DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Supplement Volume 2005
Website: http://AIMsciences.org
pp. 443–452
EXISTENCE OF A STABLE SET FOR SOME NONLINEAR
PARABOLIC EQUATION INVOLVING CRITICAL SOBOLEV
EXPONENT
Michinori Ishiwata
Department of Mathematical Sciences,
School of Science and Engineering, Waseda University,
169-8555, 4-1 Okubo 3-chome, Shinjyuku-ku, Tokyo, Japan.
Abstract. In this paper, we discuss the asymptotic behavior of some solutions for
nonlinear parabolic equation in RN involving critical Sobolev exponent.
For the subcritical problem (with bounded domain), it is well-known that the
solution which intersects the “stable set” must be a global one. But for the critical
problem, it is not known whether the same conclusion holds or not.
In this paper, we shall show that, in the critical case, the same conclusion actually
holds true. The proof requires the concentration compactness type argument.
1. Introduction and Main Results. In this paper, we are concerned with the
asymptotic behavior of some solutions for the following nonlinear parabolic equation:

∂u


= ∆p u + u|u|q−2 in Ω × (0, Tm ),
∂t
(P)
u =
0
on ∂Ω × [0, Tm ),


u =
u0
in Ω × {0},
where N ≥ 2, p ∈ (1, N ), q ≤ p∗ (p∗ := N p/(N − p) denotes the critical exponent of
k∇·kp
,→ Lq ), Ω ⊂ RN , ∆p u := div(|∇u|p−2 ∇u),
the Sobolev embedding D01,p := C0∞
1,p
2
∞
u0 ∈ X := D0 ∩ L ∩ L and Tm denotes the maximal existence time of the
solution of (P) which satisfies (1.3)-(1.5) below. In the main part of this paper, we
assume that Ω = RN and q = p∗ . In this case, the homogeneous Dirichlet boundary
condition appears in (P) should be interpreted as the condition at |x| ' ∞. Lr (Ω)norm is denoted by k · kr,Ω and the subscript Ω is often omitted when no confusion
occurs.
The following functionals J, I and the subset W of X, called “stable set”, play
an important role throughout this paper:
1
1
J(u) := k∇ukpp − kukqq , I(u) := −k∇ukpp + kukqq ,
(1.1)
p
q
¶
¾
½
µ
1 1
−
S q/(q−p) , I(u) < 0 ,
(1.2)
W := u ∈ X; J(u) <
p q
where S(:= inf u∈D1,p \{0} k∇ukpp /kukpq ) denotes the best constant of the Sobolev
0
inequality.
2000 Mathematics Subject Classification. Primary: 35B33, 35B35; Secondary: 35K65.
Key words and phrases. Stable set, critical Sobolev exponent, lack of compactness,
concentration-compactness principle.
443
444
MICHINORI ISHIWATA
It is well-known that (P) has a solution u which satisfies
∆p u, u|u|q−2 ∈ L2 (0, T ; L2 (RN )),
u∈W
1,2
2
N
(0, T ; L (R )) ∩
C([0, T ]; D01,p (RN ))
(1.3)
∞
∞
N
∩ L (0, T ; L (R )) (1.4)
for all T ∈ (0, Tm ) and
if Tm < ∞, then lim ku(t)k∞ = ∞,
t→Tm
(1.5)
see e.g. [19]. Throughout this paper, we only treat solutions of (P) which satisfy
(1.3)-(1.5). In this paper, we shall consider the asymptotic behavior of solutions
which enter the set W at some time.
Since (P) gives one of the typical models which describe various nonlinear phenomena, it is important to analyze all the possible asymptotic behavior of solutions
of (P). From this point of view, there are enormous amounts of works concerning
the asymptotics of solutions of (P), especially for the semilinear (i.e. p = 2) case.
Roughly speaking, known results mentioned above (for p = 2) can be summarized
in the following way (see e.g. [4], [8], [10], [11], [12], [14], [15], [18] [20], [21], [22],
[23], [25] and references therein).
• When the initial data u0 is “small”, the behavior of the solution is almost
governed by ut = ∆u, the linear part of (P). Especially, the solution exists
globally in time (i.e., Tm = ∞) and approaches to 0 in the appropriate sense
(e.g. in the sense of L∞ ) as t → ∞.
• When the initial data u0 is “large”, the behavior of the solution is almost
controlled by ut = u|u|q−2 which is related to the nonlinear part of (P).
Especially the maximal existence time of the solution is finite (i.e., Tm < ∞)
and the solution blows up in the appropriate sense (e.g. in the sense of L∞ ).
• If the initial data is neither “small” nor “large”, then the solution exhibits the
borderline behavior and this behavior depends delicately on the individual
nature of the equation. For example, if Ω is a bounded domain, p = 2 and
q < 2∗ , then the ω-limit set of the orbit is contained in the set which consists
of stationary solutions.
We briefly review some precise known results. At first, we recall the following:
Proposition 1.1. [11]
Let p = 2, q < 2∗ and Ω be a bounded domain in RN (N ≥ 3). Assume that there
exists t such that u(t) ∈ W . Then Tm = ∞ and u(t) → 0 in H01 as t → ∞.
The key fact for obtaining Proposition 1.1 is the following:
Lemma 1.1.
Let p = 2 and q < 2∗ . Then for any M > 0, there exists δ = δ(M ) > 0 such that
for any u0 satisfying ku0 kq ≤ M , ku(t)kq ≤ M + 1 holds for all t ∈ [0, δ].
For Lemma 1.1, see also [3], [26]. By virtue of Lemma 1.1, the asymptotic
behavior of solutions of (P) with subcritical q (i.e. q < 2∗ ) is now well-understood.
But as for (P) with the critical q (i.e. q = 2∗ ), only few results are available so
far, see e.g. [7], [9], [12]. Especially, in the critical case, using Proposition 2.1 and
Proposition 2.2 below, we can easily verify that Lemma 1.1 is no longer true for
Ω = RN . Reflecting this lack of the control of the Lq -norm of the solution u(t) in
terms of the Lq -norm of the initial data u0 , only a partial result is known for the
critical case. For example,
A STABLE SET FOR NONLINEAR PARABOLIC EQUATION
445
Proposition 1.2. [12]
Let p = 2, q = 2∗ and Ω be a bounded domain in RN (N ≥ 3). Assume that there
exists t such that u(t) ∈ W and Tm = ∞. Then u(t) → 0 in H01 as t → ∞.
As is easily seen, there exist some differences between Proposition 1.1 and Proposition 1.2 with respect to the position of the statement Tm = ∞. Namely, in Proposition 1.1, Tm = ∞ appears in the conclusion but in Proposition 1.2 it is in the
assumption. Our main result is concerned with this “difference of the position”.
Our main result is the following:
Main Theorem.
Let p ∈ (1, N ), q = p∗ and Ω = RN . Assume that there exists t such that
u(t) ∈ W . Then Tm = ∞.
Remark 1.1. The assumption Ω = RN is only required for the simplicity of the
argument. The same result should hold for general Ω under the appropriate assumption on the boundary of Ω (e.g. the smoothness of ∂Ω), especially in the case
where p = 2.
2. Preliminaries.
2.1. Scaling properties. Let u be a solution of (P), λ > 0,
α :=
p(q − 2)
,
q−p
β :=
p
.
q−p
(2.1)
Note that if q = p∗ , then α = [p(N + 2) − 2N ]/p and β = (N − p)/p.
For any x0 ∈ RN and t0 ∈ R+ , let us define y, s, v by
y := λ(x − x0 ),
s := λα (t − t0 ),
λβ v(y, s) := u(x, t).
(2.2)
The following two propositions are easily obtained by direct calculation.
Proposition 2.1.
(P) is invariant under (2.2), i.e.,
⇐⇒
∂v
= ∆p v + v|v|q−2 in (y, s) ∈ RN × [0, δ]
∂s
∂u
= ∆p u + u|u|q−2 in (x, t) ∈ RN × [t0 , t0 + δ/λα ]
∂t
(2.3)
(2.4)
holds for δ > 0.
Proposition 2.2.
Assume that a < b and q = p∗ . Then we have
° °
° °
° ∂v °
° ∂u °
° °
°
=°
,
° ∂s ° 2
° ∂t ° 2
2
N
L (a,b;L (R ))
L (t0 +a/λα ,t0 +b/λα ;L2 (RN ))
n
k∇v(s)kp = k∇u(t)kp ,
kv(s)kq = ku(t)kq .
Especially
holds.
³ ³
s ´´
J(v(s)) = J u t0 + α
λ
(2.5)
n
(2.6)
(2.7)
(2.8)
446
MICHINORI ISHIWATA
Remark 2.1. Proposition 2.1 and Proposition 2.2 assert that the invariance of
D01,p and Lq -norm (hence the invariance of the energy functional J) under scaling
(2.2) only holds for the special value of q, namely, q = p∗ , while the invariance of
(P) is always true due to the choice of α and β.
Recall that by virtue of this scale invariance of the energy structure, the stationary solution of (P) exists if and only if q = p∗ , see [16], [17].
Remark 2.2. The rescaling argument employed in [10] needs the Liouville-type
results, that is, the nonexistence of stationary solutions. Hence, taking account
into Remark 2.1, we cannot apply the argument of [10] in a direct manner in order
to obtain L∞ -bounds for solutions of (P) which enter the stable set W when q = 2∗ .
Here we emphasize again that this difficulty has a close relationship with the scale
invariance of the energy structure associated to (P).
2.2. Compactness devices. In order to carry out the scaling argument given in
[10], we have to obtain some characterization of a limit function of a sequence
formed by rescaled solutions. In the semilinear case, this characterization follows
from the Schauder-type argument. In this paper, we employ the following two types
of compactness devices.
Proposition 2.3.
Let (un ) be a family of solutions of (P) defined on the time interval [0, δ]. Assume
that for some 0 < c < c,
kun kL∞ (0,δ;L∞ (RN )) ∈ [c, c].
(2.9)
Then for some u ∈ C(RN × (0, δ)),
un → u locally uniformly in RN × (0, δ)
(2.10)
along some subsequence.
Proof.
Let (un ) be a family of solutions of (P) which satisfies (2.9). Then it is easy to see
that, for all m ∈ N, the restriction of un to Bm × (0, δ) is a bounded local solution
of (P) in Bm × (0, δ) in the sense of [6, pp. 17], where Bm := {x ∈ RN ; |x| < m}.
Take any compact subset K of RN ×(0, δ). Without loss of generality, we assume
that {0} × (0, δ) ∈ K. Take any m such that K ⊂ Km := Dm/2 × Im , where
Dm := {x ∈ Ω; |x| ≤ m} and Im := [1/m, δ − 1/m]. Let Γm be a parabolic
boundary of Bm × (0, δ). Also let
(
|p−2|/p
kun kL∞ (0,δ;L∞ (Bm )) |x − y| + |t − s|1/p if p ∈ (1, 2),
dp (X, Y : un ) :=
|p−2|/p
|x − y| + kun kL∞ (0,δ;L∞ (Bm )) |t − s|1/p if p ∈ [2, ∞)
and
p− dist(Km , Γm ; un ) :=
inf
X∈Km ,Y ∈Γm
dp (X, Y : un )
(2.11)
where X = (x, t) and Y = (y, s).
Then by (2.9),
p − dist(Km , Γm ; un )
≥
|p−2|/p
min(1, kun kL∞ (0,δ;L∞ (Bm )) )
×
≥
inf
X∈Km ,Y ∈Γm
(|x − y| + |t − s|1/p )
min(1, c|p−2|/p )(m/2 + (1/m)1/p )
=: Cm (> 0).
(2.12)
A STABLE SET FOR NONLINEAR PARABOLIC EQUATION
447
From (2.12) and [6, Theorem 1.1 in Chapter III], we have
µ
|un (X) − un (Y )|
≤
γkun kL∞ (0,δ;L∞ (Bm ))
≤
γc
max(1, c|p−2|/p )α
α
Cm
dp (X, Y ; un )
p − dist(Km , Γ; un )
×(|x − y| + |t − s|1/p )α ,
∀X, Y ∈ Km ,
¶α
(2.13)
for some α ∈ (0, 1) and γ > 0, a general constant.
Hence (un ) ⊂ C(K) forms an equicontinuous family and the conclusion follows
from the Ascoli-Arzela theorem and the usual diagonal argument.
Our next compactness device is so-called concentration compactness type one.
Proposition 2.4.
Let q = p∗ and let (un ) ⊂ D01,p be a sequence which satisfies
(un ) is bounded in L∞ and D01,p ,
fn := ∆p un + un |un |
q−2
(2.14)
2
→ 0 in L .
(2.15)
Then there exists some u ∈ D01,p such that
un * u weakly in D01,p ,
un → u strongly in Lqloc
(2.16)
(2.17)
along some subsequence. Moreover, the following alternative
u=0
or
k∇ukpp , kukqq ≥ S q/(q−p)
(2.18)
holds.
Proof.
Let η ∈ C0∞ ([0, ∞); [0, ∞)) be a function satisfying
η(t) = 0 if t ≥ 2,
η(t) = 1 if t ≤ 1,
0 ≤ η ≤ 1.
(2.19)
For a ∈ R+ , define ϕa ∈ C0∞ (RN ) by
µ
ϕa (x) := η
|x|
a
¶
.
(2.20)
Let (un ) be a sequence which appears in the assumption. Then
k∇un kp < C1 ,
(2.21)
kun k∞ < C2
(2.22)
hold for some C1 , C2 > 0.
By (2.21) and [24, pp.44] (see also [1], [2], [5], [13], [16], [17]), we can find some
u ∈ D01,p , a set S ⊂ RN which is at most countable, a family of nonnegative numbers
448
MICHINORI ISHIWATA
(µxj ) and (νxj ), nonnegative Radon measures µ and ν such that
un * u weakly in D01,p , Lq
X
|∇un |p * µ ≥ |∇u|p +
µxj δxj weakly in M (RN )
|un |q * ν = |u|q +
ε→0 n→∞
νxj = lim lim
ε→0 n→∞
(2.24)
xj ∈S
νxj δxj weakly in M (RN )
(2.25)
xj ∈S
Z
µxj = lim lim
X
(2.23)
|∇un |p ϕε,xj ,
(2.26)
|un |q ϕε,xj
(2.27)
Z
along appropriate subsequence, where M (RN ) denotes the set consists of Radon
measures on RN , δxj is a delta measure supported at xj , ϕε,xj (·) := ϕε (· − xj ) and
ϕε (·) is a function defined by (2.20) with a = ε.
By virtue of (2.22) and (2.27), we see that
Z
νxj = lim lim
|un |q ϕε,xj ≤ lim C2q |B(xj ; 2ε)| = 0
(2.28)
ε→0 n→∞
ε→0
where B(xj ; 2ε) denotes the open ball with radius 2ε centered at xj .
Then (2.28) together with (2.23) and (2.25) yields
un → u strongly in Lqloc .
(2.29)
Take any R R> 0. Let ϕR be a cut-off function defined by (2.20) with a = R. We
shall calculate (fn × un ϕR ).
By the definition of fn , we easily obtain
Z
Z
Z
fn un ϕR = ∆p un (un ϕR ) + un |un |q−2 un ϕR .
(2.30)
Therefore, integrating by parts, we deduce that
Z
Z
Z
p
q
|∇un | ϕR =
|un | ϕR − |∇un |p−2 ∇un ∇ϕR un
Z
− fn un ϕR .
(2.31)
Now observe that (p − 1)/p + 1/q + 1/N = 1 since q = p∗ . Then, by the Hölder
inequality,
Z
|2nd term in the RHS of (2.31)| ≤
|∇un |p−1 |un ||∇ϕR |
≤
k∇un kp−1
kun kq,AR k∇ϕR kN,B2R ,(2.32)
p
where AR = {x; R < |x| < 2R} and B2R = {x; |x| < 2R}.
It is easy to see that k∇ϕR kN,B2R is independent of R. Therefore by (2.21), we
can find C > 0 which does not depend on R such that
k∇un kp−1
k∇ϕR kN
p
≤
C.
(2.33)
Then (2.32), (2.33) and (2.29) yield
|2nd term in the RHS of (2.31)|
≤
→
Ckun kq,AR
Ckukq,AR as n → ∞
→
0 as R → ∞.
(2.34)
A STABLE SET FOR NONLINEAR PARABOLIC EQUATION
449
Also note that (2.22) and (2.15) imply
|3rd term in the RHS of (2.31)|
≤ kfn k2 kϕR k2 kun k∞
→ 0 as n → ∞
≡
0 as R → ∞.
Then by (2.31), (2.34) and (2.35), we have
Z
Z
p
lim lim
|∇un | ϕR = lim lim
|un |q ϕR .
R→∞ n→∞
R→∞ n→∞
(2.35)
(2.36)
Moreover by (2.24), (2.25) and (2.28), we obtain that
Z
|∇un |p ϕR → µ(ϕR ) as n → ∞
Z
Z
X
p
≥
|∇u| ϕR +
µxj ϕR (xj ) ≥ |∇u|p ϕR
xj ∈S
→ k∇ukpp as R → ∞,
Z
|un |q ϕR
(2.37)
→ ν(ϕR ) as n → ∞
Z
X
=
|u|q ϕR +
0 × ϕR (xj )
xj ∈S
→
kukqq
as R → ∞.
Therefore by (2.36)-(2.38), we deduce that
Z
Z
k∇ukpp ≤ lim lim
|∇un |p ϕR = lim lim
|un |q ϕR = kukqq .
R→∞ n→∞
R→∞ n→∞
(2.38)
(2.39)
It is easy to see that the Sobolev inequality and (2.39) yield Skukpq ≤ kukqq .
Hence if u 6= 0, we obtain kukqq ≥ S q/(q−p) and again by the Sobolev inequality, we
also have k∇ukpp ≥ S q/(q−p) .
3. Proof of Main Theorem. Now let us give the proof of Main Theorem.
It is well-known that under the assumption of Main Theorem, we have
u(t) ∈ W ∪ {0},
∀t ∈ [t, Tm ),
(3.1)
see e.g. [20], [25].
By virtue of (1.5), in order to prove Tm = ∞, it is enough to verify that
lim sup ku(t)k∞ < ∞.
(3.2)
t→Tm
We shall show that (3.2) actually holds.
Hereafter we assume that (3.2) does not hold, i.e.,
lim sup ku(t)k∞ = ∞,
(3.3)
t→Tm
and we shall deduce some contradiction.
From (3.3), we can find some (tn ) which satisfies tn → Tm , ku(tn )k∞ → ∞ and
ku(tn )k∞ ≥
1
sup ku(t)k∞ .
2 t∈[0,tn ]
(3.4)
450
MICHINORI ISHIWATA
Let (xn ) be a sequence which satisfies
1
ku(tn )k∞ ≤ |u(xn , tn )|.
2
Let α and β be numbers given by (2.1). Define (λn ), y, s, (un ) by
ku(tn )k∞ = λβn
(3.5)
(3.6)
and
y = λn (x − xn ),
s = λα
n (t − tn ),
λβn un (y, s) = u(x, t).
(3.7)
We claim the following:
Claim.
There exists δ > 0, independent of n, such that
°
°
° ∂un °
°
°
→ 0 as n → ∞
° ∂s ° 2
2
N
(3.8)
L (−1,δ;L (R ))
and
kun kL∞ (−1,δ;L∞ (RN )) ∈ [1, 2],
1
|un (0, 0)| ≥ ,
2
p
k∇un kp < S q/(q−p) , ∀s ∈ [−1, δ]
(3.9)
(3.10)
(3.11)
for all n.
Proof of Claim.
Note that by (3.6) and (3.7), we have
kun (0)k∞ = 1.
(3.12)
From this relation and some computation using Lp -Lq estimate of the heat kernel,
we can deduce the existence of δ > 0 which satisfies
kun (s)kL∞ (0,δ;L∞ (RN )) < 2.
(3.13)
By (3.4), (3.6) and (3.7), we also have
kun kL∞ (−1,0;L∞ (RN )) ≤ 2.
(3.14)
Then (3.12)-(3.14) yield (3.9).
It is easy to see that (3.10) follows from (3.5)-(3.7).
From (1.2), we can easily obtain that k∇ukpp < S q/(q−p) for u ∈ W . This fact
together with (3.1) and (2.6) lead (3.11).
Multiplying (P) by ∂u/∂t and integrating over RN , we obtain
°
°
° ∂u(t) °2
d
°
°
(3.15)
° ∂t ° = − dt J(u(t)).
2
By (1.2) and (3.1), we can easily verify that J(u(t)) ≥ 0 for all t ∈ [t, Tm ). From
this fact and (3.15),
lim J(u(t)) = c
t→Tm
holds for some c ∈ R.
(3.16)
A STABLE SET FOR NONLINEAR PARABOLIC EQUATION
451
Then (2.5), (3.15) and (3.16) yield
°
°
° °2
° ∂u °
° ∂un °2
°
°
°
= °
° ∂s ° 2
° ∂t ° 2
L (−1,δ;L2 (RN ))
L (tn −1/λα ,tn +δ/λα ;L2 (RN ))
n
=
→
n
α
J(u(tn − 1/λα
n )) − J(u(tn + δ/λn ))
c − c = 0,
(3.17)
i.e., (3.8).
By (3.9) and Proposition 2.3,
un → u in Cloc ((−1, δ) × RN )
(3.18)
holds for some u. Moreover, using (3.8), we can easily show that u is independent
of s. Combining (3.18) with (3.10), we see that
1
.
2
From (3.8) and (3.9), we can find some σ ∈ [0, δ/2] such that
u(0) ≥
∆p un (σ) + un (σ)|un (σ)|q−2 =
∂un (σ)
→ 0 in L2 ,
∂s
kun (σ)k∞ < 2.
(3.19)
(3.20)
(3.21)
Then from (3.11), (3.21), (3.20) and Proposition 2.4, we obtain v ∈ D01,p which
satisfies
un (σ) → v a.e.,
(3.22)
un (σ) * v in D01,p ,
v = 0 or k∇vkpp ≥ S q/(q−p) .
(3.23)
(3.24)
Note that u = v follows from (3.18) and (3.22). This fact together with (3.19)
implies that
v = u 6= 0.
(3.25)
Hence by (3.24), (3.23) and (3.11), we have
S q/(q−p) ≤ k∇vkpp ≤ lim k∇un (σ)kpp < S q/(q−p) ,
n→∞
(3.26)
a contradiction.
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Received September, 2004; revised April, 2005.
E-mail address: [email protected]