Journal of Mathematical Sciences, Vol. 98, No. 6, 2000
AN ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH BOUNDARY DAMPING I. N. Kostin
UDC 517.9
The existence of a global compact attractor is proved. The question on the existence of a Lyapunov functional is studied. The existence of a Lyapunov functional leads to a series of important facts on the structure of the attractor. Bibliography: 8 titles.
w Introduction For a function u = u(t, x), we consider the following semilinear equation with boundary damping: =0
ii-Au+u+f(u)
+ (vVu) = 0
inN:+ x f 2 ,
(1.1)
on R+ x 3~2
(1.2)
(the dot denotes differentiation with respect to time) in a bounded domain f2 c R n whose boundary 3 ~ is of class C 2. Let v denote the unit outward normal vector to 0f2. The nonlinear function f : IR ~ IR is of class C 1 and satisfies some growth conditions (cf. Sec. 2). The problem (1.1), (1.2) generates a semigroup of continuous solution operators St : Xo --+ Xo, t >~0, in the phase space X0 = Ht (f2) x L2 (f2) of the variables (u, ti). These solution operators are defined by the formula
= (u(t,), a(t,.)), where u is the generalized solution to the problem (1.1), (1.2) satisfying the initial conditions u(0,. ) = % fi(0,.) = V . The energy functional associated with the nonlinear wave process described by Eq. (1.1) is defined by the equality
1 .2 + lU2 +F(u) E(u,/i) = ./[llvu[2 + ~u D
3My,
(1.3)
where F is a primitive function of f . Formal integration by parts shows that the problem (l.1), (1.2) leads to the energy equation d
=
_ f gt2dx" 3f~
(1.4)
Translated from Problemy Matematicheskogo A naliza, No. 18, 1998, pp. 181-197. 1072-3374/00/9806-0753525.00
~)2000 KluwerAcademic/Plenum Publishers
753
We can expect that the solution semigroup indicates a typical behavior of dissipative dynamical systems (and, in particular, possesses a global attractor). The main goal of the paper is to establish the existence of a compact global attractor. We also discuss the question on the existence of a Lyapunov functional. By definition, a Lyapunov functional for the above-mentioned semigroup is a continuous function L: X0 ---+IR that does not increase along each trajectory and is not constant along each nonstationary trajectory. The existence of a Lyapunov functional for a semigroup implies a number of important facts about the structure of an attractor. These facts are essentially used in the further study. From (1.4) it follows that the energy functional E is a Lyapunov functional for the semigroup
St if
for any solution u to the problem (1.1), (1.2) the identity u = 0 in E+ • 3f2 implies the identity fi = 0 in 1I~+ • ~2. We show that this is true in the case n --- 1. The proof of a similar fact in the multidimensional case is not known. The asymptotic behavior of solutions to wave equations with boundary damping was studied by many authors (cf. [1-5] and the references therein). These papers are mainly concentrated on the problem of the stabilization and controllability of solutions. To establish the existence of an attractor for the problem ( 1.1 ), (1.2), we use the general scheme from [6], which was successfully applied to the study of semilinear wave equations with distributed damping
t i + t i - Au + f ( u ) =
h(x)
(cf. also [7]). The paper is organized as follows. In Sec. 2, we give a list of assumptions about f~ and f and formulate the main results. In Secs. 3 and 4, we prove the existence and uniqueness of generalized solutions to the problem (1.1), (1.2). In Secs. 5 and 6, we prove the existence of an attractor, In Sec. 7, we discuss questions concerning the existence of a Lyapunov functional.
w Assumptions and Results Throughout the paper, we assume that the following conditions hold. ( ~ I) ~ C R n is a bounded domain with boundary of class C 2. ( f l ) f is a function of class C 1 and there exists a c o n s t a n t Cf (f2) For any ~ > 0, there exists a constant
cf(e)
> 0 such that
> 0 such that
[ft(s)l ~
If(s)l <~~ls[ +cf(e)
Without loss of generality, we can also assume that the following conditions hold. (/3) IF(s) l >/- 88
2 for all s C IR, where F is a primitive function o f f .
(f4) f ( 0 ) : 0 (therefore, If(s)[ <~ Cflul). 754
for all s E IR.
We study the initial boundary-value problem
ii - Au + u + f ( u ) = 0 + (vVu) = 0 u=q~,
in IR+ • f2,
on LR+ • aft,
a=~r
(2.1) (2.2)
on{0}•
(2.3)
We introduce the function spaces
Xo = HI(~"~) x L2(k'~), X1 = { (u, v) 6 H 2 ( ~ ) • H t ( ~ ) " (VVu) + v = 0 on a ~ }
equipped with the norms
II(u,v)ll2-- IlVuH 2 + Ilull 2 + Ilvll 2,
II(u,~,)ll 2 = Ilauli 2 + Ilull 2 + [IVvll 2,
where ]l " II without indices denotes the L2 (~2)-norm. For a pair (% ~) E Xo, a function u" [0, T] x f~ ~ 1~ such that (u, ~) E L=([0, T],Xo), u(0,. ) = ~o, and the identity T
T
0 17
holds for all functions rl 6 Loo([O,T],Hl(a)) such that ~ c L~([O, TI,HI(~2)) and n ( T , - ) = 0 is called an
Xo-solution to the problem (2.1)-(2.3) on [0, T]. T h e o r e m 2.1. Let (f21), ( f 1), and (f2) hold. Then for any ((p, ~) E Xo there exists a unique Xo-solution
u to the problem (2.1)-(2.3). The corresponding solution operators St : Xo --+Xo, St (q0,~) = (u(t, - ),/t(t, - )), are Lipschitz continuous. If (% V) E X1, then St (tp, V) E L~ ([0, T], X1) for any T. For a nonempty set A E X0 and a bounded nonempty set B E X0, we introduce the nonsymmetric distance dist(B,A) = sup inf Ila - bllo. bEB aEA
A global attractor A for the semigroup {St, t 6 IR+} is a minimal closed set A c X0 such that dist(St(B),A) -+ 0 as t ---++~o for all bounded sets B C X0. As is known, if a global attractor exists, then it is unique. If it is compact, then it is invariant and connected. We refer to [7, 8] for a detailed treatment of the theory of attractors for abstract semigroups. We prove the existence of a compact global attractor for the semigroup generated by the problem (2.1)-(2.3) under the following additional assumption. 755
(f~2) There exists a vector field a = ( a t , . . . ,a~) : f~ ---+N n of class C t such that (a) diva(x) = 1 for allx E f2, (b) the symmetric part of the matrix A(x) = {Ajk (x)} = {3aj/Ox~} is positive definite, i.e., there exists a positive number pt such that Ajk(x)~k~j )
for all x E f2 and ~ E R n,
(c) (v(x)a(x)) >I 13for all x E 0~2, where ~ is a positive constant. We can choose a smaller 13 so that (d) la(x)l ~< 1/[~ on ~n. The assumption (s
requires a detailed discussion. We only note that this assumption holds if there
exist a point x0 E s and a linear operator B : 1Rn -+ ]Rn such that the symmetric part of B is positive definite and (v(x)B(x - xo)) >1 0 for all x E On. In this case, we can set =
+
trB + smes0f~
'
where q0 is the (unique up to an additive constant) solution to the problem Aq~= m e s o n (vVq)) = mesf~
in n , on ~Y~
and a is sufficiently small.
Theorem 2.2. Let (n 1), (f~2), ( f I ), and (f2) hold. Then the solution semi group {St, t E I~+ } possesses a compact global attractor. Moreovel; this attractor is bounded in the Xl-norm. A continuous function
L:Xo--+ IR is called a Lyapunov.functional
for the semigroup {St, t E R+} if
for any a E X0 the following implication holds:
L(S,(a))=L(a) Vt>~O~S,(a)--a Vt~>0. We refer to the monographs [7, 8] for the complete list of consequences of the existence of a Lyapunov functional.
Theorem 2.3. Let n be a bounded one-dimensional domain, and let ( f l ) and (f2) hold. Then the energy functional (1.3) is a Lyapunov functional for the solution semigroup {St, t E 1~+}.
w X1-Solutions Let (q0,gt) C X1, and let T be a positive number. A function u : [0, T] x f2 ~ IR such that (u, ti) E L~ (I0, T], Xl ) and Eqs. (2.1)-(2.3) hold almost everywhere is called an Xl-solution to the problem (2.1)-(2.3) on [0, T]. It is easy to see that any Xl-solution is an X0-solution. To prove the solvability of the problem (2.1)-(2.3) in X1, we consider discrete approximations with respect to time. We fix a positive integer N and set 'c = T I N . For an arbitrary sequence of functions hk, we introduce the finite difference operators Dh~ -=-h~ = "c-1 (hk -- hk-1) and D2hk =--h~ = z - t (Dhk - Dhk-1). 756
We consider a sequence of functions Uk, k = - 1,0, 1,..., such that I! u~--Auk+uk+f(uk) = 0 inf~ f o r k = 1,2,..., I
u k + (vVu~) = 0
uo=%
on Of~ for k = 1,2,...,
u~=~
(3.1) (3.2)
inf,.
(3.3)
For any k = 1,2, .... (3.1)-(3.3) is a semilinear elliptic problem, where Uk is the unknown function. This problem has a unique solution for all "c < CU. Therefore, the sequence uk is uniquely determined by (3.1)-(3.3). Moreover, uk E H2(f~) for all k >~ 1. Multiplying (3.1) by 2u~ and integrating over fL we find
Dll(u~,u'~)ll2 <~-2 f f(uk)u~ <~2Qllu~ll Ilugll 2 ~< cfll(u~,4)l[ 2. f~
Therefore, II(uk, Uk)[] , 02 ~< ( 1 - "cCf)-k[](q), ~) 11g.
(3.4)
We apply D to Eq. (3.1) and multiply the result by 2u~. Integrating over fL we find
Dlt(uk,, uk)HO H 2 ,, 2 <~2 fOf(uk)u~ <~2cAI4II II.Zll <~Cfll%,u~)llo: Hence
]I(Uk,AUE' - Uk--f(Uk))[I 2 = [[(Ulk,u~)ll0 ,ix,,2 ~ ( 1 - - ~ci)-~ll (v, Aq,- ~ - f(~0))ll 2 9
(3.5)
For any N, we introduce the functions fi', ~, ~, and ~ on [0, T] x f2 as follows: ~(t,. ) = uk on each interval ( ( k - 1)~, k'~]. g(t,. ) = u; on each interval ( ( k - l)g,k'c]. ~(t,. ) is linear on each interval [(k - 1)g, k'c] and ~((k'c,. )) = uk for all k. ~ t , . ) is linear on each interval [ ( k - 1)'r, k'c] and ~'((k'c,- )) = u~ for all k. By (3.4) and (3.5), there exists a constant C that depends on T and such that
II(q,, ~r 2 but is independent of N
IIz~ll 2 + II~ll2 ~ c,
II/,all + II~F2 ~< c,
IIV~l 2 + ]l~l 2 ~ c,
]]V~]]2 4-]]v[[ 2 ~ C,
Ilvql + I1~12 ~< c,
II~l 2 ~ c
almost everywhere on [0, T]. Moreover,
live'- V~]l 2 + I1~'- ~112 ~< c~ 2,
If~'- vii 2 ~< c~ 2.
Therefore, there exists a pair (u, v) E L~,([0, T], Xl) and a sequence of positive integers {N j} such that if--+ u, V~'---+ Vu, ~---+ v -+ u, AT -+ Au, V ---+v
weakly inHl([O,T] • f2), weakly in L2([0, T] xf2) 757
a s N ---+~ , N E {Nj}. We write (3.1)-(3.3) in the form v-&-~+~+f(~)=0, ~'+ (vVff) = 0
u=~
in [0, r] • f~,
on [0, T] x Of2, ona.
Passing to the limit, we find that u is an Xl-solution to the problem (2.1)-(2.3). We show that this solution is unique. We consider pairs (~Pl, ~1), (q~2,~2) E X1 and denote by ul and u2 the corresponding Xl-sOlutions. We multiply the difference of the equations for ul and u2 by 2w, where w = ul - u2. We find that d
l[(w, *) ll02 2 [j ( f ( u l )
f(u2) ),, <<,2cfllwll llr
f2
Hence
II(ut,
(u2,u2)ll 2 ~ efflll(~Pt,Vl)- (qo2,V2)l[~.
(3.6)
Thus, the following asserton holds. Theorem 3.1. Let (f21), (f 1), and (f2) hold. Then for any (~p,~) C X1 there exists a unique X1 -solution
u to the problem (2.1)-(2.3). w Xo-Solutions The existence of X0-solutions follows from Theorem 3.1 and the fact that X1 is everywhere dense in X0. Passing to the limit in (3.6), we conclude that any X0-solution obtained as the limit of X~-solutions satisfies (3.6). To complete the proof of Theorem 2.1, it remains to show that this solution is unique.
Let Ul and u2 be Xo-solutions with the same initial data (q0,~). We take the difference of the identities (2.4) for these two solutions and set s
1](t, x) = / w(o,x) d o for t ~< s,
q (t, x) = 0 for t >
S,
t
where w = ul - u2 and s E [0, T]. We have s
s
f f w2dxdlq-f f Ivvw--VQVTl--~T[q-TtwTI]dxdt=0, 03f2
0 f2
where y~ = w-1(f(ut) -f(u2)). Eliminating the boundary term, we obtain the inequalities s
f~ 758
O~
s
O f~
Taking into account that r I(s, x) = 0 and w(O, x) = O, we obtain the inequality S
(4.1) s
0 s
Since S
S
0
0
S
S
t
S
0
t S
S
s
~/Is-tldtfw2((y,,x)d(Y 0
=
T2
s
s2--2f w2(~'x)a~<"5- f w2(t'x)dt'
o
0
0
(4.1) implies S
fw2(s,')dx<~Cf(l+T---~)
f fw2dxdt.
s
0 s
By the Gronwall lemma, we conclude that w = 0 in [0, T] x fL Hence any X0-solution is uniquely determined by (2.4). We introduce the semigroup of continuous solution operators St : X0 --+ X0 by the formula
s,(e,v) = (u,~), where u is an Xo-solution to the problem (2.1)-(2.3).
w An Absorbing Set In this section, we show that the semigroup St has a bounded absorbing set, i.e., a set B C Xo such that St (A) C B for any bounded set A c X0 and sufficiently large t. We introduce the energy functional E : Xo ~ R+:
E(%~)-- ~1 IlV~oll2 +
~ iivll2 + ~ IIq,[[2 +
ff(w)dx. s
By (f3), we have E(q),v ) ~> 111(%v)112. Proposition 5.1. There exists a continuous functional ff~ :Xo --~ 1R+ and a positive number R such that ]E(q',t~)-/~('P,~r
~< ~1E (%V)
and for any Xo-solution u to the problem (2.1)-(2.3) we have ff.(u(t, . ), it(t,. )) <~ff.(u(O, . ), it(O,. ) )e -t/R + R(1 - e-t/R).
(5.1)
759
In the proof of Proposition 5.1, we use rather complicated estimates, In the sequel, we omit dx in integrals over f2 and OfL Proof. It suffices to prove the inequality (5.1) for the case (q),~) C X1. We multiply (2.1) by ~. We have
d E(u, it) = - f ~2.
(5.2)
We multiply (2.1) by 2(aVu), where a is the vector field defined in (f~2). We have
2f ~(aVu)-2f~u(aVu)+2fu(aVu)+2ff(u)(aVu)=0. f~
~
U2
(5.3)
f2
In view of the assumptions about a and the boundary condition a + (vVu) = 0, we can transform the terms in (5.3) as follows:
f~
f~
f~
_-2~f~(aV.)- f(aw~) f2
~2
>~2
f~
3f~
/i(aVu) + Ilfill; - ~
~f2
li2;
f~
f~
> - 2 f a(aV.) + (2~- 1 ) H v t / l l 2 + / ( v a ) t g b t { 2 3f2
Of~
1 /~i2 + ( 2 ~ -
~> - ~ 3
1)llUull2;
on
f~
~2
2ff(u)(aVu) ~2
3f~
=2f(aVF(u))= f~
0f~
-2fF(,)+2f(va)F(u). f~
Of2
Multiplying (2.1) by u, we find
f . . - J .~. + II.ll~ + f . I ( . / = f~
760
~2
f~
0
(5.4)
We have transformed the terms as follows:
f//u -- ddt Jf u u - Iltil[2;
f~
f2
Of~
On
ldf
2
= INuli N + g ~ j u
.
3~2 We multiply (5.4) by 1 - 8, where the number 8 C (O,,u) will be chosen later, and add the result to (5.3). We introduce the notation
-2-d
"
(5.5)
We have
+~-~
d p (ii)+ZSE(u,r u,
~<-2~u-8)llVu]] 2 13
u 2
0ta f2
(5.6)
Of~
Using (f2) and the Poincar6 inequality, we choose a sufficiently small 8 so that the right-hand side of (5.6) is bounded by a constant C that depends only on f and fL We can also assume that C > 13-1 + ~ 3/2. Thus, we have
~p(u, it) + 2BE(u, Li) ~
(5.7)
On We can choose a larger C so that the inequality 2p(q),~ ) ~< CE((p,~) holds. We set
E, = E + p/C and
R = 3C/(48). By (5.2) and (5.7), we find
~E(u,-d
t~)4-~E(u,~)+l.
Integrating this differential inequality, we obtain (5.1). From Proposition 5.1 it follows that the set 8 = {(~,v) ~ Xo 9e(,p,v) ~
w Asymptotic Compactness We fix a pair (% ~r c X0 and consider the linear problem i~-Av+v=0
inlR+xf2,
~)+ (vVv) = 0
on ~ + x ~U2,
v----~p, 9 = ~ g
on{O}x~. 761
By Theorem 2.1, there exists a unique generalized solution v E L=([0, T],H~) to this problem such that r E L.o([0, T] ,L2). Let Vt : Xo --+ Xo be the corresponding solution operator, Vt(q0,~) = (v(t,.), r (t,.)). The energy functional E0 : Xo ---+R+ corresponding to the problem (6.1)-(6.3) has the form
Eo(~, V)= ~ IIV,~II2 + ~ I1~r + ~ liq~tl2 Repeating the arguments from Sec. 5, we establish the following assertion. Proposition 6.1. There exists a continuous functional Eo : Xo ~ R+ and a positive number R such that
le0(~0,v) - ~0(q,, v)l ~<~1E o(,p, v) and for any Xo-solution v to the problem (2.1)-(2.3) we have ff~O(V(t,. ), 9(t,. )) ~~O(v(O,- ), "0(0,- ))e -t/R.
We introduce the operator Wt 9X0 ~ X0 as follows: Wt = St - ~5. If w is the first component of Wt (r ~), then the second component is ~ and the function w satisfies the inhomogeneous linear problem (6.1)
in ~ + x gl,
- Aw + w = - f ( u )
a+ + (vVw) -----0
on N+ x 0~,
(6.2)
w=0,
on{0}•
(6.3)
~=0
where u is a function such that (u(t,.), a(t,. )) =- St(cp, ~). Proposition 6.2. For any t E IR+, the operator Wt is compact on B. Moreover, there exists a constant C such that
Ilwt(~0,v)ll~ ~
Proof. Throughout the proof, we denote by C a constant depending only on f, EL and the representation W = S - V it follows that
II(~, v)lit.
From
II(w,w)llg ~ c. We differentiate (6.1) with respect to time and multiply the result by # (this requires the a priori regularity, which can be obtained with the help of finite difference arguments (cf. Sec. 3)). We have 1 d
2 _jr_[ fi~2
(6 .4)
0n Repeating the arguments used in the proof of Proposition 5.1 and setting _]~, we find
an 762
1"
where the functional 9 is defined by formula (5.5) and 1 1 K = ~ + 2113.
Choosing a larger K, we find that 29(q),v) ~ KII(w,V)II~
v(~0, v)
e
Xo.
Hence the functional Y. : Xo --+ ]~ defined by the equality E(~,W) = ~ll(v0,~))tl2q- 21p ( * , ~ ) satisfies the estimate _ 3 2 1 II(w,~)ll~ <~z(w,v) ~< ~ll(,+,~)llo. 4 Dividing (6.5) by K and adding (6.4) to the result, we obtain the inequalities
Z::(~,, ) d ~ ~<_ ~. ii(w, |
+ cll(~,,~)llo <
24~CZ(w, v , , 2~) - ( w+ , , ~ ) . 3 g
Integrating these differential inequalities and taking into account that vi,(0,- ) = 0 and fi~(0, 9) = - f ( u ) , we find
II(,~,vo)ll,g~ c for all t ~> 0. Finally, we note that
I[(w,w)ll~ = IIAwll2 + Ilwll2+ IIVwll2 = II~+w+f(u)ll 2 + Ilwll2 + I[Vwll 2 ~< 211~>112+ 2lie+ f(u)ll 2 + Ilwll2 + IIV~'l[2 ~< 211(w,~)ll g + II(w,~)llo2 + (1 +cf)211ull 2. This completes the proof. Propositions 5.1, 6.1, and 6.2 imply the existence of a compact attractor A for St (the corresponding abstract theorems can be found in [7] or [8]). By Proposition 6.2, this attractor is bounded in the Xl-norm. Theorem 2.2 is proved. w A Lyapunov F u n c t i o n a l
In this section, we consider the semigroup S generated by the one-dimensional problem. Without loss of generality, we can assume that f2 = [0, 1]. We write the problem in the form
i2-Uxx+U+f(u)=O
i n l R + • [0, l],
l~(t,0) - ux(t,0) = 0
for all t E N+,
#t(t,l)+u~(t,1)=O
for all t C ~+.
(7.1)
Proposition 7.1. Let a function u satisfy (7.1). If ~(t, O) = a(t, 1) ----Ofor all t >~O, then/~(t,x) = Ofor
all x E [0, 1] and t >~O. 763
Proof. We fix a positive number "c and introduce the function w(t,x) = u(t + % x ) - u(t,x).
This
function satisfies the linear equation
- Wxx + w + f w = 0,
(7.2)
--I
where f (t,x) = (f(u(t + "c,x)) - f(u(t,x)))/w(t,x), and the following boundary conditions at the left end:
w(t, O) = O,
Wx(t, 0) = 0.
(7.3)
We regard t as the spatial variable and x as the time. Equation (7.2) remains a linear hyperbolic equation, whereas the boundary conditions (7.3) become the initial conditions.
By the standard results on the
uniqueness of a solution to a linear wave equation, we have w(t,x) = 0 for all x and t/> x. We return to the traditional interpretation of the variables t and x. We consider the backwards equation, i.e., we replace t with - t . We have a linear wave equation with homogeneous Dirichlet boundary conditions. By a well-known uniqueness result, we can continue the identity w(t,x) = 0 to the whole set I~+ x [0, 1]. In view of the definition of w, we conclude that u does not depend on t. By Proposition 7.1, if the energy functional E is constant on a trajectory of the semigroup for all instants of time, then this trajectory is stationary. Theorem 2.3 is proved. The author thanks Fabric for attracting his attention to the problem and for hours of interesting discussions. The author is also grateful to Ladyzhenskaya, who suggested the idea of the proof of Proposition 7.1. This work was partially supported by the INTAS and the University of Bordeaux.
References 1. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam (1992). 2. G. Chen, "Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain," J. Math. Pure Appl., 58, 249-273 (1979). 3. V. Komomik and E. Zuazua, "A direct method for the boundary stabilization of the wave equation;' J. Math.
Pure Appl., 69, No. 1, 33-54 (1990). 4. J. Lagnese, "Decay of solutions of wave equations in a bounded region with boundary dissipation," J. Differ.
Equations, 50, No. 2, 163-182 (1983). 5. I. Lasiecka, "Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary," J. Differ. Equations, 79, No. 2, 340-381 (1989). 6. O. A. Ladyzhenskaya, "On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations;' Russian Math. Surveys, 42, No. 6, 27-73 (1987). 7. J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI (1988). 8. O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge (1991).
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