therefore, whenever
4/r(l + r ) y f c - ( 2 ^ ( l + y f ) e ^ | | V y ' ' [ | t ; ( n , ) ^ < l , which can be written
''""^' ^''.'•iGd-^o^Vr^T
for soine constant G > 1, tiien y„, converges to y in C([0. T], Wj (J2)) and «„ converges to M in i,-(to >: (0, D ) . This completes the proof
3.2 Proof of Corollary 1
Now, we prove Corollary 1 Consider the following system:
a , y - A y - ) - y ^ = 0 in £2 x (0, r / 2 ) , -V = 0 on a n x (0, T/2) y ( - . 0 ) - . y " in Q.
Recall that no blow-up phenomena occurs. We can establish by classical energy estimate tiiat y(., T/2) e H^ iQ) Funhermore, one has
'l>^''- ^^2)f„,,„, < ^11/11=.,^, < tnax L ^ .
Consequemly. applying Theorem !, we obtain the existence of u e LHii x iT/2 T)) such ihai the solution of y i • >/|
a , v - A y + y^ = l|,^i7 in Q x (T/2, T),
> = 0 on an X iT/2.T).
yi..r/2)^yi..T/2) in Q.
satisfies y'(., 7") = 0,
Put
v( t) = ly^'''^ f""" 'e(0,7-/2), 1 yi: r) for / e [r/2, T].
0 springer
r
Constnicuon of a Connpj for the Cubic Semilinear Heal Equa flien y satisfies (1) in case y = 1 witii
' • ' i ;;(-, f) for r e [ T / 2 . T]
and y(-, T ) = 0 , This c o m p l e t e s the proof of Corollary 1.
Acknowledpnenls The aulhor would like w express her graunide lo boih referees of diis journal for die valuable commenrs, important suggesnons, and correcuons of tins work which impraved luibsianoallv the first version of this anicle
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