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J. Differential Equations 257 (2014) 2136–2158
www.elsevier.com/locate/jde
Large time behavior of a solution of carbon dioxide
transport model in concrete carbonation process
Kota Kumazaki
Natural and Physical Sciences, Tomakomai National College of Technology, 443, Nishikioka, Tomakomai, Hokkaido, 059-1275, Japan
Received 29 May 2013; revised 12 August 2013
Available online 12 June 2014
Abstract
In [7] we show the global existence and uniqueness of a solution of carbon dioxide transport model in concrete carbonation process. This model is governed by a parabolic-type equation which has a non-local term depending on the unknown function itself. In this paper, we show the large time behavior of that solution.© 2014 Elsevier Inc. All rights reserved.
Keywords: Large time behavior; Nonlinear parabolic equation; Concrete carbonation process
1. Introduction
In this paper we consider the following initial boundary value problem (P) = {(1.1), (1.2),(1.3)}, which is a mathematical model of carbon dioxide transport in concrete carbonation pro-cess:
∂
∂t
[φ(1 − e− ∫ t
0 u(τ)dτ)u] − �u = −w0ue− ∫ t
0 u(τ)dτ in Q(T ) := (0, T ) × Ω, (1.1)
u = ub on S(T ) := (0, T ) × Γ, (1.2)
u(0) = u0 in Ω. (1.3)
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jde.2014.05.0320022-0396/© 2014 Elsevier Inc. All rights reserved.
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2137
Fig. 1. The graph of φ.
Here, Ω is a bounded domain of R3 with a smooth boundary Γ = ∂Ω , T > 0 is a fixed finite number, φ is a function in C1(R) satisfying φ0 ≤ φ(r) ≤ 1 for r ∈ R where φ0 is a positive constant, ub is a given function on Q(T ), and w0 and u0 also are given functions on Ω .
Eq. (1.1) is a diffusion equation derived from the following balanced law of carbon dioxide in concrete carbonation process proposed by Maekawa, Chaube and Kishi [9] and Maekawa, Ishida and Kishi [10]:
∂
∂t
{φ[(1 − S)v + Su
]} − div(φ[H1(1 − S)∇v +H2S∇u
]) = −κuw in Q(T ). (1.4)
Here, we explain briefly the physical background to (1.4) (see [2] for details). In (1.4), Ω is a domain occupied by concrete, and the unknown functions v = v(t, x) and u = u(t, x) represent the concentrations of carbon dioxide in air and in water at a time t and a position x ∈ Ω , respec-tively. In the equilibrium state, the relation v = ρ0u is known to hold for a positive constant ρ0by Henry’s law. Also, φ = φ(z) represents the porosity, which is the ratio of the volume of the voids inside of the concrete to the volume of the whole concrete (see Fig. 1), and z is the ratio of the volume of consumed calcium hydroxide to the volume of the total calcium hydroxide. Next, S represents the degree of saturation corresponding to the relative humidity; its relationship to the degree of saturation is given as a hysteresis operator in [9,10]. Accordingly, φ(1 − S)v and φSu represent the concentration of gaseous carbon dioxide in pores and the concentration of carbon dioxide dissolved in pore water, respectively.
In the diffusion term, H1 and H2 are positive constants, and H1(1 −S) and H2S represent the diffusion coefficients of gaseous carbon dioxide in pore and carbon dioxide dissolved in water. In the forcing term, κ is a reaction rate and w represents the concentration of calcium ion. The forcing term represents the consumed carbon dioxide during the carbonation of concrete, and is given by the reaction rate theory.
Eq. (1.1) is obtained from (1.4) by the following procedure. First, we use the facts that wand z (for z, see below (1.4)) are expressible in the following form:
w(t) = [Ca2+]
(0)e−κ∫ t
0 u(τ)dτ u(t), z(t) = 1 − e−κ∫ t
0 u(τ)dτ for t > 0. (1.5)
Here, κ is a positive constant, [Ca2+](0) is the concentration of calcium ion at an initial time, and u is the concentration of carbon dioxide in water, which is the unknown function. Henceafter, [a] represents the concentration of the element a. As mentioned above, the forcing term in (1.4)is given by the following form based on the reaction rate theory:
2138 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
d
dt[CaCO3] = κ
[Ca2+][
CO2−3
], (1.6)
where Ca2+ and CO2−3 are calcium ion and carbonate ion, respectively. Next, we assume that
ρ1[Ca(OH)2
] + ρ2[CaCO3] = constant. (1.7)
Here, ρ1 and ρ2 are molecule weight of calcium hydroxide and molecule weight of calcium carbonate, respectively. The relation (1.7) is based on the following chemical reaction in concrete carbonation process:
Ca(OH)2 + CO2 → CaCO3 + H2O in pore water.
Now, we consider [Ca(OH)2] and u as [Ca2+] (= w) and [CO2−3 ] in water, respectively. Then,
by the time-derivative of (1.7) we have
d
dt[CaCO3] = −ρ1
ρ2
d
dt
[Ca2+]
. (1.8)
By adding (1.6) and (1.8) we have the following ordinary differential equation:
d
dt
[Ca2+] = −κ
ρ2
ρ1
[Ca2+][
CO2−3
]
which yields from solving this equation and using the relation [CO2−3 ] = u that
w(t) = [Ca2+] = [
Ca2+](0)e−κ
∫ t0 u(τ)dτ for t > 0, (1.9)
where [Ca2+](0) is the concentration of calcium ion as an initial time, and κ = κρ1/ρ2. Also, since z is the ratio of the volume of consumed calcium hydroxide to the volume of the total calcium hydroxide, we obtain from (1.9) that
z(t) = [Ca2+](0) − [Ca2+][Ca2+](0)
= 1 − e−κ∫ t
0 u(τ)dτ for t > 0. (1.10)
Accordingly, from (1.9) and (1.10) we see that z and w are described by (1.5). Next, from (1.5)and (1.6), we see that the forcing term and the porosity φ are given by
κuw = κ[Ca2+]
(0)ue−κ∫ t
0 u(τ)dτ , (1.11)
and
φ = φ(z(t)
) = φ(1 − e−κ
∫ t0 u(τ)dτ
), (1.12)
respectively. By setting w0 = κ[Ca2+](0) and using (1.11) and (1.12) we can rewrite (1.4) by
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2139
∂
∂t
{φ(1 − e−κ
∫ t0 u(τ)dτ
)[(1 − S)ρ0u + Su
]}− div
(φ(1 − e−κ
∫ t0 u(τ)dτ
)[H1(1 − S)ρ0∇u +H2S∇u
])= −w0ue−κ
∫ t0 u(τ)dτ . (1.13)
To deal with (1.13) mathematically, we simplify by setting ρ0 =H1 =H2 = κ = 1 and φ ≡ 1in the diffusion term. This completes the physical content behind Eq. (1.1).
In regard to mathematical results, Muntean and Böhm [11] considered a mathematical model of concrete carbonation process as a one-dimensional free boundary problem for the carbon-ated front, and proved the existence and uniqueness of a solution of their problem. Also, Aiki and Muntean [4,5] considered a reducted model of Muntean–Böhm, and proved the large time behavior of the free boundary. Our aim in this study is to formulate a three-dimensional mathe-matical model of this process and to analyze the dynamics of concrete carbonation process. As a first step, Aiki and Kumazaki [1,3] proposed a mathematical model of moisture transport which involves the hysteresis operator S , and proved the existence and uniqueness of a solution of the model. Uniqueness was proved only for the one-dimensional case. As a second step, Kumazaki [7] proved the existence and uniqueness of a global solution of (P).
Here, by using the chain rule of the time derivative, we can write (1.1) as follows:
φ
(1 − exp
(−
t∫0
u(x, τ )dτ
))∂u
∂t(x, t) − �u(x, t)
+ φ′(
1 − exp
(−
t∫0
u(x, τ )dτ
))exp
(−
t∫0
u(x, τ )dτ
)u2(x, t)
= −w0(x)u(x, t) exp
(−
t∫0
u(x, τ )dτ
).
Thus, Eq. (1.1) is a nonlinear parabolic equation involving a non-local term depending on the solution. The main purpose of this paper is to show the large time behavior of a solution of (P), more precisely, to prove that a solution u of (P) converges to a solution u∞ of the steady state problem (P)∞ := {(1.14), (1.15)}:
−�u∞ + φ′(1 − w∞)w∞u2∞ = −w0u∞w∞ in Ω, (1.14)
u∞ = ub∞ on Γ, (1.15)
where ub∞ is a given function in Ω (see (A3)) and w∞ = w∞(x) is a limit of w(x, t) :=exp(−
∫ t
0 u(x, τ)dτ) as t → ∞, namely,
w(x, t) → w∞(x) as t → ∞ for each x ∈ Ω. (1.16)
Here, we note that the limit function w∞ exists, and w∞ = w∞(x) is positive and is less than 1 for each x ∈ Ω . Indeed, the solution u of (P) is non-negative on Q(T ) for any T > 0, and for each
2140 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
x ∈ Ω , w(x, t) is positive and less than 1 on [0, ∞); then, (1.16) is valid. Clearly, if ∫ t
0 u(τ)dτ
is unbounded with respect to t > 0, then w∞ = w∞(x) = 0.In this paper, we attempt to clarify the structure of u∞ and w∞ by the boundary data ub∞.
More precisely, we consider two cases for ub∞: one is the case where ub∞ does not vanish identically on Γ , and the other is the case where ub∞ = 0 a.e. on Γ .
The plan of this paper is as follows: In Section 2, we state the global existence of a solution of (P) (Theorem 1), and our main result concerning the large time behavior of this solution of (P) (Theorem 2). Our main theorem is proved in Sections 3–5. In Sections 3 and 4 we show the existence of a solution of the steady state problem (P)∞, and the global boundedness of this solution of (P). The large time behavior of (P) is proved in Section 5.
2. Our main result
In this paper, we use the following notation. In general, for Banach space X, we denote by | · |X its norm. In particular, we denote H = L2(Ω), and the norm and the inner product of Hare simply denoted by | · |H and (·,·)H , respectively. Also, H 1(Ω), H 1
0 (Ω) and H 2(Ω) are the usual Sobolev spaces.
Throughout this paper we assume the following (A1)–(A5):
(A1) Ω ⊂ R3 is a bounded domain with a smooth boundary Γ .(A2) φ is a non-decreasing function in C1(R) such that c0 = supr∈R φ′(r) < ∞ and φ0 ≤
φ(r) ≤ 1 for r ∈ R where φ0 is a positive number.(A3) ub ∈ L∞(0, ∞; H 2(Ω)) with 0 ≤ ub ≤ κ0 in Ω for a positive number κ0 and (ub)t ∈
L2(0, ∞; H 1(Ω)) ∩ L1(0, ∞; H 2(Ω)). Also, there exists ub∞ ∈ H 2(Ω) such that ub −ub∞ ∈ L2(0, ∞; H 1(Ω)).
(A4) u0 ∈ H 1(Ω) ∩ L∞(Ω), u0 ≥ 0 in Ω and u0 = ub(0) on ∂Ω .(A5) w0 ∈ L∞(Ω) and w0 > 0 in Ω .
Here, we note from 0 ≤ ub ≤ κ0 in Ω and ub − ub∞ ∈ L2(0, ∞; H 1(Ω)) that 0 ≤ ub∞ ≤ κ0in Ω . Next, we define a solution of (P) on [0, T ] in the following way:
Let u be a function on Q(T ) for 0 < T < ∞. We call the function u a solution of (P) on [0, T ]if the following conditions (S1)–(S4) hold:
(S1) u ∈ W 1,2(0, T ; H) ∩ L∞(0, T ; H 1(Ω)) ∩ L2(0, T ; H 2(Ω)), and u ≥ 0 a.e. on Q(T ).(S2) [φ(1 − e− ∫ t
0 u(τ)dτ )u]t − �u = −w0ue− ∫ t0 u(τ)dτ a.e. in Q(T ).
(S3) u = ub a.e. on S(T ).(S4) u(0) = u0 in Ω .
Our first result shows the well-posedness of (P).
Theorem 1. If (A1)–(A5) hold, then for any T > 0, (P) has one and only one solution u on [0, T ]such that
0 ≤ u ≤ u∗ := max{|u0|L∞(Ω), κ0
}a.e. on Q(T ),
where u0 is the initial data and κ0 is the same constant as in (A3).
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2141
This theorem is already proved in [7], and therefore we omit its proof in this paper. Here, we note from [7] that the existence of a solution of (P) can be proved replacing the assumptions (A1)–(A5) by (A1), (A2)′, (A3)–(A5):
(A3)′ ub ∈ W 1,2(0, T ; H 1(Ω)) ∩ L∞(0, T ; H 2(Ω)) with 0 ≤ ub ≤ κ0 in Ω for a positive num-ber κ0.
Now, we state the result concerning the large time behavior of a solution of (P).
Theorem 2. Assume (A1)–(A5) hold, and let u and u∞ be solutions of (P) and (P)∞, respectively. Then,
u(t) → u∞ strongly in H and weakly in H 1(Ω) as t → ∞.
Moreover, we obtain the following results for two cases of ub∞.
Corollary 1. Assume (A1)–(A5) hold, and let u and u∞ be solutions of (P) and (P)∞, respec-tively. Then:
(i) If ub∞ does not vanish identically on Γ , then u∞ is a solution of the following steady state problem
−�u∞ = 0 a.e. in Ω, u∞ = ub∞ a.e. on Γ.
Also, w∞ = 0 a.e. on Ω .(ii) If ub∞ = 0 a.e. on Γ , then u∞ = 0 a.e. on Ω . Moreover, if ub = ub∞ = 0 a.e. on Ω , then,
there exists a positive constant α such that
∣∣u(t)∣∣2H
≤ |u0|2H e−αt for t ≥ 0.
3. Existence of the steady state problem (P)∞
First, although we define in Introduction, here we note the definition of w(x, t) and w∞:
w(x, t) = exp
(−
t∫0
u(x, τ )dτ
),
w(x, t) → w∞(x) as t → ∞ for x ∈ Ω.
As mentioned in Introduction, the existence of w∞ is valid. In fact, by Theorem 1, since the solution u is positive a.e. on Q(T ) for any T > 0, we see that w(x, t) is positive and is less than 1 a.e. on Q(T ), and therefore, w∞ = w∞(x) is also positive and is less than 1 for a.e. x ∈ Ω . To prove Theorem 2, we show the existence of a solution of the steady state problem (P)∞ = {(3.1), (3.2)}:
2142 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
−�u∞ + φ′(1 − w∞)w∞u2∞ = −w0u∞w∞ in Ω, (3.1)
u∞ = ub∞ on Γ. (3.2)
Lemma 1. Assume (A1)–(A5) hold. Then, (P)∞ has one and only one solution u∞ ∈ H 2(Ω) ∩L∞(Ω) satisfying (3.1), (3.2), and
0 ≤ u∞ ≤ κ0 a.e. in Ω,
where κ0 is the same constant as in (A3).
Proof. First, for fixed M > 0, we consider the following function gM ∈ C1(R):
gM(r) :=
⎧⎪⎨⎪⎩
0 if r < 0,
r2 if 0 < r ≤ M,
increases monotonically if M < r ≤ 2M,
4M2 if r > 2M.
Also, we define the following convex function ψ on H :
ψ(z) :={
12 |∇z|2H if z ∈ H 1(Ω) with z = ub∞ on Γ,
+∞ otherwise.
Then, we see that ψ is proper, lower semi-continuous, convex function on H , and therefore, we can consider a subdifferential ∂ψ of ψ , which is characterized by
∂ψ(z) = −�z for z ∈ D(∂ψ) := {z ∈ H 1(Ω)
∣∣ −�z ∈ H, z − ub ∈ H 10 (Ω)
}.
Using gM and ∂ψ we consider the following equation:
∂ψ(u∞) + φ′(1 − w∞)w∞gM(u∞) = −w0u∞w∞ in H. (3.3)
Here, we can see from φ′ ≥ 0 in (A2), (A5) and the definition of gM that the function fM(z) := φ′(1 − w∞)w∞gM(z) is monotone and hemicontinuous on H . As ∂ψ is maximal monotone on H , by applying [6, Corollary 3.3], we know that ∂ψ + fM is maximal monotone on H . Moreover, for the identity mapping I from H into itself, w0w∞I is also monotone and hemicontinuous on H , we also obtain from [6, Corollary 3.3] again that w0w∞I + ∂ψ + fM is also maximal monotone on H . Now, setting B := w0w∞I + ∂ψ + fM , we see that the effec-tive domain D(B) = D(∂ψ). Then, by Minty’s theorem, we have that R(cI + B) = H for any positive constant c where R(A) is the range of an operator A. Therefore, for v ∈ H , there exists v ∈ D(∂ψ) ⊂ H such that
cv + w0w∞v + ∂ψ(v) + fM(v) = cv in H. (3.4)
Note from w0w∞ > 0 and the monotonicity of fM that the solution v of (3.4) is unique. Here, let v1 and v2 be two solutions of (3.4) for v1 and v2, respectively. By putting V = v1 − v2 and V = v1 − v2, we have
cV + w0w∞V + ∂ψ(v1) − ∂ψ(v2) + fM(v1) − fM(v2) = cV in H. (3.5)
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2143
By multiplying V to (3.5) and using the facts that w0w∞ is positive and fM is monotone, we derive
cp|V |2H ≤ c(V, V )H , (3.6)
where cP is a positive constant owing to the Poincaré inequality. By (3.6) we see
|V |2H ≤ c2
cP
|V |2H . (3.7)
Here, we set the mapping Λc : H → H given by
Λcv = v for v ∈ H,
where v is a solution of (3.5). If we take c > 0 such that it satisfies c2/cP < 1, then by (3.7), Λc is a contraction mapping. Therefore, by Banach’s fixed point theorem, we obtain v ∈ H such that Λcv = v in H . This implies that the fixed point v satisfies (3.3). Thus, we see the existence of a solution of (3.3) for any M > 0.
Next, we show the boundedness of a solution u∞ of (3.3). First, we show 0 ≤ u∞ a.e. on Ω . By multiplying [−u∞]+ to (−1) × (3.3), we have
(�u∞, [−u∞]+)
H+ (−φ′(1 − w∞)w∞gM(u∞), [−u∞]+)
H
= (w0u∞w∞, [−u∞]+)
H. (3.8)
Then, by the definition of gM , we have (−φ′(1 − w∞)w∞gM(u∞), [−u∞]+)H = 0. Since (w0u∞w∞, [−u∞]+)H ≤ 0, we obtain from (3.8) and ub∞ ≥ 0 on Γ that
∣∣∇[−u∞]+∣∣2H
= (�u∞, [−u∞]+)
H≤ 0,
which yields
∣∣[−u∞]+∣∣2H
= 0.
Therefore, we see that 0 ≤ u∞ a.e. on Ω .Next, we show that u∞ ≤ κ0 a.e. on Ω . By multiplying (3.3) by [u∞ − κ0]+, we obtain
(−�u∞, [u∞ − κ0]+)H
+ (φ′(1 − w∞)w∞gM(u∞), [u∞ − κ0]+
)H
= (−w0u∞w∞, [u∞ − κ0]+)H
. (3.9)
As φ′ and gM are positive, we see that (φ′(1 −w∞)w∞gM(u∞), [u∞ − κ0]+)H ≥ 0. Also, since u∞ ≥ 0 a.e. on Ω , then (−w0u∞w∞, [u∞ − κ0]+)H ≤ 0 holds. Therefore, from (3.9), we derive
(−�(u∞ − κ0), [u∞ − κ0]+) = (−�u∞, [u∞ − κ0]+
) ≤ 0,
H H
2144 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
which by the Poincaré inequality yields
∣∣[u∞ − κ0]+∣∣2H
≤ 0.
This means that u∞ ≤ κ0 a.e. on Ω .Finally, by taking M = κ0, we see from the definition of gM that gM(u) = u2 on Ω , and
therefore the solution u∞ satisfies (3.1) and (3.2). Therefore, the existence of a solution of (P)∞is proved. To prove Lemma 1 completely, we show the uniqueness of a solution of (P)∞. Let u∞and u∞ be solutions of (P). Then, we have
−�(u∞ − u∞) + φ′(1 − w∞)w∞(u2∞ − u2∞
) = w0(−u∞ + u∞)w∞ in Ω. (3.10)
From φ′, u∞ and u∞ being non-negative, we note that
φ′(1 − w∞)w∞(u2∞ − u2∞
)(u∞ − u∞) = φ′(1 − w∞)w∞(u∞ − u∞)2(u∞ + u∞) ≥ 0.
Hence, by multiplying (3.10) by u∞ − u∞, we obtain
∣∣∇(u∞ − u∞)∣∣2H
≤ 0.
Therefore, we see that the solution of (P) is unique. Thus, Lemma 1 is proved.
4. Global boundedness
To prove Theorem 2, we first show the following estimate.
Lemma 2. If u is a unique solution of (P), then, there exists a positive constant M such that
|ut |2L2(0,T ;H)+ sup
t∈[0,T ]|∇u|2H + |�u|2
L2(0,T ;H)≤ M for any T > 0.
Proof. First, we set u := u − ub . Then, using the definition of w, we see from (1.1) that
φ′(1 − w(t))wt(t)u(t) + φ
(1 − w(t)
)ut (t) − �u
= −w0u(t)w(t) + �ub(t) − φ(1 − w(t)
)(ub)t (t).
Next, we multiply the above equation by ut and integrate on Ω , and use Green’s formula to derive
(φ′(1 − w(t)
)wt(t)u(t), ut (t)
)H
+ φ0∣∣ut (t)
∣∣2H
+ 1
2
d
dt
∣∣∇u(t)∣∣2H
≤ (−w0u(t)w(t) + �ub(t) − φ(1 − w(t)
)(ub)t (t), ut (t)
)H
. (4.1)
Here, we note that
(�ub(t), ut (t)
)H
= d (�ub(t), u(t)
)H
− (�(ub)t (t), u(t)
)H
for a.e. t ∈ [0, T ].
dt
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2145
Hence, from (4.1), we obtain that
φ0∣∣ut (t)
∣∣2H
+ d
dt
[1
2
∣∣∇u(t)∣∣2H
− (�(ub)t (t), u(t)
)H
]
≤ (−w0u(t)w(t), ut (t))H
+ (−φ′(1 − w(t))wt(t)u(t), ut (t)
)H
− (�(ub)t (t), u(t)
)H
− (φ(1 − w(t)
)(ub)t (t)ut (t)
)H
. (4.2)
Now, we estimate the right-hand side of (4.2) separately. First, using wt(t) = −w(t)u(t) a.e. t ∈ [0, T ] and applying Young’s inequality we have
(−w0u(t)w(t), ut (t))H
+ (−φ′(1 − w(t))wt(t)u(t), ut (t)
)H
= (−w0u(t)w(t), ut (t))H
+ (−φ′(1 − w(t))w(t)u2(t), ut (t)
)H
≤ |w0|2L∞(Ω)
φ0
(u2(t),w2(t)
)H
+ c20
φ0
(u4(t),w2(t)
)H
+ φ0
2
∣∣ut (t)∣∣2H
.
Using the fact that 0 ≤ u ≤ u∗ a.e. on Q(T ), we get
(−w0u(t)w(t), ut (t))H
+ (−φ(1 − w(t)
)wt(t)u(t), ut (t)
)H
≤( |w0|2L∞(Ω)u
∗
φ0+ c2
0(u∗)3
φ0
)(u(t),w2(t)
)H
+ φ0
2
∣∣ut (t)∣∣2H
.
Let us set C0 = |w0|2L∞(Ω)u∗/φ0 + c2
0(u∗)3/φ0. Next, by (A2) and (A3), we observe
−(�(ub)t (t), u(t)
)H
− (φ(1 − w(t)
)(ub)t (t), ut (t)
)H
≤ (u∗ + κ0
)|Ω| 12∣∣�(ub)t (t)
∣∣H
+ 1
φ0
∣∣(ub)t (t)∣∣2H
+ φ0
4
∣∣ut (t)∣∣2H
, (4.3)
where |Ω| is the volume of Ω . Hence, from (4.2) and (4.3), we obtain
φ0
4
∣∣ut (t)∣∣2H
+ d
dt
[1
2
∣∣∇u(t)∣∣2H
− (�ub(t), u(t)
)H
]
≤ C0(u(t),w2(t)
)H
+ (u∗ + κ0
)|Ω| 12∣∣�(ub)t (t)
∣∣H
+ 1
φ0
∣∣(ub)t (t)∣∣2H
. (4.4)
Here, we note that
1
2
∣∣∇u(t)∣∣2H
+ (−�ub(t), u(t))H
= 1
2
∫Ω
∇u(t) · ∇u(t)dx +∫Ω
∇ub(t) · ∇u(t)dx = 1
2
(∣∣∇u(t)∣∣2H
− ∣∣∇ub(t)∣∣2H
), (4.5)
and
2146 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
−1
2
d
dt
∫Ω
w2(t)dx = −1
2
d
dt
∫Ω
e−2∫ t
0 u(τ)dτ dx
=∫Ω
u(t)e−2∫ t
0 u(τ)dτ dx = (u(t),w2(t)
)H
for a.e. t ∈ (0, T ). (4.6)
Now, by applying (4.5) and (4.6) to (4.4), we have
φ0
4
∣∣ut (t)∣∣2H
+ 1
2
d
dt
(∣∣∇u(t)∣∣2H
− ∣∣∇ub(t)∣∣2H
) + C0
2
d
dt
∫Ω
w2(t)dx
≤ (u∗ + κ0
)|Ω| 12∣∣�(ub)t (t)
∣∣H
+ 1
φ0
∣∣(ub)t (t)∣∣2H
. (4.7)
Accordingly, by integrating (4.7) over [0, t], we find
φ0
4
t∫0
∣∣ut (s)∣∣2H
ds + 1
2
∣∣∇u(t)∣∣2H
≤ 1
2
∣∣∇ub(t)∣∣2H
+ 1
2
(∣∣∇u(0)∣∣2 − ∣∣∇ub(0)
∣∣2H
) + C0
2|Ω|
+ (u∗ + κ0
)|Ω| 12
t∫0
∣∣�(ub)t (s)∣∣H
ds + 1
φ0
t∫0
∣∣(ub)t (s)∣∣2H
ds. (4.8)
Then, since the right-hand side of (4.8) is bounded because of (A3), we see that there exists C > 0 such that
t∫0
∣∣ut (s)∣∣2H
ds + supt∈[0,T ]
∣∣∇u(t)∣∣2H
≤ C for any t > 0. (4.9)
Also, from (4.9) and (ub)t ∈ L2(0, ∞; H 1(Ω)) in (A3), we see that
t∫0
∣∣ut (s)∣∣2H
ds ≤ 2C + 2
t∫0
∣∣(ub)t (s)∣∣2H
ds < +∞ for any t > 0. (4.10)
Next, we establish the global estimate of �u using (4.10). To do this, we multiply (1.1) by −�u
and integrate, thereby obtaining
(φ(1 − w(t)
)ut (t),−�u(t)
)H
+ (φ′(1 − w(t)
)wt(t)u(t),−�u(t)
)H
+ ∣∣�u(t)∣∣2H
= (−w0u(t)w(t),−�u(t))H
. (4.11)
Transporting two terms of the left-hand side in (4.11) to the right-hand side and applying the Schwarz and Young inequalities to each term, we get
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2147
−(φ(1 − w(t)
)ut (t),−�u(t)
)H
≤ ∣∣ut (t)∣∣2H
+ 1
4
∣∣�u(t)∣∣2H
,
−(φ′(1 − w(t)
)wt(t)u(t),−�u(t)
)H
= −(φ′(1 − w(t)
)w(t)u2(t),−�u(t)
)H
≤ c20
(u4(t),w2(t)
)H
+ 1
4
∣∣�u(t)∣∣2H
,
and
(−w0u(t)w(t),−�u(t))H
≤ |w0|2L∞(Ω)
(u2(t),w2(t)
)H
+ 1
4
∣∣�u(t)∣∣2H
,
where c0 is the same constant as in (A2). Adding the above estimate, we have from (4.11) that
1
4
∣∣�u(t)∣∣2H
≤ c20
(u4(t),w2(t)
)H
+ |w0|2L∞(Ω)
(u2(t),w2(t)
)H
+ ∣∣ut (t)∣∣2H
.
Therefore, using the fact that 0 ≤ u ≤ u∗ a.e. on Q(T ), we have
1
4
∣∣�u(t)∣∣2H
≤ (c2
0
(u∗)3 + |w0|2L∞(Ω)u
∗)(u(t),w2(t))H
+ ∣∣ut (t)∣∣2H
.
Using (4.6) we have from the above inequality
1
4
∣∣�u(t)∣∣2H
+ C0φ0
2
d
dt
∫Ω
w2(t)dx ≤ ∣∣ut (t)∣∣2H
, (4.12)
where φ0 and C0 are the same constants as in (A2) and (4.3), respectively. Therefore, by integrat-ing (4.12) over [0, t] with (4.10) we find �u is bounded in L2(0, T ; H) for any T > 0. Finally, by (4.9), (4.10) and (4.12) we have proved Lemma 2.
Next, we show the following lemma.
Lemma 3. Let u and u∞ be solutions of (P) and (P)∞, respectively. Then, there exists C > 0such that
supt≥0
t+1∫t
∣∣u(s) − ub(s) − (u∞ − ub∞)∣∣2H 1
0 (Ω)ds ≤ C.
Proof. First, we formulate (1.1) and (3.1) as follows:
φ(1 − w(t)
)(ut (t) − (ub)t (t)
) + φ(1 − w(t)
)(ub)t (t)
− φ′(1 − w(t))wt(t)u(t) − �
(u(t) − ub(t)
) = −w0u(t)w(t) + �ub(t),
and
−�(u∞ − ub∞) + φ′(1 − w∞)w∞u2 = −w0u∞w∞ + �ub∞.
∞
2148 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
Here, we note that wt(t) = −w(t)u(t) a.e. t ∈ [0, T ]. By the subtraction of the above two equa-tions, we have
φ(1 − w(t)
)(ut (t) − (ub)t (t)
) + φ(1 − w(t)
)(ub)t (t)
+ (φ′(1 − w(t)
)w(t)u2(t) − φ′(1 − w∞)w∞u2∞
)− �
(u(t) − ub(t) − (u∞ − ub∞)
)= −w0u(t)w(t) + w0u∞w∞ + �
(ub(t) − ub∞
).
Next, by multiplying the above equation by u − ub − (u∞ − ub∞), integrating on Ω and using Green’s formula, we obtain
(φ(1 − w(t)
)(ut (t) − (ub)t (t)
), u(t) − ub(t) − (u∞ − ub∞)
)H
+ (φ(1 − w(t)
)(ub)t (t), u(t) − ub(t) − (u∞ − ub∞)
)H
+ (φ′(1 − w(t)
)w(t)u2(t) − φ′(1 − w∞)w∞u2∞, u(t) − ub(t) − (u∞ − ub∞)
)H
+ ∣∣∇(u(t) − ub(t) − (u∞ − ub∞)
)∣∣2H
= (−w0u(t)w(t) + w0u∞w∞ + �(ub(t) − ub∞
), u(t) − ub(t) − (u∞ − ub∞)
)H
. (4.13)
Now, we set U(t) = u(t) − ub(t) for t ∈ [0, T ] and U∞ = u∞ − ub∞. Here, we note that
(φ(1 − w(t)
)Ut(t),U(t) − U∞
)H
= 1
2
d
dt
∫Ω
φ(1 − w(t)
)∣∣U(t) − U∞∣∣2
dx
− 1
2
∫Ω
φ′(1 − w(t))w(t)u(t)
∣∣U(t) − U∞∣∣2
dx. (4.14)
Also, we separate the third term of the left-hand side of (4.13) as follows:
(φ′(1 − w(t)
)w(t)u2(t) − φ′(1 − w∞)w∞u2∞,U(t) − U∞
)H
=∫Ω
[φ′(1 − w(t)
) − φ′(1 − w∞)]w(t)u2(t)
(U(t) − U∞
)dx
+∫Ω
φ′(1 − w∞)(w(t) − w∞
)u2(t)
(U(t) − U∞
)dx
+∫
φ′(1 − w∞)w∞(u2(t) − u2∞
)(U(t) − U∞
)dx. (4.15)
Ω
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2149
In particular, for the third term of (4.15), it follows from (A2) that
∫Ω
φ′(1 − w∞)w∞(u2(t) − u2∞
)(U(t) − U∞
)dx
≥∫Ω
φ′(1 − w∞)w∞(u(t) + u∞
)(u(t) − u∞
)(−ub(t) + ub∞)dx
=∫Ω
φ′(1 − w∞)w∞(u(t) + u∞
)(U(t) − U∞
)(−ub(t) + ub∞)dx
−∫Ω
φ′(1 − w∞)w∞(u(t) + u∞
)∣∣ub(t) − ub∞∣∣2
dx.
Therefore, using (4.14) and (4.15), we see that (4.13) becomes
1
2
d
dt
∫Ω
φ(1 − w(t)
)∣∣U(t) − U∞(t)∣∣2
dx + ∣∣∇(U(t) − U∞
)∣∣2H
≤ 1
2
∫Ω
φ′(1 − w(t))w(t)u(t)
∣∣U(t) − U∞∣∣2
dx
+∫Ω
φ(1 − w(t)
)∣∣(ub)t (t)∣∣∣∣U(t) − U∞
∣∣dx
+∫Ω
∣∣φ′(1 − w(t)) − φ′(1 − w∞)
∣∣w(t)u2(t)∣∣U(t) − U∞
∣∣dx
+∫Ω
φ′(1 − w∞)∣∣w(t) − w∞
∣∣u2(t)∣∣U(t) − U∞
∣∣dx
+∫Ω
φ′(1 − w∞)w∞(u(t) + u∞
)(U(t) − U∞
)(ub(t) − ub∞
)dx
+∫Ω
φ′(1 − w∞)w∞(u(t) + u∞
)∣∣ub(t) − ub∞∣∣2
dx
+∫Ω
(−w0u(t)w(t) + w0u∞w∞ + �(ub(t) − ub∞
))(U(t) − U∞
)dx. (4.16)
Now, we estimate each term on the right-hand side of (4.16). First, we observe from 0 ≤ u ≤ u∗on Q(T ) and 0 ≤ u∞ ≤ κ0 on Ω that
2150 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
∫Ω
φ′(1 − w(t))w(t)u(t)
∣∣U(t) − U∞∣∣2
dx
≤ c20u
∗
2η
(u(t),w2(t)
)H
+ (u∗ + 3κ0)2
2CP η
∣∣∇(U(t) − U∞
)∣∣2H
(4.17)
where η is any positive constant, and CP is a positive constant owing to the Poincaré inequality. Hereafter, CP denotes the same constant as in (4.17). Next, using (A2), we estimate the second and the third term of the right-hand side of (4.16) as follows:
∫Ω
φ(1 − w(t)
)∣∣(ub)t (t)∣∣∣∣U(t) − U∞
∣∣dx
≤ 1
2η
∣∣(ub)t (t)∣∣2H
+ ηCP
2
∣∣∇(U(t) − U∞
)∣∣2H
, (4.18)
and
∫Ω
∣∣φ′(1 − w(t)) − φ′(1 − w∞)
∣∣w(t)u2(t)∣∣U(t) − U∞
∣∣dx
≤ 2c0
∫Ω
w(t)u2(t)∣∣U(t) − U∞
∣∣dx
≤ 2c20(u
∗)3
η
(u(t),w2(t)
)H
+ ηCP
2
∣∣∇(U(t) − U∞
)∣∣2H
. (4.19)
For the fourth term of (4.16), we find
∫Ω
φ′(1 − w∞)∣∣w(t) − w∞
∣∣u2(t)∣∣U(t) − U∞
∣∣dx
≤ c20(u
∗)4
2η
∣∣w(t) − w∞∣∣2H
+ ηCP
2
∣∣∇(U(t) − U∞
)∣∣2H
(4.20)
holds. Also, we estimate the fifth and the sixth term of (4.16) as follows:
∫Ω
φ′(1 − w∞)w∞(u(t) + u∞
)(U(t) − U∞
)(ub(t) − ub∞
)dx
≤ c20
2η
(u∗ + κ0
)2∣∣ub(t) − ub∞∣∣2H
+ ηCP
2
∣∣∇(U(t) − U∞
)∣∣2H
, (4.21)
and
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2151
∫Ω
φ′(1 − w∞)w∞(u(t) + u∞
)∣∣ub(t) − ub∞∣∣2
dx
≤ c0(u∗ + κ0
)∣∣ub(t) − ub∞∣∣2H
. (4.22)
For the last term of the right-hand side of (4.16), we derive∫Ω
(−w0u(t)w(t) + w0u∞w∞ + �(ub(t) − ub∞
))(U(t) − U∞
)dx
=∫Ω
(−w0u(t)w(t) + w0u∞w∞)(
U(t) − U∞)dx
−∫Ω
∇(ub(t) − ub∞
) · ∇(U(t) − U∞
)dx
≤ 1
2η
∣∣−w0u(t)w(t) + w0u∞w∞∣∣2H
+ ηCP
2
∣∣∇(U(t) − U∞
)∣∣2H
+ 1
2
∣∣∇(ub(t) − ub∞
)∣∣2H
+ 1
2
∣∣∇(U(t) − U∞
)∣∣2H
. (4.23)
By adding (4.17)–(4.23), we have from (4.16), for a.e. t ∈ (0, T ),
1
2
d
dt
∫Ω
φ(1 − w(t)
)∣∣U(t) − U∞∣∣2
dx
+(
1
2− 5
2ηCP − (u∗ + 3κ0)
2CP
4η
)∣∣∇(U(t) − U∞
)∣∣2H
≤(
c20u
∗
4η+ 2c2
0(u∗)3
η
)(u(t),w2(t)
)H
+ 1
2η
∣∣(ub)t (t)∣∣2H
+ c20(u
∗)4
2η
∣∣w(t) − w∞∣∣2H
+(
c20
2η
(u∗ + κ0
)2 + c0(u∗ + κ0
))∣∣ub(t) − ub∞∣∣2H
+ 1
2η
∣∣−w0u(t)w(t) + w0u∞w∞∣∣2H
+ 1
2
∣∣∇(ub(t) − ub∞
)∣∣2H
. (4.24)
Here, we take η0 > 0 such that δ0 := 1/2 − (5η0CP )/2 − ((u∗ + 3κ0)2CP η0)/4 > 0, and put
m0 := (c20(u
∗)4 + 1)/2η0 + (c20(u
∗ + κ0)2)/2η0 + c0(u
∗ + κ0). Then, using (4.5) and integrating (4.24) over [t, t + 1] for any t > 0
t+1∫t
∣∣∇(U(t) − U∞
)∣∣2H
+(
c20u
∗
8η0+ c2
0(u∗)3
η0
)∫Ω
w2(t + 1)dx
≤ m0
δ0
( t+1∫ ∣∣(ub)t (t)∣∣2H
ds +t+1∫ ∣∣w(s) − w∞
∣∣2H
ds +t+1∫ ∣∣ub(t) − ub∞
∣∣2H
ds
t t t
2152 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
+t+1∫t
∣∣−w0u(t)w(s) + w0u∞w∞∣∣2H
ds +t+1∫t
∣∣∇(ub(t) − ub∞(t)
)∣∣2H
ds
)
+ φ0
∫Ω
∣∣u0 − ub(0) − (u∞ − ub∞)∣∣2
dx +(
c20u
∗
8η0+ c2
0(u∗)3
η0
)∫Ω
w2(t)dx. (4.25)
Consequently, the right-hand side of (4.25) is bounded by (A3)–(A5), w∞ < 1, 0 ≤ u ≤ u∗ a.e. on Q(T ) and u∞ ≤ κ0 a.e. on Q(T ), so that Lemma 2 is proved.
5. Proof of Theorem 2
Now, for every (tn) ⊂ [0, ∞) satisfying that tn is non-decreasing with tn → ∞ as n → ∞, we define
wn(t) := w(tn + t), un(t) := u(tn + t), ubn(t) := ub(tn + t) for any t ∈ [0,1],
where u is a unique solution of (P) and ub. Then, from the definition of un, we see that un is a solution of the following problem:
∂
∂t
[φ(1 − wn(t)
)un(t)
] − �un(t) = −w0un(t)wn(t) a.e. in Q(1),
un = ubn a.e. on S(1),
un(0) = u(tn) in Ω.
By Lemmas 2 and 3, we see that
1∫0
∣∣un(s) − ubn(s) − (u∞ − ub∞)∣∣2H 1
0 (Ω)ds
=tn+1∫tn
∣∣un(t) − ub(t) − (u∞ − ub∞)∣∣2H 1
0 (Ω)dt ≤ C1, (5.1)
and
|unt |2L2(0,1;H)+ sup
t∈[0,1]
∣∣∇un(t)∣∣2H
+ |�un|2L2(0,1;H)≤ C2. (5.2)
Here, we note from [8, Lemma 3.7.1] that the following inequality holds:
|z|H 2(Ω) ≤ C(|�z|H + |z|H
)for z ∈ H 2(Ω),
where C is a positive constant. Using this inequality, we have
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2153
1∫0
∣∣un(s)∣∣2H 2(Ω)
ds =tn+1∫tn
∣∣u(t)∣∣2H 2(Ω)
dt
≤ 2C2
( tn+1∫tn
∣∣�u(t)∣∣2H
dt +tn+1∫tn
∣∣u(t)∣∣2H
dt
)≤ 2C2(C2 + (
u∗)2|Ω|).
Hence, we see that un is bounded in L2(0, 1; H 2(Ω)). By (5.1), (5.2), and the above result, we can take a subsequence {unk} ⊂ {un}, for which the following convergences hold as k → ∞:
u(tnk) → ξ strongly in H and weakly in H 1(Ω),⎧⎨⎩
(unk)t → 0 strongly in L2(0,1;H),
unk → ξ strongly in C([0,1];H )
, weakly in W 1,2(0,1;H) ∩ L2(0,1;H 2(Ω)
),
unk(t) → ξ strongly in H, weakly in H 1(Ω) and uniformly in t ∈ [0,1],unk − ubnk − (u∞ − ub∞) → u∗ weakly in L2(0,1;H 1
0 (Ω)).
Indeed, the strong convergence of (un)t holds as obtained by the following:
1∫0
∣∣(un)t (t)∣∣2H
dt =tn+1∫tn
∣∣ut (τ )∣∣2H
dτ
=tn+1∫0
∣∣ut (τ )∣∣2H
dτ −tn∫
0
∣∣ut (τ )∣∣2H
dτ.
Here, by setting f (t) := ∫ t
0 |ut (s)|2H ds for any t > 0 f is non-decreasing function with respect to t and is bounded by Lemma 2. Hence, for the above sequences {tn} and {tn + 1} there exista := limn→+∞ f (tn) and b := limn→+∞ f (tn + 1). Then, we easily see that a ≤ b. Also, since for any n ∈ N , there exists k ∈ N such that
f (tn + 1) =tn+1∫0
∣∣ut (τ )∣∣2H
dτ ≤tk∫
0
∣∣ut (τ )∣∣2H
dτ ≤ a,
then b ≤ a holds, and hence a = b. Finally, by n → ∞, we obtain that (un)t → 0 strongly in L2(0, 1; H).
Next, the strong convergence in C([0, 1]; H) of un is obtained from the Ascoli–Arzela the-orem. Also, for the weak convergence in H 1(Ω) of un(t), we first see from (5.2) that for each t ∈ [0, 1], there exists ξ (t) ∈ H 1(Ω) such that
un(t) → ξ (t) strongly in H, weakly in H 1(Ω) as n → ∞.
2154 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
Then, using the strong convergence of u(tn), for each t ∈ [0, 1] and any v ∈ H , we have
(v, ξ (t) − ξ
)H
≤ (v, ξ (t) − un(t)
)H
+ (v,un(t) − u(tn)
) + (v,u(tn) − ξ
)H
≤ ∣∣(v, ξ (t) − un(t))H
∣∣ + |v|Htn+1∫tn
∣∣ut (τ )∣∣2H
dτ + ∣∣(v,u(tn) − ξ)H
∣∣.
Therefore, by n → ∞ we see that ξ (t) = ξ in H for each t ∈ [0, 1]. Moreover, similarly to the above argument, for any v ∈ H , we have
∣∣(v,un(t) − ξ)H
∣∣ ≤ |v|Htn+1∫tn
∣∣ut (τ )∣∣2H
dτ + ∣∣(v,u(tn) − ξ)H
∣∣→ 0 as n → ∞.
Here, using the fact that H 1(Ω) is dense in H and considering the dual space H ∗ of H as H , we see that H is dense in (H 1(Ω))∗. Therefore, from the above convergence, we obtain the uniformly weak convergence of un in H 1(Ω).
In each convergence of un we can see from (A3) and the strong convergence of C([0, 1]; H)
that u∗ = ξ − u∞ in L2(0, 1; H). Therefore, by letting k → ∞, we have
unk − ubnk → (u∞ − ub∞) + u∗ = ξ − ub∞ weakly in L2(0,1;H 10 (Ω)
). (5.3)
Also, since wn(t) → w∞ = w∞(x) as n → ∞ for each t ∈ (0, 1) and x ∈ Ω , by the Lebesgue convergence theorem we have
e− ∫ tn+·0 u(τ,·)dτ → w∞ strongly in L2(Q(1)
). (5.4)
Now, we prove that ξ = u∞ in H . For v ∈ C∞0 (Q(1)) we have
1∫0
(φ(1 − wnk(t)
)(unk)t (t), v(t)
)H
dt +1∫
0
(φ′(1 − wnk(t)
)wnk(t)u
2nk(t), v(t)
)H
dt
−1∫
0
(�
(unk(t) − ubnk(t)
), v(t)
)H
dt = −1∫
0
(w0unk(t)wnk(t), v(t)
)H
dt
+1∫
0
(�ubnk(t), v(t)
)H
dt. (5.5)
In (5.5), the first term on the left-hand side converges to 0 by φ ≤ 1 and the strong convergence of (unk)t . For the second term on the left-hand side of (5.5), we have
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2155
∣∣∣∣∣1∫
0
(φ′(1 − wnk(t)
)wnk(t)u
2nk(t) − φ′(1 − w∞)w∞ξ2, v(t)
)H
dt
∣∣∣∣∣
≤∣∣∣∣∣
1∫0
((φ′(1 − wnk(t)
) − φ′(1 − w∞))wnk(t)u
2nk(t), v(t)
)H
dt
∣∣∣∣∣
+∣∣∣∣∣
1∫0
(φ′(1 − w∞)
(wnk(t) − w∞
)u2
nk(t), v(t))H
dt
∣∣∣∣∣
+∣∣∣∣∣
1∫0
(φ′(1 − w∞)w∞
(u2
nk(t) − ξ2), v(t))H
dt
∣∣∣∣∣. (5.6)
As to the first term of the right-hand side in (5.6), from (5.4), we see that wnk → w∞ a.e. in Q(1)
as k → ∞ and hence, by φ ∈ C1 in (A2),
φ′(1 − wnk) → φ′(1 − w∞) a.e. in Q(1) as k → ∞.
Therefore, since wnk and unk are bounded, by the Lebesgue convergence theorem we have
∣∣∣∣∣1∫
0
((φ′(1 − wnk(t)
) − φ′(1 − w∞))wnk(t)u
2nk(t), v(t)
)H
dt
∣∣∣∣∣≤
∫Q(1)
∣∣(φ′(1 − wnk(t)) − φ′(1 − w∞)
)wnk(t)u
2nk(t)
∣∣v(t)dxdt
≤ (u∗)2 sup
K
|v|∫K
∣∣φ′(1 − wnk(t)) − φ′(1 − w∞)
∣∣dxdt
≤ (u∗)2 sup
K
|v|∫
Q(1)
∣∣φ′(1 − wnk(t)) − φ′(1 − w∞)
∣∣dxdt → 0 as k → ∞,
where K is the compact support of the function v. For the second and the third term of the right-hand side in (5.6), we can see that two terms converge to 0 as k → ∞ by using (5.4) and by the strong convergence of unk in C([0, 1]; H), respectively. Therefore, from the above results, we obtain
1∫0
(φ′(1 − wnk(t)
)wnk(t)u
2nk(t), v(t)
)H
dt
→1∫ (
φ′(1 − w∞)w∞ξ2, v(t))H
dt as k → ∞. (5.7)
0
2156 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
Next, for the third integral of the left-hand side in (5.5), we see from (5.4) that
−1∫
0
(�
(unk(t) − ubnk(t)
), v(t)
)H
dt =1∫
0
∫Ω
∇(unk(t) − ubnk(t)
) · ∇v(t)dxdt
→1∫
0
∫Ω
∇(ξ − ub∞) · ∇v(t)dxdt = −1∫
0
(�(ξ − ub∞), v(t)
)H
dt as k → ∞. (5.8)
Also, from the strong convergence of unk , (5.4) and ub −ub∞ ∈ L2(0, ∞; H 2(Ω)), the following convergence for two terms of the right-hand side in (5.5) holds:
−1∫
0
(w0unk(t)wnk(t), v(t)
)H
dt +1∫
0
(�ubnk(t), v(t)
)H
dt
→ −1∫
0
(w0ξw∞, v(t)
)H
dt +1∫
0
(�ub∞, v(t)
)H
dt as k → ∞. (5.9)
Finally, by k → ∞, we derive from (5.7)–(5.9) that ξ − ub∞ ∈ H 10 (Ω) and
1∫0
(−�(ξ − ub∞), v(t))H
dt +1∫
0
(φ′(1 − w∞)ξ2w∞, v(t)
)H
dt
=1∫
0
(−w0ξw∞ + �ub∞, v(t))H
dt.
This implies that
−�ξ + φ′(1 − w∞)ξ2w∞ = −w0ξw∞ a.e. on Ω, (5.10)
and
ξ = ub∞ a.e. on Γ.
Finally, by the uniqueness of (P)∞ in Lemma 1, we see that ξ = u∞ in H , and also see that the whole sequence un converges to ξ in H . Thus, Theorem 2 is proved.
Now, we prove Corollary 1. For the above sequence {uk} := {unk}, using Lemma 2 we can extract a subsequence {ukm} such that
�ukm → 0 strongly in L2(0,1;H) as m → ∞. (5.11)
K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158 2157
Therefore, by combining it with (5.10) and (5.11), and the other convergence of un, we see that u∞ satisfies
φ′(1 − w∞)u2∞w∞ = −w0u∞w∞ a.e. on Ω. (5.12)
Now, we first consider non-vanishing ub∞ on Γ . By (5.10) and (5.12), we conclude that u∞is a solution of the following Dirichlet problem:
−�u∞ = 0 in Ω, u∞ = ub∞ on Γ.
Also, by (5.12) and (A2), we see that u∞ and w∞ satisfy
u∞w∞ = 0 or u∞ = − w0
φ′(1 − w∞)a.e. in Ω.
Then, since u∞ and φ′ are positive, and w0 is strictly positive, we get u∞w∞ = 0, i.e., u∞ = 0 or w∞ = 0. Now, we assume that u∞ = 0 a.e. in Ω . From the result of boundary value problems of elliptic type, we see that u∞ ∈ H 2(Ω), and therefore u∞ ∈ C(Ω). Now, since u∞ = 0 a.e. in Ω , we see that u∞ = 0 in Ω . Therefore, we have u∞ ∈ H 1
0 (Ω) and ub∞ = 0 on Γ by u∞ − ub∞ ∈H 1
0 (Ω). This contradicts the condition assumed for ub∞. Therefore, we conclude that w∞ = 0. This implies that
∫ t
0 u(τ)dτ keeps increasing along with time t , and does not remain finite for any x ∈ Ω .
In contrast, for ub∞ = 0 a.e. on Γ , we see from (5.10) and (5.12) that u∞ is a solution of the following Dirichlet problem:
−�u∞ = 0 in Ω, u∞ = 0 on Γ. (5.13)
Therefore, we see that u∞ = 0.To prove statement (ii) of Corollary 1 completely we present the decay estimate. Using (4.16)
with ub = ub∞ = u∞ = u∞w∞ = 0, we get
(φ(1 − w(t)
)ut (t), u(t)
)H
+ (φ′(1 − w(t)
)w(t)u2(t), u(t)
)H
+ ∣∣∇u(t)∣∣2H
= (−w0u(t)w(t), u(t))H
for a.e. t > 0. (5.14)
Note that the right-hand side is negative since u is positive in Q(T ). Therefore, we obtain
1
2
d
dt
∫Ω
φ(1 − w(t)
)∣∣u(t)∣∣2
dx
+ 1
2
∫Ω
φ(1 − w(t)
)w(t)
∣∣u(t)∣∣2
dx + ∣∣∇u(t)∣∣2H
≤ 0 for a.e. t > 0. (5.15)
Since the second term on the left-hand side of (5.15) is positive because of (A2) and the positive-ness of u, from (5.15) we have
2158 K. Kumazaki / J. Differential Equations 257 (2014) 2136–2158
1
2
d
dt
∫Ω
φ(1 − w(t)
)∣∣u(t)∣∣2
dx + ∣∣∇u(t)∣∣2H
≤ 0. (5.16)
Here, we remark that the second term on the left-hand side of (5.16) becomes
∣∣∇u(t)∣∣2H
≥ 1
cP
∣∣u(t)∣∣2H
≥ 1
cP
∫Ω
φ(1 − w(t)
)∣∣u(t)∣∣2H
.
Now, by setting
I (t) :=∫Ω
φ(1 − w(t)
)∣∣u(t)∣∣2
dx for t > 0,
we see from (5.16) that
d
dt
(1
2I (t)e
2cP
t)
≤ 0 for a.e. t > 0. (5.17)
Consequently, by integrating (5.17) over [0, t], we obtain
I (t) ≤ I (0)e− 2
cPt = |u0|2H e
− 2cP
tfor t ≥ 0.
Therefore, we obtain the decay estimate for statement (ii) of Corollary 1. Thus, statements (i) and (ii) of Corollary 1 are completely proved.
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