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Upper semicontinuity of attractors for the reaction diffusion equation
38
Communications
in Nonlinear
Science
Upper Semicontinuity Diffusion Equation 1
& Numerical
Simulation
of Attractors
Boling GUO CFr. Bixiang WANG (Institute of Applied Physics and Computational 100088, China)
Vol.1,
No.2
(Apr.
1996)
for the Reaction
Mathematics,
P.O. Box 8009, Beijing
This paper deals with the reaction diffusion equation in domain R = R or R = (-L, L) with L < co. Let AL and A be the global attractor of this equation corresponding to R = (-L, L) and R = R, respectively. We show that the global attractor A is upper semicontinuity at 0 with respect to the sets {AL} in some sense. Key Words: global attractor , upper semicontinuity, reaction diffusion equation.
Abstract:
Introduction The study of the long time behaviour of the solutions for nonlinear evolution partial differential equations is a interesting question. It is well known that the long time behaviour of dynamical systems can be described by the existence of the global attractors. In the present paper, we show that the global attractor of the reaction diffusion equation is upper semicontinuous in some sense.
1. Main
Results
In the present paper, we consider the following reaction diffusion equation : ut - YAU + f(u) + Xou + g(z) = 0, u(x,O) = Q(X), and for R = (-n,n)
V(x, t) E R x R+,
Vx E R,
(1.1) (14
with n E N;
&IL, - UAU, + f(un) + Xozl, + g(x) = 0, %(X,0) l&(-n,
= ‘110,(x),
V(z, t) E R x R+,
v’z E 0,
t) = u,(n, t) = 0,
(1.3) (1.4) (1.5)
where v and X0 are positive numbers, g(x) E L’(R), f : R + R is a smooth function which satisfies f(u)u 2 0, Vu E R, (1.6) f(0) = 0, f’(0) = 0, f’(u) > -C, Vu E R, (1.7) If’b-4 5 CC1 + b-4”) > P > 0, v’u E R U.8) In the following, we denote by H = L’(R) with the usual inner product (., .) and norm // . /I . For any Banach space X, denote its norm by 11. 11~. We introduce the semigroup S(t) : Ill(R) + H’(R), S(t)uo = u(t), where u(t) is the solution of problem (l.l)-(1.2). S,(t) : Hk(-n,n) -+ Hi(--n,n),S,(t)uo = un(t), where u,(t) is the solution of problem (1.3)-(1.5). It is well known that V’n, S,(t) has a compact global attractor A, in Hi(--n, n) (see [l] and [2]). Since the dynamical system S(t) is not compact in H’ (R), we can not establish the existence of the global attractor of S(t) in Ill(R). In [3], th e authors construct the global weak attractor A for S(t) in H’(R). Our aim here is to show that the weak attractor A is upper semicontinuous at 0 with respect to the sets {,4n}nEN in H1. More precisely, we will show that ‘The
paper
was received
on Mar.
5, 1996
Guo,
Theorem
Upper
Assume that (1.6)-(1.8)
0.1
ilW
2. The
etc.:
proof
PHI
of Theorem
Semicontinuity
39
of . .
hold. Then we have
(dn, d) = 0,
VR c R bounded,
1.1
To establish Theorem 1.1, we first recall that 0.1 Assume that (1.6)-(1.8) hold, g(x) E H’(R), problem (l.l)-(1.2), the following holds:
then for any solution u(t)
Lemma
llu(t)ilH3(R)
5
cl,
‘dt
2
of
tl,
where Cr is a positive number depending only on u, Xo, and 11g11Hz;tl depends on V, Xc, 11gj1H2 and p when ]]us]l 5 p. The proof of this lemma is standard (see [3] ), and therefore is omitted here. For problem (1.3)-(1.5), we can deduce similarly that there exists a constant Cz such that any solution un(t) of (1.3)-(1.5) satisfies
Proof
ll%@NlH~(-n,n) L c2> where C2 depends only on V, xc, ]]g]]H2(n);
vt
L
(2.1)
t2,
t2 depends only on V, Xc, ]]g]/H2 and p when
llwhllL P.Let A, = n
u S(t)&
820t>s -
C
where the closure is taken with respect to the topology of Ht(-n, BI = {u E Hi(-n,n)
(2.2)
Bl,
n), and
: II~IIH;(-~,~J I G},
where C2 is as in (2.1). Then it follows from the results of [I] that A, is the compact global Let attractor of S,(t) in H,‘(-n,n). A = n u S(t)B C B, s>o t>s where the closure is taken with respect to the weak topology of H’(R),
(2.3) and
B = {u E H’(R) : II~IH~R)5 G>, where Cr is as in Lemma 2.1. In [3], the authors show that A is the global weak attractor of S(t) in H1(R). By definition (2.3), it is easy to see that (see [2] ): Proposition
wn. E H1(R) Lemma such
0.1 A point w E A if and only such that S(t,)w, -+ w in H’(R)
if
there exist t, -+ m and a bounded sequence weakly.
0.2 Assume that uon E A,, then there exists a subsequence usnk that uOnb converges to u in H1(R) for all bounded Cl c R.
of
uen and IJ E A
40
Communications
Proof
Since ~0~ E A,, by (2.2) we see that
in Nonlinear
Science
& Numerical
11~0nllH+n,n)
Simulation
5
Vol.1,
(32.
No.2
(Apr.
1996)
(2.4)
where C2 is the constant in (2.1) and independent of n. Denote by un(t) = S(t)uon, the solution of problem (1.3)-( 1.5). Due to ~0~ E A,, we know that un(t) is defined on R. Then by (2.1), we find that llun(t)ll~~(--n,n) I C, V’t E R. (2.5)
Ilw&)Il~-(-n,n) I C, vt E R.
(2.6)
By (1.3) and (2.5) it comes that,
Let Jk and Rk, Ic, m E N, be sequences of compact intervals of R such that Jk C Jk+l, 0, C R m+l and UJk = UO, = R.Vm fixed, when n large enough such that R, c (-n,n) the k
solution un(t) is defimnedin 0,. (2.7) we see that
And we can consider the sequence {u,(t)In,}.
By (2.5) and
(2.8) This implies that UtER UlzEN un(t) is a precompact set in Hi (n,) and the family of mapBy Ascoli’s theorem, pings u,(t) E C”(R,H~(fL)) ’1s eq uicontinuous from R into Hi(&). there exists a subsequence u,, of U, such that u,, converges to u in Co ( JO, Hi (0,)); and using Ascoli’s theorem again, we show, by induction that there is a subsequence u,,+, of u,, such that u,,+, converges to w in Co (Jk+l , Hi (0,)). Finally taking a diagonal subsequence in the usual way, there exists a subsequence u,, of u, such that u,, -+ w in C”( J, HA (0,)) for any compact interval J c R, and v E C”(R, Hi (n,)). Due to (2.8) we have
Il4t)llH1(~,) 5 C, b”tE R.
(2.9)
where C is a constant independent of m. Let m change from 0 to o, and take a diagonal subsequence as above, then we fmd that there exists a subsequence 2~1,of uL1,such that ul, --+ ‘u in C”( J, H,‘(o))
for any compact interval J and R.
(2.10)
In addition, sup Ilw(t)llH;p)
< C,
VR c R bounded,
(2.11)
tER
where C is a constant independent v(t) E H1(R)
of R. And hence we see that and Il~(t)jlHl(RJ 5 C, Vt E R.
(2.12)
In the following, we show that w(t) is a solution of problem (l.l)-(1.2). For the sake of simplicity, we denote by un(t) the sequence of ul, (t) in (2.10). Let J and 0 be fixed compact intervals, then by (2.5) and (2.7), when n large enough such that Szc (-n, n) we have Au, -+ Au 2
in Lw(R, L2(Q)),
+ f$ in Lw(R,L2(R)).
(2.13) (2.14)
Guo,
etc.:
Upper
Semicontinuity
of
41
.
Due to un(t) satisfies dtu, - YAU, + f(~n) + XOZL~+ g(z) = 0, V(z, t) E R x R. And therefore VW E C,-(n),
and V’1c,E C,-(J),
it follows that
J
(Au,,$(t)wP J U(G),G(t)w)dt
$(t)w)dt =0. +x0 +J(9, sp:$(t)w)dt at
(2.15)
J
Taking the limit of (2.15), by (2.13)-(2.15) we find that the following holds in the sense of distribution: bv (2.16) - - UAV + f(u) + Xov + g(z) = 0, V(z, t) E R x J. Since J and R are arbitrary, we see that (2.16) is valid for all (2, t) E R that v(t) is a solution of (l.l)-(1.2) with initial value ~(2) = v(z,O). Taking t, + DC),by (2.12) we know that
x
R, which shows
(2.17) And then by Proposition
2.1 it follows that
vo = S(t,)S(-t,)vo E A. This together with (2.10) concludes Lemma 2.2. We are now in a position to complete the proof of Theorem 1.1. Proof of Theorem 1.1. Assume that Theorem 1.1 is not true, then there exist a bounded interval R c R1 and 6 > 0, and a sequence u,~ E A,, such that dw (~1 (u,, , A) 2 6 > 0.
(2.18)
However, by Lemma 2.2 we know that there exists a subsequence u,,., of u,, such that
which contradicts (2.18). And then Theorem 1.1 is proved. Remark 2.1. We here only deal with the one-dimensional reaction diffusion equation with the Dirichlet boundary condition. In fact, for multidimensional case and other boundary conditions, Theorem 1.1 is also valid. The proof is similar and hence we do not repeat it again.
References [l]
Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics; Springer-Verlag, New York, 1988. [2] Babin, A.V. and Vishik, MI., Attractors of Evolutionary Equations, Nauka, MOSCOW, 1989. [3] Babin, -4.V. and Vishik, M.I., Attractors of partial differential evolution equations in an unbounded domain, Proceedings of the Royal Society of Edinburgh, 116A( 1990); 221.-243.
Communications
in Nonlinear
Science
Upper Semicontinuity Diffusion Equation 1
& Numerical
Simulation
of Attractors
Boling GUO CFr. Bixiang WANG (Institute of Applied Physics and Computational 100088, China)
Vol.1,
No.2
(Apr.
1996)
for the Reaction
Mathematics,
P.O. Box 8009, Beijing
This paper deals with the reaction diffusion equation in domain R = R or R = (-L, L) with L < co. Let AL and A be the global attractor of this equation corresponding to R = (-L, L) and R = R, respectively. We show that the global attractor A is upper semicontinuity at 0 with respect to the sets {AL} in some sense. Key Words: global attractor , upper semicontinuity, reaction diffusion equation.
Abstract:
Introduction The study of the long time behaviour of the solutions for nonlinear evolution partial differential equations is a interesting question. It is well known that the long time behaviour of dynamical systems can be described by the existence of the global attractors. In the present paper, we show that the global attractor of the reaction diffusion equation is upper semicontinuous in some sense.
1. Main
Results
In the present paper, we consider the following reaction diffusion equation : ut - YAU + f(u) + Xou + g(z) = 0, u(x,O) = Q(X), and for R = (-n,n)
V(x, t) E R x R+,
Vx E R,
(1.1) (14
with n E N;
&IL, - UAU, + f(un) + Xozl, + g(x) = 0, %(X,0) l&(-n,
= ‘110,(x),
V(z, t) E R x R+,
v’z E 0,
t) = u,(n, t) = 0,
(1.3) (1.4) (1.5)
where v and X0 are positive numbers, g(x) E L’(R), f : R + R is a smooth function which satisfies f(u)u 2 0, Vu E R, (1.6) f(0) = 0, f’(0) = 0, f’(u) > -C, Vu E R, (1.7) If’b-4 5 CC1 + b-4”) > P > 0, v’u E R U.8) In the following, we denote by H = L’(R) with the usual inner product (., .) and norm // . /I . For any Banach space X, denote its norm by 11. 11~. We introduce the semigroup S(t) : Ill(R) + H’(R), S(t)uo = u(t), where u(t) is the solution of problem (l.l)-(1.2). S,(t) : Hk(-n,n) -+ Hi(--n,n),S,(t)uo = un(t), where u,(t) is the solution of problem (1.3)-(1.5). It is well known that V’n, S,(t) has a compact global attractor A, in Hi(--n, n) (see [l] and [2]). Since the dynamical system S(t) is not compact in H’ (R), we can not establish the existence of the global attractor of S(t) in Ill(R). In [3], th e authors construct the global weak attractor A for S(t) in H’(R). Our aim here is to show that the weak attractor A is upper semicontinuous at 0 with respect to the sets {,4n}nEN in H1. More precisely, we will show that ‘The
paper
was received
on Mar.
5, 1996
Guo,
Theorem
Upper
Assume that (1.6)-(1.8)
0.1
ilW
2. The
etc.:
proof
PHI
of Theorem
Semicontinuity
39
of . .
hold. Then we have
(dn, d) = 0,
VR c R bounded,
1.1
To establish Theorem 1.1, we first recall that 0.1 Assume that (1.6)-(1.8) hold, g(x) E H’(R), problem (l.l)-(1.2), the following holds:
then for any solution u(t)
Lemma
llu(t)ilH3(R)
5
cl,
‘dt
2
of
tl,
where Cr is a positive number depending only on u, Xo, and 11g11Hz;tl depends on V, Xc, 11gj1H2 and p when ]]us]l 5 p. The proof of this lemma is standard (see [3] ), and therefore is omitted here. For problem (1.3)-(1.5), we can deduce similarly that there exists a constant Cz such that any solution un(t) of (1.3)-(1.5) satisfies
Proof
ll%@NlH~(-n,n) L c2> where C2 depends only on V, xc, ]]g]]H2(n);
vt
L
(2.1)
t2,
t2 depends only on V, Xc, ]]g]/H2 and p when
llwhllL P.Let A, = n
u S(t)&
820t>s -
C
where the closure is taken with respect to the topology of Ht(-n, BI = {u E Hi(-n,n)
(2.2)
Bl,
n), and
: II~IIH;(-~,~J I G},
where C2 is as in (2.1). Then it follows from the results of [I] that A, is the compact global Let attractor of S,(t) in H,‘(-n,n). A = n u S(t)B C B, s>o t>s where the closure is taken with respect to the weak topology of H’(R),
(2.3) and
B = {u E H’(R) : II~IH~R)5 G>, where Cr is as in Lemma 2.1. In [3], the authors show that A is the global weak attractor of S(t) in H1(R). By definition (2.3), it is easy to see that (see [2] ): Proposition
wn. E H1(R) Lemma such
0.1 A point w E A if and only such that S(t,)w, -+ w in H’(R)
if
there exist t, -+ m and a bounded sequence weakly.
0.2 Assume that uon E A,, then there exists a subsequence usnk that uOnb converges to u in H1(R) for all bounded Cl c R.
of
uen and IJ E A
40
Communications
Proof
Since ~0~ E A,, by (2.2) we see that
in Nonlinear
Science
& Numerical
11~0nllH+n,n)
Simulation
5
Vol.1,
(32.
No.2
(Apr.
1996)
(2.4)
where C2 is the constant in (2.1) and independent of n. Denote by un(t) = S(t)uon, the solution of problem (1.3)-( 1.5). Due to ~0~ E A,, we know that un(t) is defined on R. Then by (2.1), we find that llun(t)ll~~(--n,n) I C, V’t E R. (2.5)
Ilw&)Il~-(-n,n) I C, vt E R.
(2.6)
By (1.3) and (2.5) it comes that,
Let Jk and Rk, Ic, m E N, be sequences of compact intervals of R such that Jk C Jk+l, 0, C R m+l and UJk = UO, = R.Vm fixed, when n large enough such that R, c (-n,n) the k
solution un(t) is defimnedin 0,. (2.7) we see that
And we can consider the sequence {u,(t)In,}.
By (2.5) and
(2.8) This implies that UtER UlzEN un(t) is a precompact set in Hi (n,) and the family of mapBy Ascoli’s theorem, pings u,(t) E C”(R,H~(fL)) ’1s eq uicontinuous from R into Hi(&). there exists a subsequence u,, of U, such that u,, converges to u in Co ( JO, Hi (0,)); and using Ascoli’s theorem again, we show, by induction that there is a subsequence u,,+, of u,, such that u,,+, converges to w in Co (Jk+l , Hi (0,)). Finally taking a diagonal subsequence in the usual way, there exists a subsequence u,, of u, such that u,, -+ w in C”( J, HA (0,)) for any compact interval J c R, and v E C”(R, Hi (n,)). Due to (2.8) we have
Il4t)llH1(~,) 5 C, b”tE R.
(2.9)
where C is a constant independent of m. Let m change from 0 to o, and take a diagonal subsequence as above, then we fmd that there exists a subsequence 2~1,of uL1,such that ul, --+ ‘u in C”( J, H,‘(o))
for any compact interval J and R.
(2.10)
In addition, sup Ilw(t)llH;p)
< C,
VR c R bounded,
(2.11)
tER
where C is a constant independent v(t) E H1(R)
of R. And hence we see that and Il~(t)jlHl(RJ 5 C, Vt E R.
(2.12)
In the following, we show that w(t) is a solution of problem (l.l)-(1.2). For the sake of simplicity, we denote by un(t) the sequence of ul, (t) in (2.10). Let J and 0 be fixed compact intervals, then by (2.5) and (2.7), when n large enough such that Szc (-n, n) we have Au, -+ Au 2
in Lw(R, L2(Q)),
+ f$ in Lw(R,L2(R)).
(2.13) (2.14)
Guo,
etc.:
Upper
Semicontinuity
of
41
.
Due to un(t) satisfies dtu, - YAU, + f(~n) + XOZL~+ g(z) = 0, V(z, t) E R x R. And therefore VW E C,-(n),
and V’1c,E C,-(J),
it follows that
J
(Au,,$(t)wP J U(G),G(t)w)dt
$(t)w)dt =0. +x0 +J(9, sp:$(t)w)dt at
(2.15)
J
Taking the limit of (2.15), by (2.13)-(2.15) we find that the following holds in the sense of distribution: bv (2.16) - - UAV + f(u) + Xov + g(z) = 0, V(z, t) E R x J. Since J and R are arbitrary, we see that (2.16) is valid for all (2, t) E R that v(t) is a solution of (l.l)-(1.2) with initial value ~(2) = v(z,O). Taking t, + DC),by (2.12) we know that
x
R, which shows
(2.17) And then by Proposition
2.1 it follows that
vo = S(t,)S(-t,)vo E A. This together with (2.10) concludes Lemma 2.2. We are now in a position to complete the proof of Theorem 1.1. Proof of Theorem 1.1. Assume that Theorem 1.1 is not true, then there exist a bounded interval R c R1 and 6 > 0, and a sequence u,~ E A,, such that dw (~1 (u,, , A) 2 6 > 0.
(2.18)
However, by Lemma 2.2 we know that there exists a subsequence u,,., of u,, such that
which contradicts (2.18). And then Theorem 1.1 is proved. Remark 2.1. We here only deal with the one-dimensional reaction diffusion equation with the Dirichlet boundary condition. In fact, for multidimensional case and other boundary conditions, Theorem 1.1 is also valid. The proof is similar and hence we do not repeat it again.
References [l]
Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics; Springer-Verlag, New York, 1988. [2] Babin, A.V. and Vishik, MI., Attractors of Evolutionary Equations, Nauka, MOSCOW, 1989. [3] Babin, -4.V. and Vishik, M.I., Attractors of partial differential evolution equations in an unbounded domain, Proceedings of the Royal Society of Edinburgh, 116A( 1990); 221.-243.