Manuscript submitted to
AIMS’ Journals
Volume X, Number 0X, XX 200X
Website: http://AIMsciences.org
pp. X–XX
CONVERGENCE TO STATIONARY SOLUTIONS
FOR A PARABOLIC-HYPERBOLIC PHASE-FIELD SYSTEM
Maurizio Grasselli
Dipartimento di Matematica “F.Brioschi”
Politecnico di Milano
Via Bonardi, 9
I-20133 Milano, Italy
Hana Petzeltová
Mathematical Institute AS CR
Žitná, 25
CZ-115 67 Praha, Czech Republic
Giulio Schimperna
Dipartimento di Matematica “F.Casorati”
Università di Pavia
Via Ferrata, 1
I-27100 Pavia, Italy
(Communicated by . . . )
Abstract. A parabolic-hyperbolic nonconserved phase-field model is here analyzed. This is an evolution system consisting of a parabolic equation for the
relative temperature ϑ which is nonlinearly coupled with a semilinear damped
wave equation governing the order parameter χ. The latter equation is characterized by a nonlinearity φ(χ) with cubic growth. Assuming homogeneous
Dirichlet and Neumann boundary conditions for ϑ and χ, we prove that any
weak solution has an ω-limit set consisting of one point only. This is achieved
by means of adapting a method based on the Lojasiewicz-Simon inequality.
We also obtain an estimate of the decay rate to equilibrium.
1. Introduction. Let Ω ⊂ R3 be a bounded domain with smooth boundary ∂Ω.
Suppose that a two-phase-material, which occupies Ω for any time t ≥ 0, is subject
to temperature variations only. Denote by ϑ its relative temperature with respect
2000 Mathematics Subject Classification. 35B40, 35Q99, 80A22.
Key words and phrases. Phase-field models, convergence to stationary solutions, LojasiewiczSimon inequality.
The first and third authors were supported by the Italian MIUR PRIN Research Project Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera. The second author was
supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No.
AV0Z10190503 and by Grant A1019302 of GA AV CR. The third author was also supported by
the HYKE Research Training Network.
1
2
M. GRASSELLI, H. PETZELTOVÁ, G. SCHIMPERNA
to some given critical temperature at which the two phases coexist and indicate by
χ the order parameter. Consider then the evolution system for the pair (ϑ, χ)
(
(ϑ + λ(χ))t − ∆ϑ = f,
(1.1)
εχtt + χt − ∆χ + χ + φ(χ) − λ0 (χ)ϑ = 0,
in Ω×(0, ∞). Here λ and φ are smooth functions, the former with quadratic growth
and the latter with cubic growth, f is a time dependent heat source, and ε > 0 is
a (small) inertial parameter.
System (1.1) endowed with the boundary conditions
ϑ = 0,
χn = 0,
on ∂Ω × (0, ∞),
(1.2)
where the subscript n stands for the outward normal derivative, reduces, if ε = 0,
to the well-known nonconserved Caginalp system (see, e.g., [6]). However, there
are rapid phase transformation processes in nonequilibrium dynamics for which
the inertial term εχtt must be taken into account (see, e.g., [9] and references
therein). In [10], problem (1.1)-(1.2) was analyzed within the theory of dissipative
dynamical systems, obtaining the existence of a global attractor. Supposing f ≡ 0,
a further and deeper analysis was carried out in [11], proving in particular some
smoothness and stability properties of the global attractor as well as the existence of
an exponential attractor. Here we are interested in studying the behavior of single
smooth trajectories. It is not difficult to realize that, if f ≡ 0, then a stationary
solution of problem (1.1)-(1.2) is a pair (0, χ∞ ) where χ∞ solves
(
−∆χ∞ + χ∞ + φ(χ∞ ) = 0, in Ω,
(1.3)
χn = 0,
on ∂Ω.
It is well known that the structure of the set of solutions to (1.3) for a multidimensional domain may be quite complicated. In particular, this set may contain
a continuum of nonradial solutions if Ω is a ball or an annulus (see, for instance, [16]
and references therein). If this is the case, it is highly nontrivial to decide whether
or not a given trajectory converges to a single stationary state. Moreover, this might
not happen even for finite-dimensional dynamical systems (cf. [5]). It is also worth
recalling the following negative result for semilinear parabolic equations [25] (cf. also
[24]). Namely, there exists a function f (x, u) of class C ∞ such the parabolic equation
ut −∆u = f (x, u) has a bounded solution whose ω-limit set is a continuum. In 1983,
L. Simon [28] developed a method to study the longtime behavior of gradient-like
dynamical systems based on a deep result from the theory of analytic functions of
several variables due to S. Lojasiewicz [22, 23]. Roughly speaking, by this method
one can show that any sufficiently smooth trajectory converges to a stationary state,
provided that the nonlinearity is analytic. The cornerstone is a generalized version
of the Lojasiewicz theorem applicable to analytic functionals on Banach spaces.
Later on, several contributions simplified considerably Simon’s original approach
(cf., e.g., [7, 14, 17, 20, 21, 26, 31]), making it accessible for application to a broad
class of semilinear problems with variational structure. However, in some cases,
Simon’s approach can also be used to handle problems with only a partial variational
structure. Typical examples are just phase field systems like (1.1) with ε = 0 which
have been examined in [1, 2, 12, 32] (see also [3, 4, 13] for the conserved case). The
main goal of this paper is to demonstrate that those results can be extended to (1.1)
with ε > 0. In order to do that, we need to use the approach developed in [17, 21]
to deal with damped semilinear hyperbolic equations. More precisely, we will prove
CONVERGENCE TO STATIONARY SOLUTIONS
3
that, if φ is real analytic and satisfies suitable growth and coercivity assumptions,
then any (weak) solution converges to a single stationary state. Taking advantage
of recent extensions of the Simon’s method to the asymptotically autonomous case
(see [8, 19]), we can handle a heat source term f which enjoys a sort of integral
decay condition as t goes to ∞. In addition, we can still show the (algebraic) decay
rate to equilibrium in the spirit of [18].
2. Notation and preliminary results. Set H = L2 (Ω) and denote by h·, ·i and
k · k the canonical inner product and the norm in H, respectively. Let us then
introduce the linear positive operator A = −∆ + I : D(A) ⊂ H → H with domain
D(A) = {v ∈ H 2 (Ω) : vn = 0,
on ∂Ω}
2
and B = −∆ : D(B) ⊂ H → H with D(B) = H (Ω) ∩ H01 (Ω). For any r ∈ R, set
V r = D(Ar/2 ) and V0r = D(B r/2 ) endowed with the inner products
hv1 , v2 iV r = hAr/2 v1 , Ar/2 v2 i,
hv1 , v2 iV0r = hB r/2 v1 , B r/2 v2 i.
Clearly, we have V 0 = V00 ≡ H. Identifying H with its dual space H ∗ , there holds
(V r )∗ = V −r and (V0r )∗ = V0−r . We also recall that V r1 ,→ V r2 and V0r1 ,→ V0r2
with compact injection for any r1 , r2 such that r1 > r2 . Moreover, we need to
introduce the Banach spaces
V r = V0r × V r+1 × V r .
Our assumptions on the nonlinearities are (cf. also [11, Remarks 3.1 and 3.2])
λ ∈ C 2 (R),
|λ00 (y)| ≤ c0 ,
∀y ∈ R,
2
φ ∈ C (R),
(2.2)
00
|φ (y)| ≤ c1 (1 + |y|),
lim inf
|y|→∞
∀y ∈ R,
(2.3)
φ(y)
> −α1 ,
y
(2.4)
for some positive constants c0 and c1 , where α1 is the first eigenvalue of A.
Regarding f , we suppose that it is translation bounded in V0−1 , namely
Z t+1
sup
kf (s)k2V −1 ds < ∞.
t≥0
(2.1)
(2.5)
0
t
We now rewrite system (1.1) in the following abstract form
(
(ϑ + λ(χ))t + Bϑ = f,
in (0, ∞),
εχtt + χt + Aχ + φ(χ) − λ0 (χ)ϑ = 0,
in (0, ∞),
(2.6)
endowed with initial conditions
ϑ(0) = ϑ0 , χ(0) = χ0 , χt (0) = χ1 ,
in Ω.
(2.7)
We begin with
Theorem 2.1. Let (2.1)-(2.5) hold. Then, for any ε ∈ (0, 1], system (2.6) generates
a dissipative process Uε (t, τ ) on V 0 . Let (ϑ(t), χ(t), χt (t)) = Uε (t, 0)(ϑ0 , χ0 , χ1 ) be
the solution to the Cauchy problem (2.6)-(2.7). If (2.5) is replaced by
Z ∞
kf (s)k2V −1 ds < ∞,
(2.8)
0
0
4
M. GRASSELLI, H. PETZELTOVÁ, G. SCHIMPERNA
then t≥0 (ϑ(t), χ(t), χt (t)) is bounded and precompact in V 0 . If (ϑ0 , χ0 , χ1 ) ∈ V 1
S
and f is translation bounded in H, then t≥0 (ϑ(t), χ(t), χt (t)) is bounded in V 1 .
S
Proof. The existence of the dissipative process Uε (t, τ ) has been proved in [10].
Let us prove that it is also asymptotically compact (see, e.g., [15, 16, 29]). More
precisely, following [11, Section 5], we can decompose the trajectory with initial
data z0 = (ϑ0 , χ0 , χ1 ) ∈ V 0 as
z = z d + z c = (ϑd , χd , χdt ) + (ϑc , χc , χct ),
where
ϑdt + Bϑd = f,
εχdtt + χdt + Aχd + φ0 (χd ) = 0,
z d (0) = z0 ,
and
ϑct + Bϑc = −λ0 (χ)χt ,
εχctt + χct + Aχc + φ0 (χ) + φ1 (χ) − φ0 (χd ) = λ0 (χ)ϑ,
z c (0) = 0.
Here φ0 and φ1 are such that φ = φ0 + φ1 and (cf. [11, Remark 3.1])
yφ0 (y) ≥ 0,
∀ y ∈ R,
φ1 (y)
> −α1 ,
y
|φ000 (y)| ≤ c(1 + |y|),
lim inf
|y|→∞
|φ01 (y)|
γ
≤ c(1 + |y| ),
∀ r ∈ R,
γ ∈ [0, 2), ∀ r ∈ R.
Then, on account of the above properties and (2.8), it is not hard to prove that
kz d (t)kV 0 → 0
as t goes to ∞. On the other hand, arguing as in [11], we can find that
kz c (t)kV 1 ≤ c,
∀ t ≥ 0.
We thus conclude that the orbit originating from z0 is precompact in V 0 . Assume
now that (ϑ0 , χ0 , χ1 ) ∈ V 1 . Then an easy adaptation of [11, Thm. 3.4, Cor. 4.4,
Thm. 4.6] (see also [10, Thm. 4.1]) implies the boundedness of the orbit in V 1 .
Remark 2.2. Notice that we cannot take advantage of the Webb’s compactness
principle [30] (as in [17]) since φ has critical growth in three dimensions (see also
Remark 3.5 below).
Let us introduce the set
S = v∞ ∈ V 1 : G(v∞ ) = 0 ,
where
G(v) = Av + φ(v),
∀v ∈ V 1 ,
and define, for any v ∈ V 1 ,
E(v) =
1
kvk2V 1 + hΦ(v), 1i,
2
CONVERGENCE TO STATIONARY SOLUTIONS
5
Ry
where Φ(y) = 0 φ(ζ)dζ. Note that, due to the assumptions (2.2)-(2.4), the set S
is bounded in V 2 , hence in L∞ (Ω).
Let us prove now the following
Lemma 2.3. Let (2.1)-(2.4) and (2.8) hold. For any fixed ε ∈ (0, 1], consider
(ϑ0 , χ0 , χ1 ) ∈ V 0 and the corresponding trajectory (ϑ(t), χ(t), χt (t)) = Uε (t, 0)(ϑ0 , χ0 , χ1 ).
Then we have
Z ∞
(kϑ(t)k2V 1 + kχt (t)k2 ) dt ≤ M,
(2.9)
0
0
for some positive constant M . Also, there hold
ϑ(t) → 0,
strongly in H, as t → ∞,
χt (t) → 0,
strongly in H, as t → ∞,
ωε (ϑ0 , χ0 , χ1 ) ⊆ (v1 , v2 , v3 ) ∈ V 1 : v1 ≡ 0, v2 ∈ S, v3 ≡ 0 ,
(2.10)
(2.11)
(2.12)
and E is constant on the set {χ∞ ∈ V 1 : (0, χ∞ , 0) ∈ ωε (ϑ0 , χ0 , χ1 )}. If, in
addition, (ϑ0 , χ0 , χ1 ) ∈ V 1 and
Z
sup
t≥0
t+1
kf (s)k2V 1 ds < ∞,
(2.13)
t
then we also have
ϑ(t) → 0,
strongly in C 0 (Ω̄) ∩ V01 , as t → ∞.
(2.14)
Proof. In the sequel of the paper, we will denote with c a generic positive constant,
independent of ε, which may vary even in the same line.
We define
1
Z(t) =
kϑ(t)k2 + εkχt (t)k2 + E(χ(t)),
(2.15)
2
and we observe that
d
Z(t) = −kϑ(t)k2V 1 − kχt (t)k2 + hf, ϑi.
0
dt
(2.16)
Then, on account of Theorem 2.1 and (2.8), we easily deduce (2.9) as well as (2.10)
and (2.11).
Consequently, any point of ωε (ϑ0 , χ0 , χ1 ) is of the form (0, χ∞ , 0). Let {tn }n∈N
be an unbounded increasing sequence such that χ(tn ) → χ∞ in V 1 , as n goes to
∞. Then, for any s ∈ [0, 1], χ(tn + s) → χ∞ in V 1 , so that G(χ(tn + s)) → G(χ∞ )
in V −1 , as n tends to ∞. Therefore, using the second equation of system (2.6) and
denoting by hh·, ·ii the duality pairing between V −1 and V 1 , we have (cf. (2.10) and
6
M. GRASSELLI, H. PETZELTOVÁ, G. SCHIMPERNA
(2.11))
Z
hhG(χ∞ ), vii = hh
1
G(χ∞ )ds, vii
0
1
Z
hhG(χ(tn + s)), viids
= lim
n→∞
0
1
Z
hhλ0 (χ(tn + s))ϑ(tn + s) − εχtt (tn + s) − χt (tn + s), viids
= lim
n→∞
0
1
Z
hhλ0 (χ(tn + s))ϑ(tn + s) − χt (tn + s), viids
= lim
n→∞
0
− lim εhhχt (tn + 1) − χt (tn ), vii
n→∞
= 0,
for any v ∈ V 1 . Thus we deduce χ∞ ∈ S. In addition, observe that
lim Z(t) = Z∞ ,
t→∞
where E(χ∞ ) = Z∞ for any χ∞ such that (0, χ∞ , 0) ∈ ωε (ϑ0 , χ0 , χ1 ).
Finally, on account of (2.13) and using the boundedness of the trajectory in V 1 ,
we can obtain (see [12, (3.21)])
kϑ(t)kV 1+2η ≤ c,
∀ t > 0,
0
for all η ∈ (0, 41 ), and a further argument (cf. [12, (3.28)-(3.29)]) leads to
1 (Ω) ≤ c,
kϑ(t)kV02κ +B6,2
∀ t > 0,
1
for κ ∈ ( 34 , 1). Since V01+2η is compactly embedded in V01 and V02κ + B6,2
(Ω) is
0
compactly embedded in C (Ω̄), using again (2.9), we deduce (2.14).
Remark 2.4. If λ is linear, (2.10) can be substituted with
ϑ(t) → 0,
strongly in V01 , as t → ∞.
(2.17)
In fact, observe that
ϑt + Bϑ = f − χt .
Therefore, setting, for all t > 0
h(t) = kB 1/2 ϑ(t)k2 ,
it is easy to realize that
h0 (t) ≤ ck(f − χt )(t)k2 ,
for all t > 0. Then, on account of (2.8) and (2.9), we can argue as in [33,
Lemma 6.2.1] (see also [27]) to deduce (2.17).
3. Main result. We first report the main tool of this section, namely the wellknown Lojasiewicz-Simon inequality. We will use it in the form proved by Haraux
and Jendoubi [17, Thm. 2.2 and Prop. 5.3.1].
Lemma 3.1. Suppose that φ is real analytic and assume (2.3)-(2.4). Let v∞ ∈ S.
Then there exist ρ ∈ (0, 21 ), σ > 0, and a positive constant C0 such that
kG(v)kV −1 ≥ C0 |E(v) − E(v∞ )|1−ρ ,
for all v ∈ V 1 such that kv − v∞ kV 1 < σ.
(3.1)
CONVERGENCE TO STATIONARY SOLUTIONS
7
Remark 3.2. If ρ0 < ρ, then we can always find σ0 ≤ σ such that inequality (3.1)
holds with ρ and σ replaced by ρ0 and σ0 , respectively.
Our main result is
Theorem 3.3. Let (2.1) and (2.13) hold. Suppose also, in place of (2.8),
Z ∞
sup t1+δ
kf (s)k2 ds < ∞,
t≥0
(3.2)
t
for some δ > 0, and let φ be real analytic satisfying (2.3)-(2.4). Take (ϑ0 , χ0 , χ1 ) ∈
V 0 and set
(ϑ(t), χ(t), χt (t)) = Uε (t, 0)(ϑ0 , χ0 , χ1 ),
∀ t ≥ 0.
Then, ωε (ϑ0 , χ0 , χ1 ) consists of a single point (0, χ∞ , 0) and, as t goes to ∞,
χ(t) → χ∞ ,
strongly in V 1 .
(3.3)
If
2ρ
,
(3.4)
1 − 2ρ
then one can find t∗ = t∗ (ε) > 0 and a positive constant C1 , independent of ε, such
that
ρ
kχ(t) − χ∞ k ≤ C1 t− 1−2ρ ,
∀ t ≥ t∗ .
(3.5)
Otherwise, one can find ρ0 ∈ (0, ρ) so that
2ρ0
δ>
,
(3.6)
1 − 2ρ0
δ>
a time t∗∗ = t∗∗ (ε) > 0 and a positive constant C2 , independent of ε, such that
ρ0
kχ(t) − χ∞ k ≤ C2 t− 1−2ρ0 ,
∀ t ≥ t∗∗ .
(3.7)
1
Remark 3.4. If, for instance, (ϑ0 , χ0 , χ1 ) ∈ V , then, in particular, χ(t) is uniformly bounded in L∞ (Ω). Therefore, on account of the fact that S is a bounded
subset of L∞ (Ω), the assumption on the analyticity of φ can be slightly relaxed
just by supposing that φ is analytic on a suitable bounded interval [−M, M ] with
M > 0 such that supv∞ ∈S kv∞ kL∞ (Ω) < M . In this case, however, we have to use a
localized version of Lojasiewicz-Simon inequality in place of Lemma 3.1 (see [2, 14]).
Remark 3.5. Consider a Cauchy-Neumann (or Dirichlet) problem for the hyperbolic damped semilinear wave equation with critical growth in dimension three.
Suppose that the nonlinearity is real analytic. Then, as a by-product of Theorem 3.3
(take λ ≡ 0), we have that any weak solution to the above problem converges to
a single steady state (compare with [17] where the subcritical case is analyzed by
means of the Webb’s principle).
Remark 3.6. In the case f ≡ 0, from [11] we know that the semigroup S(t) =
Uε (t, 0) defines a dynamical system on V 0 which possesses a global connected attractor Aε bounded in V 1 . Since the system has a global Lyapunov functional, then
it is well known that
n
Aε = (ϑ0 , χ0 , χ1 ) ∈ V 1 : the trajectory z(t) originated from (ϑ0 , χ0 , χ1 )
is complete and bounded in V 1 , and
o
lim z(t) = (0, χ∞ , 0), in V 0 , for some χ∞ ∈ S .
t→−∞
8
M. GRASSELLI, H. PETZELTOVÁ, G. SCHIMPERNA
If, in addition, we suppose that φ is real analytic, then Theorem 3.3 entails that Aε
is the union of all the constant and heteroclinic orbits.
Proof. Let (ϑ0 , χ0 , χ1 ) ∈ V 0 and consider (0, χ∞ , 0) ∈ ωε (ϑ0 , χ0 , χ1 ). In order to
prove (3.3), we proceed along the lines of [8, Proof of Thm. 2.3]. First let us assume
(3.4). We recall that δ comes from (3.2), while ρ comes from (3.1). Let us introduce
the unbounded set
σ
Σ = t ≥ 0 : kχ(t) − χ∞ kV 1 ≤
3
where σ is given by Lemma 3.1. For every t ∈ Σ, define
τ (t) = sup t0 ≥ t : sup kχ(s) − χ∞ kV 1 ≤ σ ,
s∈[t,t0 ]
and observe that τ (t) > t, for every t ∈ Σ. Notice that Σ and, consequently, τ (t)
depend on ε which is henceforth fixed.
Thanks to Lemma 2.3, we can take t0 ∈ Σ large enough such that
kϑ(t)k + kχt (t)k ≤ 1,
∀ t ≥ t0 ,
and set
J = [t0 , τ (t0 )),
(
Z
∞
t ∈ J : N (ϑ, χ)(t) >
J1 =
1−ρ )
kf (s)k ds
,
2
t
J2 = J \ J1 ,
J3 = {t ≥ 0 : βkϑ(t)k ≤ kf (t)k} ,
where
N (ϑ, χ)(t) = kϑ(t)k + kχt (t)k + kG(χ(t))kV −1 ,
∀ t > 0,
and β > 0 is to be fixed below.
Then, we introduce the functional
1
kϑ(t)k2 + εkχt (t)k2 + E(χ(t)) − E(χ∞ )
L0 (t) =
2
Z τ (t0 )
+
hf (s), ϑ(s)iΥJ3 (s) ds + αεhχt (t), A−1 G(χ(t))i,
t
for every t ∈ J, where ΥJ3 is the characteristic function of J3 .
Observe now that
d
L0 = −kϑk2V 1 − kχt k2 + hf, ϑi(1 − ΥJ3 ) + αεhχt , [A−1 G(χ)]t i
0
dt
− αhχt , A−1 G(χ)i − αkG(χ)k2V −1 + αhλ0 (χ)ϑ, A−1 G(χ)i.
(3.8)
On the other hand, using the Poincaré and the Young inequalities, we have
hf, ϑi(1 − ΥJ3 ) ≤ cβkϑk2V 1 ,
0
α
− αhχt , A−1 G(χ)i ≤ kG(χ)k2V −1 + cαkχt k2 .
4
Observe now that, owing to the Hölder inequality and using the embedding V 1 ,→
L6 (Ω), we have
hχt , A−1 [φ0 (χ)χt ]i ≤ kχt kkA−1 [φ0 (χ)χt ]k
≤ ckχt kkφ0 (χ)χt kL1 (Ω) ≤ ckχt k2 1 + kχk3L6 (Ω) ≤ ckχt k2 ,
CONVERGENCE TO STATIONARY SOLUTIONS
9
which entails
αεhχt , [A−1 G(χ)]t i ≤ cαkχt k2 .
On the other hand, also using the Poincaré inequality, we have
α
αhλ0 (χ)ϑ, A−1 G(χ)i ≤ kG(χ)k2V −1 + cαkλ0 (χ)ϑk2V −1
4
α
≤ kG(χ)k2V −1 + cαkλ0 (χ)ϑk2L6/5 (Ω)
4
α
≤ kG(χ)k2V −1 + cα 1 + kχk2L6 (Ω) kϑk2L3/2 (Ω)
4
α
≤ kG(χ)k2V −1 + cαkϑk2V 1 .
0
4
Collecting the above estimates, from (3.8) we deduce
α
d
L0 ≤ −(1 − cα − cβ)kϑk2V 1 − (1 − cα)kχt k2 − kG(χ)k2V −1 .
0
dt
2
Therefore, choosing α and β small enough and using again the Poincaré inequality,
we find
d
L0 ≤ −γ1 N (ϑ, χ)2 ,
(3.9)
dt
for some γ1 = γ1 (α, β) > 0.
Thus L0 is decreasing. Since
d
d
|L0 (t)|ρ sgn L0 (t) = ρ|L0 (t)|ρ−1 L0 (t),
t ∈ J,
(3.10)
dt
dt
it is clear that |L0 |ρ sgn L0 is decreasing as well.
Observe now that, for every t ∈ J1 , using (3.1), we have
|L0 (t)|1−ρ ≤ cN (ϑ, χ)(t).
Consequently, thanks to (3.10), we infer
Z
Z τ (t0 )
d
N (ϑ, χ)(t)dt ≤ −c
|L0 (t)|ρ sgn L0 (t)
dt
J1
t
0
≤ c |L0 (t0 )|ρ + |L0 (τ (t0 ))|ρ ,
where we intend that |L0 (τ (t0 ))| = 0 if τ (t0 ) = ∞.
On the other hand, if t ∈ J2 , by definition of J2 and (3.2), we deduce
Z ∞
1−ρ
2
N (ϑ, χ)(t) ≤
kf (s)k ds
≤ ct−(1+δ)(1−ρ) .
(3.11)
(3.12)
(3.13)
t
Therefore, thanks to (3.4), we can integrate N (ϑ, χ) over J2 to get
Z
N (ϑ, χ)(t)dt ≤ ct0−δ+ρ+ρδ .
J2
Thus kχt (·)k is integrable over J and, due to Lemma 2.3 and (3.2),
Z τ (t0 )
0 ≤ lim sup
kχt (t)kdt
t0 ∈Σ, t0 →∞
≤ c lim sup
t0 ∈Σ, t0 →∞
t0
|L0 (t0 )|ρ + |L0 (τ (t0 ))|ρ + t0−δ+ρ+ρδ = 0.
(3.14)
10
M. GRASSELLI, H. PETZELTOVÁ, G. SCHIMPERNA
Notice that, for every t ∈ J,
Z
t
kχ(t) − χ∞ k ≤
kχt (s)kds + kχ(t0 ) − χ∞ k.
(3.15)
t0
Suppose now that τ (t0 ) < ∞ for any t0 ∈ Σ. By definition, we have
kχ(τ (t0 )) − χ∞ kV 1 = σ,
∀ t0 ∈ Σ.
Consider an unbounded sequence {tn }n∈N ⊂ Σ such that
lim kχ(tn ) − χ∞ kV 1 = 0.
n→∞
By compactness, we can find a subsequence {tnk }k∈N and an element χ̃∞ ∈ S such
that kχ̃∞ − χ∞ kV 1 = σ and
lim kχ(τ (tnk )) − χ̃∞ kV 1 = 0.
k→∞
Then, owing to (3.14) and (3.15), we deduce the contradiction
!
Z τ (tn )
k
0 < kχ̃∞ − χ∞ k ≤ lim sup
kχt (s)kds + kχ(tnk ) − χ∞ k = 0.
k→∞
tnk
Hence, τ (t0 ) = ∞ for some t0 > 0 large enough. We can thus say that kχt (·)k is
indeed integrable over (t0 , ∞). Hence, by compactness, (3.3) follows. Note now that
L0 (t) goes to 0 as t goes to ∞ and, being decreasing, it follows that is nonnegative
on (t0 , ∞).
To obtain (3.5), let us suppose first that either [tn0 , ∞) ⊂ J1 or [tn0 , ∞) ⊂ J2 for
some n0 ∈ N. In the former case we have, for any t > tn0 ,
(L0 (t))1−ρ ≤ cN (ϑ, χ)(t),
and, on account of (3.9), we deduce
d
2(1−ρ)
L0 + cL0
≤ 0,
dt
which yields
1
L0 (t) ≤ c(1 + t)− 1−2ρ .
Using (3.12) we infer
Z ∞
ρ
N (ϑ, χ)(s) ds ≤ c(1 + t)− 1−2ρ ,
(3.16)
t
and this gives (3.5) since
Z ∞
Z
kχ(t) − χ∞ k ≤
kχt (s)k ds ≤
t
∞
ρ
N (ϑ, χ)(s) ds ≤ c(1 + t)− 1−2ρ .
t
On the other hand, if [tn0 , ∞) ⊂ J2 , then, on account of (3.13), we get
N (ϑ, χ)(t) ≤ c t−(1+δ)(1−ρ) ,
so that
Z
∀ t > t n0 ,
(3.17)
∞
N (ϑ, χ)(s) ds ≤ ct−δ+ρ+ρδ ,
∀ t > t n0 ,
t
and, arguing as above,
kχ(t) − χ∞ k ≤ ct−δ+ρ+ρδ .
Therefore (3.5) also holds in this case.
In order to complete the proof, it remains to handle where neither J1 nor J2
contain a half-line. Since, by construction, J1 is an open set, then there exists a
CONVERGENCE TO STATIONARY SOLUTIONS
countable family of disjoint open sets (an , bn ) such that J1 =
first note that, for any n ∈ N, there holds
1−ρ
Z ∞
.
kf (s)k2 ds
N (ϑ, χ)(an ) =
11
S∞
n=0
(an , bn ). Let us
an
Thus, on account of (3.2) and (3.11), we easily obtain
L0 (an ) ≤ c an −(1+δ) .
(3.18)
Consider t ∈ J1 and denote by n∗ (t) the integer such that t ∈ (an∗ , bn∗ ). Since
(3.16) holds everywhere in J1 , on account of (3.18), we deduce
− 1
L0 (t) ≤ c(1 − 2ρ)(t − an∗ ) + (L0 (an∗ ))−1+2ρ 1−2ρ
1
h
i− 1−2ρ
(1+δ)(1−2ρ)
≤ c(1 − 2ρ)(t − an∗ ) + can∗
.
Observe that t can be chosen in such a way that n∗ is arbitrarily large. In particular,
since (1 + δ)(1 − 2ρ) > 1, we can find an integer n∗ such that
1
− 1−2ρ
L0 (t) ≤ [c(1 − 2ρ)t]
∀ t ∈ (an∗ , bn∗ ).
,
Fixed this n∗ , it is clear that the above inequality still holds for any t ∈ J1 , t > bn∗ ,
namely,
1
L0 (t) ≤ c t− 1−2ρ ,
∀ t ∈ J1 ∩ (an∗ , ∞).
(3.19)
Recall now that, on J1 , we have
−c
d
(L0 (t))ρ ≥ N (ϑ, χ)(t),
dt
(3.20)
while (3.17) holds on J2 . Thus, observing that J2 is measurable and using (3.17),
(3.19), (3.20), we deduce, for any t > min{t0 , an∗ },
Z ∞
Z ∞
kχ(t) − χ∞ k ≤
kχt (s)k ds ≤
N (ϑ, χ)(s) ds
t
t
Z
Z
=
N (ϑ, χ)(s) ds +
N (ϑ, χ)(s) ds
(t,∞)∩J1
Z ∞
d
(L0 (s))ρ ds + c
≤ −c
ds
t
ρ
≤ c t− 1−2ρ + t−δ+ρ+ρδ ,
(t,∞)∩J2
∞
−(1+δ)(1−ρ)
Z
s
ds
t
which entails (3.5), provided that (3.4) holds. Otherwise, we can find ρ0 ∈ (0, ρ)
such that (3.6) is satisfied with ρ replaced by ρ0 (cf. Remark 3.2). Then we proceed
as before to obtain (3.7).
Acknowledgments. The authors thank Sergey Zelik and the anonymous referee
for their useful remarks which contributed to improve the results contained in this
paper.
12
M. GRASSELLI, H. PETZELTOVÁ, G. SCHIMPERNA
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