Citation: C Singh, X Wang, Y Morsi, CS Wong. Importance of stent-graft design for aortic arch aneurysm repair[J]. AIMS Bioengineering, 2017, 4(1): 133-150. doi: 10.3934/bioeng.2017.1.133
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Both natural rubber and synthetic elastomers are widely used in engineering applications, such as connecting components for offshore wind farms that are nowadays intensively used to harvesting energy. Due to the growing importance of good theoretical estimations of durability properties of elastomeric parts, a well-trusted modelling of the material behaviour is an essential prerequisite so that numerical prediction is available for large times. Rubber-like materials exhibit a strongly non-linear behaviour characterized by a joint large strain and a non-linear stress-strain response [3]. In the presence of aggressive environments (like those where large bodies of water are present), the nonlinear behavior of elastomers, which can be in fact studied in controlled laboratory conditions, alters non-intuitively [2]. The main reason for this situation is that the porous structure of rubber-like materials allows small particles to diffuse inside. Ingressing particles, often ionically charged, accumulate and interact with the internal solid fabric. Such interactions lead to unwanted alterations of the originally designed mechanical behavior; see e.g., [8] for more context on this physical problem. From the modeling and mathematical upscaling point of view, the derivation of the correct model equations for such a problem setting is open.
In our recent works [1,5], and [6], we considered a class of one-dimensional free boundary problems to model the diffusants penetration in dense and foam rubbers. The use of kinetic-type free boundary conditions is meant to avoid the explicit description of the mechanics of the involved materials and how the constitutive laws (in particular, the stress tensor) are affected by the presence of diffusants. One can use such free boundary formulations to derive theoretically-grounded practical estimations of the position of the diffusants penetration front and compare them with laboratory experiments; e.g., [7] where a parameter identification exercise has been performed in this context.
The main novelty introduced in the present paper is that the growth rate of the region filled with diffusants is described by an ordinary differential equation that includes the effect of breaking its growth. The breaking mechanism should be seen here as the hyperelastic response to a too fast diffusion penetration. Such a mechanism introduces non-monotonicity in our problem formulation. This makes us wonder whether this type of problems has a chance to be well-posed and, if true, in which sense?
To address the question, we consider the following one-dimensional free boundary problem show in Eqs (1.1)–(1.5). Let t∈[0,T] be the time variable for T>0. The problem is to find a curve z=s(t) on [0,T] and a function u defined on the set Qs(T):={(t,z)|0<t<T, 0<z<s(t)} such that the following system of equations is satisfied:
ut−uzz=0 for (t,z)∈Qs(T), | (1.1) |
−uz(t,0)=β(b(t)−γu(t,0)) for t∈(0,T), | (1.2) |
−uz(t,s(t))=σ(u(t,s(t)))st(t) for t∈(0,T), | (1.3) |
st(t)=a0(σ(u(t,s(t)))−αs(t)) for t∈(0,T), | (1.4) |
s(0)=s0,u(0,z)=u0(z) for z∈[0,s0], | (1.5) |
where β, γ, a0 and α are given positive constants, b is a given function on [0,T] and s0 and u0 are the initial data. In Eq (1.3) and Eq (1.4), σ is the function on R called the positive part, i.e. it is given by Eq (1.6),
σ(r)={r if r≥0,0 if r<0. | (1.6) |
The problem in Eqs (1.1)–(1.5), which is denoted here by (P)(u0,s0,b), was proposed by Nepal S et al., in [7] as a mathematical model describing the migration of diffusants into rubber. The set [0,s(t)] represents the region occupied by a solvent occupying the one-dimensional pore [0,∞), where s=s(t) is the position of moving interface of the region and u=u(t,z) represents the content of the diffusant at the place z∈[0,s(t)] at time t>0.
From the mathematical point of view, (P)(u0,s0,b) is a one-phase free boundary problem with Robin-type boundary conditions at both boundaries. Such kind of problem structure has been considered in [5,6], and [1]. In our recent work [1], as a simplified setting for (P)(u0,s0,b), we consider the case α=0 and prove the existence and uniqueness of a globally-in-time solution. Furthermore, what concerns the large time behavior of a solution, we show that the free boundary s goes to infinity as time elapses.
In this paper, we consider the case α>0. As already anticipated, the difficulty in this case is the lack of the monotonicity of the boundary condition imposed on the moving boundary. Indeed, similarly to the proof of the existence result in [1], by introducing a variable ˜u(t,y)=u(t,ys(t)) for (t,y)∈Q(T):=(0,T)×(0,1) we transform (P)(u0,s0,b) into a problem on the fixed domain Q(T). Let denote this problem by (PC)(˜u0,s0,b), where ˜u0(y)=u0(s0y) for y∈[0,1]. Moreover, to find a solution (PC)(˜u0,s0,b) we first consider the following auxiliary problem (AP)(˜u0,s,b): For a given function s(t) on [0,T], find ˜u(t,y) satisfying:
˜ut(t,y)−1s2(t)˜uyy(t,y)=yst(t)s(t)˜uy(t,y) for (t,y)∈Q(T),−1s(t)˜uy(t,0)=β(b(t)−γ˜u(t,0)) for t∈(0,T),−1s(t)˜uy(t,1)=a0σ(˜u(t,1))(σ(˜u(t,1))−αs(t)) for t∈(0,T),˜u(0,y)=˜u0(y) for y∈[0,1]. |
Here, we set gα(r)=a0σ(r)(σ(r)−αs(t)) for r∈R and t∈(0,T). In the case α=0, due to the function σ, g0(r)=a0(σ(r))2 is monotonically increasing with respect to r. Due to this fact, we can use the theory of evolution equations governed by subdifferentials of convex functions (cf. [4] and references therein) and find a solution ˜u of (AP)(˜u0,s,b) in the case α=0. However, looking now at the case α>0, since gα(r) is not monotonic anymore with respect to r, we cannot apply the theory of evolution equations directly. Hence, the existence of a solution to (AP)(˜u0,s,b) in the case α>0 is not at all clear. As far as we are aware, the type of free boundary problems is novel. We refer the reader to [9] (and related references) for explanations of how and why free-boundary problems can be used to model fast transitions in materials.
The purpose of this paper is to establish a methodology to deal with the existence of locally-in-time solutions to (PC)(u0,s0,b) in the case α>0. For this to happen, we consider weak solutions to (PC)(˜u0,s0,b). The definition of our concept of weak solutions is explained in Section 2. As next step, we proceed in the following way: For a given function η on Q(T) and ε>0, we consider in Section 3 a smooth approximation ηε of η and construct a solution ˜u to the following auxiliary problem: For given s on [0,T], find ˜u(t,y) such that it holds
˜ut(t,y)−1s2(t)˜uyy(t,y)=yst(t)s(t)ηy(t,y) for (t,y)∈Q(T),−1s(t)˜uy(t,0)=β(b(t)−γ˜u(t,0)) for t∈(0,T),−1s(t)˜uy(t,1)=a0(σ(˜u(t,1)))2−ασ(ηε(t))s(t)) for t∈(0,T),˜u(0,y)=˜u0(y) for y∈[0,1]. |
The plan is to obtain uniform estimates of solutions with respect to ε. After that, by the limiting process ε→0 and benefitting of Banach's fixed point theorem, we wish to construct a weak solution ˜u of (AP)(˜u0,s,b). In Section 4, we define a solution mapping ΓT between s and the weak solution ˜u of (AP)(˜u0,s,b). We show that, for some T′≤T, the mapping ΓT′ is a contraction mapping on a suitable function space. Finally, using Banach's fixed point theorem, we prove the existence and uniqueness of a weak solution to the coupled problem (s,˜u) of (PC)(˜u0,s0,b) on [0,T′]. Additionally, we show that the maximal length of the free boundary s is a priori determined by given parameters, the time derivative of the function b, and initial data s0. Moreover, we guarantee that the solution ˜u is non-negative and bounded on Q(T).
In this paper, we use the following notations. We denote by |⋅|X the norm for a Banach space X. The norm and the inner product of a Hilbert space H are denoted by |⋅|H and (⋅,⋅)H, respectively. Particularly, for Ω⊂R, we use the notation of the usual Hilbert spaces L2(Ω), H1(Ω), and H2(Ω). Throughout this paper, we assume the following parameters and functions:
(A1) a0, α, γ, β and T are positive constants.
(A2) s0>0 and u0∈L∞(0,s0) such that u0≥0 on [0,s0].
(A3) b∈W1,2(0,T) with b≥0 on (0,T). Also, we set
b∗=max{max0≤t≤Tb(t),γ|u0|L∞(0,s0)}. |
It is convenient to consider (P)(u0,s0,b) transformed into a non-cylindrical domain. Let T>0. For given s∈W1,2(0,T) with s(t)>0 on [0,T], we introduce the following new function obtained by the change of variables and fix the moving domain:
˜u(t,y)=u(t,ys(t)) for (t,y)∈Q(T):=(0,T)×(0,1). | (2.1) |
By using the function ˜u, (P)(u0,s0,b) becomes the following problem (PC)(˜u0,s0,b) posed on the non-cylindrical domain Q(T):
˜ut(t,y)−1s2(t)˜uyy(t,y)=yst(t)s(t)˜uy(t,y) for (t,y)∈Q(T), | (2.2) |
−1s(t)˜uy(t,0)=β(b(t)−γ˜u(t,0)) for t∈(0,T), | (2.3) |
−1s(t)˜uy(t,1)=σ(˜u(t,1))st(t) for t∈(0,T), | (2.4) |
st(t)=a0(σ(˜u(t,1))−αs(t)) for t∈(0,T), | (2.5) |
s(0)=s0, | (2.6) |
˜u(0,y)=u0(ys(0))(:=˜u0(y)) for y∈[0,1]. | (2.7) |
Here, we introduce the following function space: For T>0, we put H=L2(0,1), X=H1(0,1), V(T)=L∞(0,T;H)∩L2(0,T;X) and |z|V(T)=|z|L∞(0,T;H)+|zy|L2(0,T;H) for z∈V(T). Note that V(T) is a Banach space with the norm |⋅|V(T). Also, we denote by X∗ and ⟨⋅,⋅⟩X the dual space of X and the duality pairing between X and X∗, respectively.
We define now our concept of solutions to (PC)(˜u0,s0,b) on [0,T] in the following way:
Definition 2.1. For T>0, let s be a function on [0,T] and ˜u be a function on Q(T), respectively. We call that a pair (s,˜u) is a solution of (P)(˜u0,s0,b) on [0,T] if the next conditions (S1)–(S4) hold:
(S1) s∈W1,∞(0,T), s>0 on [0,T], ˜u∈W1,2(0,T;X∗)∩V(T).
(S2)
∫T0⟨˜ut(t),z(t)⟩Xdt+∫Q(T)1s2(t)˜uy(t)zy(t)dydt+∫T01s(t)σ(˜u(t,1))st(t)z(t,1)dt−∫T01s(t)β(b(t)−γ˜u(t,0))z(t,0)dt=∫Q(T)yst(t)s(t)˜uy(t)z(t)dydt for z∈V(T). |
(S3) st(t)=a0(σ(˜u(t,1))−αs(t)) for a.e. t∈(0,T).
(S4) s(0)=s0 and ˜u(0,y)=˜u0(y) for a.e. y∈[0,1].
The main result of this paper is the existence and uniqueness of a locally-in-time solution of (PC)(˜u0,s0,b). We state this result in the next theorem.
Theorem 2.2. Let T>0. If (A1)–(A3) hold, then there exists T∗≤T such that (PC)(˜u0,s0,b) has a unique solution (s,˜u) on [0,T∗]. Moreover, the function ˜u is non-negative and bounded on Q(T).
In this section,
we consider the following auxiliary problem (AP1)(˜u0,s,b): For T>0 and s∈W1,2(0,T) with s>0 on [0,T]
˜ut(t,y)−1s2(t)˜uyy(t,y)=yst(t)s(t)˜uy(t,y) for (t,y)∈Q(T), | (3.1) |
−1s(t)˜uy(t,0)=β(b(t)−γ˜u(t,0)) for t∈(0,T), | (3.2) |
−1s(t)˜uy(t,1)=a0σ(˜u(t,1))(σ(˜u(t,1))−αs(t)) for t∈(0,T), | (3.3) |
˜u(0,y)=˜u0(y) for y∈[0,1], | (3.4) |
where σ is the same function as in Eq (1.6).
Definition 3.1. For T>0, let ˜u be a function on Q(T), respectively. We call that a function ˜u is a solution of (AP1)(˜u0,s,b) on [0,T] if the conditions (S'1)–(S'3) hold:
(S'1) ˜u∈W1,2(0,T;X∗)∩V(T).
(S'2)
∫T0⟨˜ut(t),z(t)⟩Xdt+∫Q(T)1s2(t)˜uy(t)zy(t)dydt+∫T0a0s(t)σ(˜u(t,1))(σ(˜u(t,1))−αs(t))z(t,1)dt−∫T01s(t)β(b(t)−γ˜u(t,0))z(t,0)dt=∫Q(T)yst(t)s(t)˜uy(t)z(t)dydt for z∈V(T). |
(S'3) ˜u(0,y)=˜u0(y) for y∈[0,1].
Now, we introduce the following problem (AP2)(˜u0,s,η,b): For s∈W1,∞(0,T) with s>0 on [0,T] and η∈V(T)
˜ut(t,y)−1s2(t)˜uyy(t,y)=yst(t)s(t)ηy(t,y) for (t,y)∈Q(T), | (3.5) |
−1s(t)˜uy(t,0)=β(b(t)−γ˜u(t,0)) for t∈(0,T), | (3.6) |
−1s(t)˜uy(t,1)=a0((σ(˜u(t,1))2−ασ(η(t,1))s(t)) for t∈(0,T), | (3.7) |
˜u(0,y)=˜u0(y) for y∈[0,1], | (3.8) |
The definition of solutions of (AP2)(˜u0,s,η,b) is Definition 3.1 with (S'2) replaced by (S''2), which now reads
(S''2):
∫T0⟨˜ut(t),z(t)⟩Xdt+∫Q(T)1s2(t)˜uy(t)zy(t)dydt+∫T0a0s(t)((σ(˜u(t,1))2−ασ(η(t,1))s(t))z(t,1)dt−∫T01s(t)β(b(t)−γ˜u(t,0))z(t,0)dt=∫Q(T)yst(t)s(t)ηy(t)z(t)dydt for z∈V(T). |
First, we construct a solution ˜u of (AP2)(˜u0,s,η,b) on [0,T]. To do so, for each ε>0 we solve the following problem (AP2)ε(˜u0ε,s,η,b):
˜ut(t,y)−1s2(t)˜uyy(t,y)=yst(t)s(t)ηy(t,y) for (t,y)∈Q(T),−1s(t)˜uy(t,0)=β(b(t)−γ˜u(t,0)) for t∈(0,T),−1s(t)˜uy(t,1)=a0((σ(˜u(t,1))2−ασ((ρε∗η)(t,1))s(t)) for t∈(0,T),˜u(0,y)=˜u0ε(y) for y∈[0,1]. |
Here ρε is a mollifier with support [−ε,ε] in time and ρε∗η is the convolution of ρε with η:
(ρε∗η)(t,1)=∫∞−∞ρε(t−s)¯η(s,1)ds for t∈[0,T], | (3.9) |
where ¯η(t,1)=η(t,1) for t∈(0,T) and vanishes otherwise. Also, ˜u0ε is an approximation function of ˜u0 such that {˜u0ε}⊂X, |˜u0ε|H≤|˜u0|H+1 and ˜u0ε→˜u0 in H as ε→0.
Now, we define a family {ψt}t∈[0,T] of time-dependent functionals ψt:H→R∪{+∞} for t∈[0,T] as follows:
ψt(u):={12s2(t)∫10|uy(y)|2dy+1s(t)∫u(1)0a0((σ(ξ))2dξ−a0αu(1)σ((ρε∗η)(t,1))−1s(t)∫u(0)0β(b(t)−γξ)dξ if u∈D(ψt),+∞otherwise, |
where D(ψt)=X for t∈[0,T]. Here, we show the property of ψt.
Lemma 3.2. Let s∈W1,2(0,T) with s>0 on [0,T], η∈V(T) and assume (A1)–(A3). Then the following statements hold:
(1) There exists positive constants C′0, C′1 and C″ such that the following inequalities hold:
(i) |u(y)|2≤C′y(ψt(u)+1)foru∈D(ψt)andy=0,1,(ii) 12s2(t)|uy|2H≤C″(ψt(u)+1)foru∈D(ψt), |
(2) For t∈[0,T], the functional ψt is proper, lower semi-continuous, and convex on H.
Proof. We fix ε>0 and let t∈[0,T], u∈D(ψt) and put l=max0≤t≤T|s(t)| and ηε(t)=(ρε∗η)(t,1) for t∈[0,T]. If u(1)<0, 1s(t)∫u(1)0a0(σ(ξ))2dξ=0. If u(1)≥0, then it holds
1s(t)∫u(1)0a0(σ(ξ))2dξ−a0αu(1)σ(ηε(t))=a0s(t)13u3(1)−a0αu(1)σ(ηε(t))≥a03s(t)u3(1)−13δ3u3(1)−23δ3/2(a0ασ(ηε(t)))32, |
where δ is an arbitrary positive number. By using the fact that σ(r)≤|r| for r∈R and taking a suitable δ=δ0 we have
1s(t)∫u(1)0a0(σ(ξ))2dξ−a0αu(1)ηε(t)≥−23δ3/20(a0αη∗ε)32, | (3.10) |
where η∗ε=max0≤t≤T|ηε(t)|. Moreover, for both cases u(0)<0 and u(0)≥0, we observe that
−1s(t)∫u(0)0β(b(t)−γξ)dξ=βs(t)[γ2u2(0)−b(t)u(0)]≥βγ2lu2(0)−βb∗au(0)≥βγ4lu2(0)−βlγ(b∗a)2. | (3.11) |
Accordingly, if u(1)<0, then we have that,
ψt(u)≥12s2(t)∫10|uy(y)|2dy+βγ4lu2(0)−βlγ(b∗a)2. | (3.12) |
If u(1)≥0, then, by Eq (3.10) and Eq (3.11) we also have that:
ψt(u)≥12s2(t)∫10|uy(y)|2dy−23δ3/20(a0αη∗ε)32+βγ4lu2(0)−βlγ(b∗a)2. | (3.13) |
In Eq (3.12) and Eq (3.13), since the first term in the right-hand side is non-negative we can find a positive constant C′0 that (i) of Lemma 3.2 for y=0 holds. In addition, by βγ4lu2(0)≥0 we also see that (ii) of Lemma 3.2 holds. Moreover, it holds that:
|u(1)|2=|∫10uy(y)dy+u(0)|2≤2(∫10|uy(y)|2dy+|u(0)|2)≤2(2l22s2(t)∫10|uy(y)|2dy+|u(0)|2). | (3.14) |
Therefore, by Eq (3.14) and the result for |u(0)|2 and |uy|2H we see that there exists a positive constant C′1 such that (i) of Lemma 3.2 hold for y=1.
Next, we prove statement (2). For t∈[0,T] and r∈R, put
g1(s(t),ηε(t),r)=1s(t)∫r0a0(σ(ξ))2dξ−a0αrσ(ηε(t)),g2(s(t),b(t),r)=−1s(t)∫r0β(b(t)−γξ)dξ. |
Then, by g1(s(t),ηε(t),r)=−a0αrσ(ηε(t)) for r≤0, g1(s(t),ηε(t),r) is a linear decreasing function for r≤0. Also, by s(t)>0 we see that,
∂2∂r2g1(s(t),ηε(t),r)=2a0s(t)r>0 for r>0,∂2∂r2g2(s(t),b(t),r)=βγs(t)>0 for r∈R. |
This means that ψt is convex on H. By using (i) and (ii) of Lemma 3.2 together with Sobolev's embedding X↪C([0,1]) in one dimensional case, it is easy to prove that the level set of ψt is closed in H, fact which ensures to the lower semi-continuity of ψt. Thus, we see that statement (2) holds.
Lemma 3.3 guarantees the existence of a solution to (AP2)ε(˜u0ε,s,η,b).
Lemma 3.3. Let T>0 and ε>0. If (A1)–(A3) hold, then, for given s∈W1,∞(0,T) with s>0 on [0,T] and η∈V(T), the problem (AP2)ε(˜u0ε,s,η,b) admits a unique solution ˜u on [0,T] such that ˜u∈W1,2(0,T;H)∩L∞(0,T;X). Moreover, the function t→ψt(˜u(t)) is absolutely continuous on [0,T].
Proof. Let ε>0 be arbitrarily fixed. By Lemma 3.2, for t∈[0,T] ψt is a proper lower semi-continuous convex function on H. From the definition of the subdifferential of ψt, for t∈[0,T], z∗∈∂ψt(u) is characterized by u, z∗∈H,
z∗=−1s2(t)uyy on (0,1),−1s(t)uy(0)=β(b(t)−γu(0)), −1s(t)uy(1)=a0((σ(u(1))2−ασ((ρε∗η)(t,1))s(t)). |
Namely, ∂ψt is single-valued. Also, we see that there exists a positive constant C such that for each t1, t2∈[0,T] with t1≤t2, and for any u∈D(ψt1), there exists ˉu∈D(ψt2) satisfying the inequality
|ψt2(ˉu)−ψt1(u)|≤C(|s(t1)−s(t2)|+|b(t1)−b(t2)|+|ηε(t1)−ηε(t2)|(1+|ψt1(u)|), | (3.15) |
where ηε(t)=(ρε∗η)(t). Indeed, we take ˉu:=u. Then, ˉu∈D(ψt2), and by the definition of ψt, it holds
ψt2(ˉu)−ψt1(u)=(12s2(t2)−12s2(t1))∫10|uy(y)|2dy+(1s(t2)−1s(t1))∫u(1)0a0(σ(ξ))2dξ+a0αu(1)(ηε(t1)−ηε(t2))−(1s(t2)−1s(t1))∫u(0)0β(b(t2)−γξ)dξ−βs(t1)(b(t2)−b(t1))u(0). | (3.16) |
We denote each term in the right-hand side by I1, I2, I3, I4 and I5. Let put l=max0≤t≤T|s(t)| and a=min0≤t≤T|s(t)|. For other than I2, by the fact that |u(y)|≤12(1+u2(y)) for y=0, 1 and Lemma 3.2 it easy to see that,
|I1|≤C1|s(t1)−s(t2)|(|ψt1(u)|+1), | (3.17) |
5∑k=3|Ik|≤C2(|ηε(t1)−ηε(t2)|+|s(t2)−s(t1)|+|b(t2)−b(t1)|)(|ψt1(u)|+1), | (3.18) |
where C1 and C2 are positive constants. For I2 by the definition of ψt it holds that,
1s(t1)∫u(1)0a0(σ(ξ))2dξ=ψt1(u)−12s2(t1)∫10|uy(y)|2dy+a0αu(1)ηε(t1)+1s(t1)∫u(0)0β(b(t1)−γξ)dξ. |
Here, we note that
|a0αu(1)ηε(t1)|≤a0αη∗ε2(1+u2(1)),|1s(t1)∫u(0)0β(b(t1)−γξ)dξ|≤1a(βb∗|u(0)|+γ2u2(0))≤1a(βb∗2+(βb∗2+γ2)u2(0)). |
where η∗ε=max0≤t≤T|ηε(t)|. Hence, by Lemma 3.2 we have that,
|I2|≤C3|s(t1)−s(t2)|(|ψt1(u)|+1), | (3.19) |
where C3 is a positive constant. As a consequence, by Eqs (3.17)–(3.19) we infer that there exists a positive constant C such that Eq (3.15) holds.
Now, (AP2)ε(˜u0ε,s,η,b) can be written into the following Cauchy problem (CP)ε:
˜ut+∂ψt(˜u(t))=yst(t)s(t)ηy(t) in H,˜u(0,y)=˜u0ε(y) for y∈[0,1]. |
Since ystsηy∈L2(0,T;H), by the general theory of evolution equations governed by time dependent subdifferentials (cf. [4]) we see that (CP)ε has a solution ˜u on [0,T] such that ˜u∈W1,2(Q(T)), ψt(˜u(t))∈L∞(0,T) and t→ψt(˜u(t)) is absolutely continuous on [0,T]. This implies that ˜u is a unique solution of (AP2)ε(˜u0ε,s,η,b) on [0,T].
As next step, we provide an uniform estimate with respect to ε on a solution ˜u of (AP2)ε(˜u0ε,s,η,b).
Lemma 3.4. Let T>0, s∈W1,∞(0,T) with s>0 on [0,T], η∈V(T) and ˜uε be a solution of (AP2)ε(˜u0ε,s,η,b) on [0,T] for each ε>0. Then, it holds that
|˜uε(t)|2H+∫t0|˜uεy(τ)|2Hdy≤M(1+|η|2V(T))fort∈[0,T]andε∈(0,1], | (3.20) |
where M=M(a0,a,β,b∗,T) is a positive constant which is independent of ε and depends on a0, a, β, b∗ and T, a=min0≤t≤Ts(t).
Proof. Let ˜uε be a solution of (AP2)ε(˜u0ε,s,η,b) on [0,T] for each ε>0.
First, it holds that
12ddt|˜uε(t)|2H−∫101s2(t)˜uεyy(t)˜uε(t)dy=∫10yst(t)s(t)ηy(t)˜uε(t)dy. | (3.21) |
The second term on the left-hand side is as follows:
−∫101s2(t)˜uεyy(t)˜uε(t)dy=a0s(t)((σ(˜uε(t,1))2−ασ(ηε(t))s(t))˜uε(t,1)−1s(t)β(b(t)−γ˜uε(t,0))˜uε(t,0)+1s2(t)∫10|˜uεy(t)|2dy≥−a0α|ηε(t)||˜uε(t,1)|−1s(t)β(b(t)−γ˜uε(t,0))˜uε(t,0)+1s2(t)∫10|˜uεy(t)|2dy, |
where ηε(t)=(ρε∗η)(t,1). From the above, we obtain that
12ddt|˜uε(t)|2H+1s2(t)∫10|˜uεy(t)|2dy≤∫10yst(t)s(t)ηy(t)˜uε(t)dy+a0α|ηε(t)||˜uε(t,1)|+1s(t)β(b(t)−γ˜uε(t,0))˜uε(t,0) for t∈[0,T]. | (3.22) |
We estimate the right-hand side of Eq (3.22). First, by Young's inequality we have that,
∫10yst(t)s(t)ηy(t)˜uε(t)dy≤14s2(t)∫10|ηy(t)|2dy+|st(t)|2∫10|˜uε(t)|2dy. | (3.23) |
Here, by Sobolev's embedding theorem in one dimension, we note that it holds that,
|z(y)|2≤Ce|z|X|z|H for z∈X and y∈[0,1], | (3.24) |
where Ce is a positive constant defined from Sobolev's embedding theorem. By Eq (3.24) and s≥a on [0,T], we obtain
1s(t)β(b(t)−γ˜uε(t,0))˜uε(t,0)≤βb∗s(t)|˜uε(t,0)|≤βb∗Ce2s(t)(|˜uεy(t)|H|˜uε(t)|H+|˜uε(t)|2H)+βb∗2s(t)≤14s2(t)|˜uεy(t)|2H+((βb∗Ce)24+βb∗Ce2a)|˜uε(t)|2H+βb∗2a, | (3.25) |
and
a0α|ηε(t)||˜uε(t,1)|≤a0α2(Ce(|˜uεy(t)|H|˜uε(t)|H+|˜uε(t)|2H)+|ηε(t)|2)≤14s2(t)|˜uεy(t)|2H+(s2(t)(a0α2Ce)2+a0α2Ce)|˜uε(t)|2H+a0α2|ηε(t)|2. | (3.26) |
From Eqs (3.21)–(3.26), we have that for t∈[0,T]
12ddt|˜uε(t)|2H+12s2(t)∫10|˜uεy(τ)|2dy≤(|st(t)|2+(βb∗Ce)24+βb∗Ce2a)|˜uε(t)|2H+βb∗2a+(s2(t)(a0α2Ce)2+a0α2Ce)|˜uε(t)|2H+14s2(t)|ηy(t)|2H+a0α2|ηε(t)|2. | (3.27) |
Denote F(t) the coefficient of |˜uε|2H in the right-hand side. As s∈W1,2(0,T), we observe that F∈L1(0,T). Then, by Gronwall's inequality, we have for t1∈[0,T],
12|˜uε(t1)|2H+12l2∫t10|˜uεy(τ)|2Hdy≤(Gε(T)∫t10F(t)dt)e∫t10F(t)dt | (3.28) |
where Gε(t)=12(|˜u0|2H+1+βb∗aT+12a2∫t0|ηy(t)|2Hdt+a0α∫t0|ηε(t)|2dt) for t∈[0,T]. We note that it holds that
∫T0|ηε(τ,1)|2dτ≤∫T0|η(τ,1)|2dτ≤Ce∫T0(|ηy(τ)|H|η(τ)|H+|η(τ)|2H)dt≤Ce(|η|L∞(0,T;H)T1/2(∫T0|ηy(τ)|2Hdτ)1/2+T|η|2L∞(0,T;H))≤CeT1/2(1+T1/2)|η|2V(T1). | (3.29) |
Therefore, by Eq (3.28) and Eq (3.29) we see that there exists a positive constant M=M(a0,a,β,b∗,T) such that Lemma 3.4 holds.
Lemma 3.5. Let T>0, s∈W1,∞(0,T) with s>0 on [0,T] and η∈V(T). If (A1)–(A3) hold, then, (AP2)(˜u0,s,η,b) has a unique solution ˜u on [0,T].
Proof. Let s∈W1,∞(0,T) with s>0 on [0,T]. Then, we already have a solution ˜uε of (AP2)ε(˜u0ε,s,η,b) on [0,T] for each ε>0. By letting ε→0 we show the existence of a solution ˜u of (AP2)(˜u0,s,η,b) on [0,T]. First, by Lemma 3.4 we see that {˜uε} is bounded in L∞(0,T;H)∩L2(0,T;X). Next, for z∈X, it holds that,
|∫10˜uεt(t)zdy|=|−1s2(t)(∫10˜uεy(t)zydy)−a0s(t)((σ(˜uε(t,1))2−ασ((ρε∗η)(t,1))s(t))z(1)+1s(t)β(b(t)−γ˜uε(t,0))z(0)+∫10yst(t)s(t)ηy(t)zdy|≤1a2|˜uεy(t)|H|zy|H+a0a|˜uε(t,1)|2|z(1)|+a0α|(ρε∗η)(t,1)||z(1)|+βb∗a|z(0)|+βγa|˜uε(t,0)||z(0)|+|st(t)|a|ηy(t)|H|z|H for a.e. t∈[0,T]. | (3.30) |
By the estimate Eq (3.30) and Eq (3.24) we infer that {˜uεt} is bounded in L2(0,T;X∗). Therefore, we take a subsequence {εi}⊂{ε} such that for some ˜u∈W1,2(0,T;X∗)∩L∞(0,T;H)∩L2(0,T;X), ˜uεi→˜u weakly in W1,2(0,T;X∗)∩L2(0,T;X), weakly-* in L∞(0,T;H) as i→∞. Also, by Aubin's compactness theorem, we see that ˜uεi→˜u in L2(0,T;H) as i→∞.
Now, we prove that the limit function ˜u is a solution of (AP2)(˜u0,s,η,b) on [0,T] satisfying ˜u∈W1,2(0,T;X∗)∩V(T), (S''2) and (S3). Let z∈V(T). Then, it holds that,
∫T0∫10˜uεt(t)z(t)dydt+∫T01s2(t)(∫10˜uεy(t)zy(t)dy)dt+∫T0a0s(t)((σ(˜uε(t,1))2−ασ((ρε∗η)(t,1))s(t))z(t,1)dt−∫T01s(t)β(b(t)−γ˜uε(t,0))z(t,0)dt=∫T0∫10yst(t)s(t)ηy(t)z(t)dydt. | (3.31) |
From the weak convergence, it is easy to see that,
∫T0∫10˜uεit(t)z(t)dydt→∫T0⟨˜ut(t),z(t)⟩Xdt,∫T01s2(t)(∫10˜uεiy(t)zy(t)dy)dt→∫T01s2(t)(∫10˜uy(t)zy(t)dy)dt as i→∞. |
The third term of the left-hand side of Eq (3.31) is as follows:
|∫T0a0s(t)((σ(˜uε(t,1))2−ασ((ρε∗η)(t,1))s(t))z(t,1)dt−∫T0a0s(t)((σ(˜u(t,1))2−ασ(η(t,1))s(t))z(t,1)dt|≤a0a(∫T0|˜uε(t,1)−˜u(t,1)|2dt)1/2(∫T02(|˜uε(t,1)|2+|˜u(t)|2)|z(t,1)|2dt)1/2.+a0α∫T0|(ρε∗η)(t,1)−η(t,1)||z(t,1)|dt |
Here, by Eq (3.24) we note that it holds that,
∫T0|˜uε(t,z)−˜u(t,z)|2dt≤Ce(∫T0|˜uε(t)−˜u(t)|2X)1/2(∫T0|˜uε(t)−˜u(t)|2H)1/2 for z=0,1, | (3.32) |
and
∫T0|˜uε(t,1)|2|z(t,1)|2dt≤C2e∫T0|˜uε(t)|X|˜uε(t)|H|z(t)|X|z(t)|Hdt≤C2e|˜uε|L∞(0,T;H)|z|L∞(0,T;H)(∫T0|˜uε(t)|2X)1/2(∫T0|z(t)|2X)1/2. |
Since (ρε∗η)(t,1)→η(t,1) in L2(0,T) as ε→0, by the boundedness of ˜uε in L2(0,T;X), ˜u∈L2(0,T;X) and the strong convergence in L2(0,T;H) we see that
∫T0a0s(t)((σ(˜uεi(t,1))2−ασ((ρεi∗η)(t,1))s(t))z(t,1)dt→∫T0a0s(t)((σ(˜u(t,1))2−ασ(η(t,1))s(t))z(t,1)dt as i→∞. |
What concerns the forth term on the left-hand side of Eq (3.31), by Eq (3.32) it follows that,
∫T01s(t)β(b(t)−γ˜uεi(t,0))z(t,0)dt→∫T01s(t)β(b(t)−γ˜u(t,0))z(t,0)dt as i→∞. |
Therefore, by the limiting process i→∞ in Eq (3.31) we see that the limit function ˜u is a solution of (AP2)(˜u0,s,η,b) on [0,T]. Also, the solution ˜u is unique. Indeed, let ˜u1 and ˜u2 be a solution of (AP2)(˜u0,s,η,b) on [0,T] and put ˜u=˜u1−˜u2. Then, (S''2) implies that
⟨˜ut,z⟩X+1s2(∫10˜uyzydy)+a0s((σ(˜u1(⋅,1))2−ασ(η(⋅,1))s)−((σ(˜u2(⋅,1))2−ασ(η(⋅,1))s))z(1)+1sβγ(˜u1(⋅,0)−˜u2(⋅,0))z(0)=0 for z∈X a.e. on [0,T]. | (3.33) |
We take z=˜u in Eq (3.33). Then, by the monotonicity of σ we have
12ddt|˜u(t)|2H+1s2(t)∫10|˜uy(t)|2dy≤0 for a.e. t∈[0,T]. | (3.34) |
By Eq (3.34), we have the uniqueness of a solution ˜u to (AP2)(˜u0,s,η,b) on [0,T].
Lemma 3.6. Let T>0 and s∈W1,∞(0,T) with s>0 on [0,T]. Then, (AP1)(˜u0,s,b) has a unique solution ˜u on [0,T].
Proof. From Lemma 3.5, we see that (AP2)(˜u0,s,η,b) has a solution ˜u on [0,T] such that ˜u∈W1,2(0,T;X∗)∩L∞(0,T;H)∩L2(0,T;X). Define a solution operator δT(η)=˜u, where ˜u is a unique solution of (AP2)(˜u0,s,η,b) for given η∈V(T). Let put δT(ηi)=˜ui for i=1,2 and η=η1−η2 and ˜u=˜u1−˜u2. Then, by (S''2) it holds that
12ddt|˜u(t)|2H+1s2∫10|˜uy(t)|2dy+a0s(t)((σ(˜u1(t,1))2−ασ(η1(t,1))s(t))−((σ(˜u2(t,1))2−ασ(η2(t,1))s(t)))˜u(t,1)+βγs(t)|˜u(t,0)|2=∫10yst(t)s(t)ηy(t)˜u(t)dy for a.e. t∈[0,T]. | (3.35) |
For the third term I3 in the left-hand side of Eq (3.35), by the monotonicity of σ we have that I3≥−a0α|η(t,1)||˜u(t,1)|. From Eq (3.24) it holds that,
|η(t,1)||˜u(t,1)|≤C1/2e|η(t,1)|(|˜uy(t)|1/2H|˜u(t)|1/2H+|˜u(t)|H) |
and
∫t0|η(τ,1)||˜u(τ,1)|dτ≤C1/2e(|˜u|1/2L∞(0,T1;H)∫t0|η(t,1)||˜uy(τ)|1/2Hdτ+|˜u|L∞(0,T1;H)∫t0|η(t,1)|dτ)≤C1/2e(|˜u|1/2L∞(0,T1;H)T1/41(∫t0|˜uy(τ)|2H)1/4+|˜u|L∞(0,T1;H)T1/21)(∫t0|η(t,1)|2)1/2. | (3.36) |
Let T1∈(0,T] and we integrate Eq (3.35) over [0,t] for any t∈[0,T1]. Then, by Eq (3.36) we obtain,
δ(|˜u(t)|2H+∫t0|˜uy(τ)|2Hdτ)≤a0αC1/2eT1/41(1+T1/41)(∫t0|η(τ,1)|2dτ)1/2|˜u|V(T1)+|st|L∞(0,T1)aT1/21(∫t0|ηy(τ)|2Hdτ)1/2|˜u|V(T1) | (3.37) |
and δ=min{1/2,1/l2}, where l=max0≤t≤T|s(t)|. Finally, by Eq (3.37) we have
δ|˜u|V(T1)≤[a0αC1/2eT1/41(1+T1/41)+|st|L∞(0,T1)aT1/21]|η|V(T1). | (3.38) |
From Eq (3.38) we see that there exists T1≤T such that δT1 is a contraction on V(T1). Hence, Banach's fixed point theorem guarantees that there exists ˜u∈V(T1) such that δT1(˜u)=˜u. Thus, we see that (AP1)(˜u0,s,b) has a solution ˜u on [0,T1]. Here, T1 is independent of the choice of the initial data. Therefore, by repeating the local existence argument, we have a unique solution ˜u of (AP1)(˜u0,s,b) on the whole interval [0,T]. Thus, we see that Lemma 3.6 holds.
Here, for given s∈W1,∞(0,T) with s>0 on [0,T] we show that a solution ˜u of (AP1)(˜u0,s,b) is non-negative and bounded on Q(T).
Lemma 3.7. Let T>0, s∈W1,∞(0,T) with s>0 on [0,T] and ˜u be a solution of (AP1)(˜u0,s,b) on [0,T]. Then, it holds that
0≤˜u(t)≤u∗(T):=max{αl(T),b∗γ}on[0,1]fort∈[0,T], |
where l(T)=max0≤t≤T|s(t)|.
Proof. By (S'2), we note that it holds that,
⟨˜ut,z⟩X+∫101s2˜uyzydy+a0sσ(˜u(⋅,1))(σ(˜u(⋅,1))−αs)z(1)−1sβ(b(⋅)−γ˜u(⋅,0))z(0)=∫10ysts˜uyzdy for z∈X a.e. on [0,T]. | (3.39) |
First, we prove that ˜u(t)≥0 on [0,1] for t∈[0,T]. By taking z=−[−˜u]+ in Eq (3.39) we have,
12ddt|[−˜u(t)]+|2H+1s2(t)∫10|[−˜u(t)]+y|2dy−a0s(t)σ(˜u(⋅,1))(σ(˜u(⋅,1))−αs(t))[−˜u(t,1)]++1s(t)β(b(t)−γ˜u(⋅,0))[−˜u(t,0)]+=−∫10yst(t)s(t)˜uy(t)[−˜u(t)]+dy a.e. on [0,T]. | (3.40) |
Here, the third term in the left-hand side of Eq (3.40) is equal to 0 and the forth term in the left-hand side of Eq (3.40) is non-negative. Also, we obtain that
∫10yst(t)s(t)[−˜u(t)]+y[−˜u(t)]+dy≤12s2(t)|[−˜u(t)]+y|2H+(st(t))22|[−˜u(t)]+|2H, |
Then, we have
12ddt|[−˜u(t)]+|2H+12s2(t)∫10|[−˜u(t)]+y|2dy≤(st(t))22|[−˜u(t)]+|2H for a.e. t∈[0,T]. |
Therefore, by Gronwall's inequality and the assumption that ˜u0≥0 on [0,1], we conclude that ˜u(t)≥0 on [0,1] for t∈[0,T].
Next, we show that a solution ˜u of (AP1)(˜u0,s,b) has a upper bound u∗(T). Put U(t,y)=[˜u(t,y)−u∗(T)]+ for y∈[0,1] and t∈[0,T]. Then, it holds that
12ddt|U(t)|2H+1s2(t)∫10|Uy(t)|2dy+a0s(t)σ(˜u(⋅,1))(σ(˜u(⋅,1))−αs(t))U(t,1)−1s(t)β(b(t)−γ˜u(⋅,0))U(t,0)=∫10yst(t)s(t)˜uy(t)U(t)dy for a.e. t∈[0,T]. | (3.41) |
Here, by ˜u(t)≥0 on [0,1] for t∈[0,T] we note that a0σ(˜u(⋅,1))(σ(˜u(t,1))−αs(t))=a0˜u(t,1)(˜u(t,1)−αs(t)). Then, by u∗(T)≥αl(T)≥αs(t) for t∈[0,T], it holds that
a0s(t)˜u(t,1)(˜u(t,1)−αs(t))U(t,1)≥a0s(t)˜u(t,1)(u∗(T)−αs(t))U(t,1)≥0. |
Also, by Eq (1.3) and b≤b∗, we observe that,
−1s(t)β(b(t)−γ˜u(t,0))U(t,0)=1s(t)β(γ˜u(t,0)−b∗+b∗−b(t))U(t,0)≥βγs(t)|U(t,0)|2+βs(t)(b∗−b(t))U(t,0)≥0. | (3.42) |
By applying the above two results to Eq (3.41) we obtain that,
12ddt∫10|U(t)|2dy+12s2(t)∫10|Uy(t)|2dy≤(st(t))22|U(t)|2H for a.e. t∈[0,T]. |
This result and the assumption that ˜u0≤b∗/γ on [0,1] implies that ˜u(t)≤u∗(T) on [0,1] for t∈[0,T]. Thus, Lemma 3.7 is proven.
At the end of this section, we relax the condition s∈W1,∞(0,T), namely, for given s∈W1,2(0,T) with s>0 on [0,T], we construct a solution to (AP1)(˜u0,s,b).
Lemma 3.8. Let T>0 and s∈W1,2(0,T) with s>0 on [0,T]. If (A1)–(A3) hold, then, (AP1)(˜u0,s,b) has a unique solution ˜u on [0,T].
Proof. For given s∈W1,2(0,T) with s>0 on [0,T], we choose a sequence {sn}⊂W1,∞(0,T) and l, a>0 satisfying a≤sn≤l on [0,T] for each n∈N, sn→s in W1,2(0,T) as n→∞. By Lemma 3.6 we can take a sequence {˜un} of solutions to (AP1)(˜u0,sn,b) on [0,T]. Let z∈X. Then, it holds that
⟨˜unt,z⟩X+1s2n(∫10˜unyzydy)+a0snσ(˜un(⋅,1))(σ(˜un(⋅,1))−αsn)z(1)−1snβ(b−γ˜un(⋅,0))z(0)=∫10ysntsn˜unyzdy a.e. on [0,T]. | (3.43) |
We take z=˜un in Eq (3.43). Then, similarly to the proof of Lemma 3.4 we derive
12ddt|˜un(t)|2H+14s2n(t)∫10|˜uny(τ)|2dy≤(|snt(t)|2+(βb∗Ce)24+βb∗Ce2a)|˜un(t)|2H+βb∗2a+(s2n(t)(a0αCe)2+a0αCe)|˜un(t)|2H for a.e. t∈[0,T]. |
From this, we infer that {˜un} is bounded in L∞(0,T;H)∩L2(0,T;X). Also, by referring to the derivation of Eq (3.30) we obtain from Eq (3.43) that
|∫10˜unt(t)zdy|≤1a2|˜uny(t)|H|zy|H+a0a|˜un(t,1)|2|z(1)|+a0α|˜un(t,1)||z(1)|+βb∗a|z(0)|+βγa|˜un(t,0)||z(0)|+|snt(t)|a(|˜un(t,1)||z(1)|+|˜un(t)|H(|zy|H+|z|H)). |
Here, we note that
∫10ysnt(t)sn(t)˜uny(t)zdy=snt(t)sn(t)(˜un(t,1)z(1)−∫10˜un(t)(yzy+z)dy). |
Here, by Lemma 3.7, we note that 0≤˜un≤u∗ on Q(T) for each n∈N, where u∗=max{αl,b∗γ}. Using this result and the boundedness of {˜un} in V(T), we see that {˜unt} is bounded in L2(0,T;X∗). Therefore, we take a subsequence {nj}⊂{n} such that for some ˜u∈W1,2(0,T;X∗)∩V(T), ˜unj→˜u strongly in L2(0,T;H), weakly in W1,2(0,T;X∗)∩L2(0,T;X), weakly-* in L∞(0,T;H), and ˜unj(⋅,x)→˜u(⋅,x) in L2(0,T) at x=0, 1 as j→∞.
Now, we consider the limiting process j→∞ in the following way:
∫T0⟨˜unt(t),z(t)⟩Xdt+∫T01s2n(t)(∫10˜uny(t)zy(t)dy)dt+∫T0a0sn(t)σ(˜un(t,1))(σ(˜un(t,1))−αsn(t))z(t,1)dt−∫T01sn(t)β(b(t)−γ˜un(t,0))z(t,0)dt=∫T0∫10ysnt(t)sn(t)˜uny(t)z(t)dydt for z∈V(T). | (3.44) |
Note that by snj→s in W1,2(0,T) as j→∞, it holds that snj→s in C([0,T]) as j→∞. From the convergence of ˜unj and snj, Eq (3.32) and a≤snj≤l on [0,T], it is clear that
∫T0⟨˜unjt(t),z(t)⟩Xdt→∫T0⟨˜ut(t),z(t)⟩Xdt,∫T0a0snj(t)σ(˜unj(t,1))(σ(˜unj(t,1))−αsnj(t))z(t,1)dt→∫T0a0s(t)σ(˜u(t,1))(σ(˜u(t,1))−αs(t))z(t,1)dt |
and
∫T01snj(t)β(b(t)−γ˜unj(t,0))z(t,0)dt→∫T01s(t)β(b(t)−γ˜u(t,0))z(t,0)dt |
as j→∞. For the second term in the left-hand side of Eq (3.44), it follows that,
|∫T01s2nj(t)(∫10˜unjy(t)zy(t)dy)dt−∫T01s2(t)(∫10˜uy(t)zy(t)dy)dt|≤∫T0|1s2nj(t)−1s2(t)||˜unjy(t)|H|zy(t)|Hdt+|∫T01s2(t)(∫10(˜unjy(t)−˜uy(t))zy(t)dy)dt|≤2la2|snj−s|C([0,T])|˜unjy|L2(0,T;H)|zy|L2(0,T;H)+|∫T0(˜unjy(t)−˜uy(t),1s2(t)zy(t))Hdt| |
Hence, we observe that
∫T01s2nj(t)(∫10˜unjy(t)zy(t)dy)dt→∫T01s2(t)(∫10˜uy(t)zy(t)dy)dt as j→∞. |
Also, the right-hand side of Eq (3.44) is as follows:
|∫T0∫10ysnjt(t)snj(t)˜unjy(t)z(t)dydt−∫T0∫10yst(t)s(t)˜uy(t)z(t)dydt|≤∫T0|snjt(t)snj(t)−st(t)s(t)||˜unjy(t)|H|z(t)|Hdt+|∫T0(˜unjy(t)−˜uy(t),yst(t)s(t)z(t))Hdt|≤1a|snjt−st|L2(0,T)|˜unjy|L2(0,T;H)|z|L∞(0,T;H)+1a2|snj−s|C([0,T])|st|L2(0,T)|˜unjy|L2(0,T;H)|z|L∞(0,T;H)+|∫T0(˜unjy(t)−˜uy(t),yst(t)s(t)z(t))Hdt| |
From this, we have that
∫T0∫10ysnjt(t)snj(t)˜unjy(t)z(t)dydt→∫T0∫10yst(t)s(t)˜uy(t)z(t)dydt as j→∞. |
Finally, by letting j→∞ we see that ˜u is a solution of (AP)(˜u0,s,b) on [0,T]. Uniqueness is proved by the same argument of the proof of Lemma 3.5.
In this section, using the results obtained in Section 3, we establish the existence of a locally-in-time solution (PC)(˜u0,s0,b). Throughout of this section, we assume (A1)–(A3). First, for T>0, l>0 and a>0 such that a<s0<l we set
M(T,a,l):={s∈W1,2(0,T)|a≤s≤l on [0,T],s(0)=s0}. |
Also, for given s∈M(T,a,l), we define two solution mappings as follows: Ψ:M(T,a,l)→W1,2(0,T;X∗)∩L∞(0,T;H)∩L2(0,T;X) by Ψ(s)=˜u, where ˜u is a unique solution of (AP1)(˜u0,s,b) on [0,T] and ΓT:M(T,a,l)→W1,2(0,T) by ΓT(s)=s0+∫t0a0(σ(Ψ(s)(τ,1))−αs(τ))dτ for t∈[0,T]. Moreover, for any K>0 we put
MK(T):={s∈M(T,a,l)| |s|W1,2(0,T)≤K}. |
Now, we show that for some T>0, ΓT is a contraction mapping on the closed set of MK(T) for any K>0.
Lemma 4.1. Let K>0. Then, there exists a positive constant T∗≤T such that the mapping ΓT∗ is a contraction on the closed set MK(T∗) in W1,2(0,T∗).
Proof. For T>0, a>0 and l>0 such that a<s0<l, let s∈M(T,a,l) and ˜u=Ψ(s). First, from the proof of Lemma 3.8 we note that it holds
|Ψ(s)|W1,2(0,T;X∗)+|Ψ(s)|L∞(0,T;H)+|Ψ(s)|L2(0,T;X)≤C for s∈MK(T), | (4.1) |
where C=C(T,˜u0,K,l,b∗,β,s0) is a positive constant depending on T, ˜u0, K, l, b∗, β and s0.
First we show that there exists T0≤T such that ΓT0:MK(T0)→MK(T0) is well-defined. Let K>0 and s∈MK(T). By the definition of σ and Ψ(s)=˜u is a solution of (AP1)(˜u0,s,b), we observe that
ΓT(s)(t)=s0+∫t0a0(σ(Ψ(s)(τ,1))−αs(τ))dτ≥s0−a0αlt for t∈[0,T]. | (4.2) |
Also, by Eq (3.24) and Eq (4.1), it holds that
ΓT(s)(t)≤s0+a0T1/2(∫t0|˜u(τ,1)|2dτ)1/2≤s0+a0T1/2(Ce|˜u|L∞(0,t;H)∫t0|˜u(τ)|X)1/2≤s0+a0T1/2(T1/4C1/2eC),∫t0|ΓT(s)(τ)|2dτ≤2s20T+4a20T∫t0(|˜u(τ,1)|2+(αs(t))2)dτ≤2s20T+4a20T(Ce|˜u|L∞(0,t;H)∫t0|˜u(τ)|X+(αl)2T)≤2s20T+4a20T(T1/2CeC2+(αl)2T), | (4.3) |
and
∫t0|Γ′T(s)(τ)|2dτ≤a20∫t0|σ(Ψ(s)(τ,1))−αs(τ)|2dτ≤2a20(T1/2CeC2+(αl)2T), | (4.4) |
where C is the same positive constant as in (4.1). Therefore, by Eqs (4.2)–(4.4) we see that there exists T0≤T such that ΓT0(s)∈MK(T0).
Next, for s1 and s2∈MK(T0), let ˜u1=Ψ(s1) and ˜u2=Ψ(s2) and set ˜u=˜u1−˜u2, s=s1−s2. Then, it holds that
⟨˜ut,z⟩X+∫10(1s21˜u1y−1s22˜u2y)zydy+a0(1s1σ(˜u1(⋅,1))(σ(˜u1(⋅,1))−αs1)−1s2σ(˜u2(⋅,1))(σ(˜u2(⋅,1))−αs2))z(1)−(1s1β(b(t)−γ˜u1(⋅,0))−1s2β(b(t)−γ˜u2(⋅,0)))z(0)=∫10(ys1ts1˜u1y−ys2ts2˜u2y)zdy for z∈X a.e. on [0,T]. | (4.5) |
By taking z=˜u in Eq (4.5) we have
12ddt|˜u|2H+∫10(1s21˜u1y−1s22˜u2y)˜uydy+a0(1s1σ(˜u1(⋅,1))(σ(˜u1(⋅,1))−αs1))−1s2σ(˜u2(⋅,1))(σ(˜u2(⋅,1))−αs2))˜u(⋅,1)−(1s1β(b(t)−γ˜u1(⋅,0))−1s2β(b(t)−γ˜u2(⋅,0))˜u(⋅,0)=∫10(ys1ts1˜u1y−ys2ts2˜u2y)˜udy a.e. on [0,T]. | (4.6) |
For the second term of the left-hand side of Eq (4.6), we observe that
∫10(1s21(t)˜u1y(t)−1s22(t)˜u2y(t))˜uy(t)dy=1s21(t)|˜uy(t)|2H+∫10(1s21(t)−1s22(t))˜u2y(t)˜uy(t)dy≥1s21(t)|˜uy(t)|2H−2l|s(t)|a3s1(t)|˜u2y(t)|H|˜uy(t)|H≥(1−η2)1s21(t)|˜uy(t)|2H−12η(2la3)2|s(t)|2|˜u2y(t)|2H, | (4.7) |
where η is arbitrary positive number. Next, the third term in the left-hand side of Eq (4.6) is as follows:
a0(σ(˜u1(t,1))(σ(˜u1(t,1))−αs1(t))s1(t)−σ(˜u2(t,1))(σ(˜u2(t,1))−αs2(t))s2(t))˜u(t,1)=a0[1s1(t)(σ(˜u1(t,1))(σ(˜u1(t,1))−αs1(t))−σ(˜u2(t,1))(σ(˜u2(t,1))−αs2(t)))+(1s1(t)−1s2(t))σ(˜u2(t,1))(σ(˜u2(t,1))−αs2(t))]˜u(t,1)=a0[1s1(t)(σ(˜u1(t,1))−σ(˜u2(t,1)))(σ(˜u1(t,1))−αs1(t))]˜u(t,1)+σ(˜u2(t,1))s1(t)(σ(˜u1(t,1))−αs1(t)−(σ(˜u2(t,1))−αs2(t)))˜u(t,1)+(1s1(t)−1s2(t))σ(˜u2(t,1))(σ(˜u2(t,1))−αs2(t))˜u(t,1):=I1+I2+I3. |
By the monotonicity of σ(r) and Eq (3.24) we have that
I1≥−a0s1(t)(|˜u1(t,1)|+αl)|Ce|˜u(t)|X|˜u(t)|H | (4.8) |
and
I2=a0s1(t)σ(˜u2(t,1))(σ(˜u1(t,1))−αs1(t)−(σ(˜u2(t,1))−αs2(t)))˜u(t,1)≥−a0|˜u2(t,1)|s1(t)α|s(t)||˜u(t,1)|≥−Ce(a0α˜u2(t,1))22s21(t)|˜u(t)|X|˜u(t)|H−12|s(t)|2. | (4.9) |
Also, using the fact that σ(r)≤|r| for r∈R and Eq (3.24), we have the following estimate:
|I3|≤(|s(t)|s1(t)s2(t))a0σ(˜u2(t,1))(σ(˜u2(t,1))+αl)|˜u(t,1)|≤Ce(a0˜u2(t,1))22a2s21(t)|˜u(t)|X|˜u(t)|H+12˜u22(t,1)|s(t)|2+Ce(a0αl˜u2(t,1))22a2s21(t)|˜u(t)|X|˜u(t)|H+12|s(t)|2. | (4.10) |
Here, we put L(1)s1(t)=a0(|˜u1(t,1)|+αl)Ce and L(1)s2(t)=Ce(a0α˜u2(t,1))2/2+Ce(a0˜u2(t,1))2/2a2+Ce(a0αl˜u2(t,1))2/2a2. Next, by using Eq (3.24) and (A3), we have
−β(1s1(t)−1s2(t))b(t)˜u(t,0)+βγ(1s1(t)˜u1(t,0)−1s2(t)˜u2(t,0))˜u(t,0)=−β(1s1(t)−1s2(t))b(t)˜u(t,0)+βγ1s1(t)|˜u(t,0)|2+βγ(1s1(t)−1s2(t))˜u2(t,0)˜u(t,0)≥−((βb∗)2Ce2a2s21(t)+(βγ|˜u2(t,0)|)2Ce2a2s21(t))|˜u(t)|X|˜u(t)|H−|s(t)|2 for t∈[0,T0]. | (4.11) |
For the right-hand side of Eq (4.6), we separate as follows:
∫10(ys1t(t)s1(t)˜u1y(t)−ys2t(t)s2(t)˜u2y(t))˜u(t)dy=∫10ys1t(t)s1(t)˜uy(t)˜u(t)dy+∫10yst(t)s1(t)˜u2y(t)˜u(t)dy+∫10(1s1(t)−1s2(t))ys2t(t)˜u2y(t)˜u(t)dy:=I4+I5+I6. |
Then, the three terms are estimated in the following way:
I4≤η2s21(t)|˜uy(t)|2H+12η|s1t(t)|2|˜u(t)|2H,I5≤12a(|st(t)|2+|˜u2y(t)|2H|˜u(t)|2H),I6≤12a2(|s(t)|2|˜u2y(t)|2H+|s2t(t)|2|˜u(t)|2H). |
From Eq (4.6) and all estimates we derive the following inequality:
12ddt|˜u(t)|2H+(1−η)1s21(t)|˜uy(t)|2H≤1s1(t)L(1)s1(t)|˜u(t)|X|˜u(t)|H+1s21(t)(L(1)s2(t)+(βγ|˜u2(t,0)|)2Ce2a2+(βb∗)2Ce2a2)|˜u(t)|X|˜u(t)|H+(12η|s1t(t)|2+12a|˜u2y(t)|2H+12a2|s2t(t)|2)|˜u(t)|2H+(12a2|˜u2y(t)|2H+12η(2la3)2|˜u2y(t)|2H+12˜u22(t,1)+2)|s(t)|2+12a|st(t)|2. | (4.12) |
We put C5(t)=((βγ|˜u2(t,0)|)2Ce)/2a2+((βb∗)2Ce)/2a2. By Young's inequality we have that
1s1(t)L(1)s1(t)|˜u(t)|X|˜u(t)|H≤1s1(t)L(1)s1(t)(|˜uy(t)|H|˜u(t)|H+|˜u(t)|2H)≤η2s21(t)|˜uy(t)|2H+((L(1)s1(t))22η+1aL(1)s1(t))|˜u(t)|2H, |
and
(L(1)s2(t)+C5(t))1s21(t)|˜u(t)|X|˜u(t)|H≤(L(1)s2(t)+C5(t))1s21(t)(|˜uy(t)|H|˜u(t)|H+|˜u(t)|2H)≤1s21(t)η2|˜uy(t)|2H+1a2((L(1)s2(t)+C5(t))22η+(L(1)s2(t)+C5(t)))|˜u(t)|2H. |
Hence, by applying these results to Eq (4.12) and taking a suitable η=η0, we obtain
12ddt|˜u(t)|2H+121s21(t)|˜uy(t)|2H≤((L(1)s1(t))22η0+1aL(1)s1(t))|˜u(t)|2H+1a2((L(1)s2(t)+C5(t))22η0+(L(1)s2(t)+C5(t)))|˜u(t)|2H.+(12η0|s1t(t)|2+12a|˜u2y(t)|2H+12a2|s2t(t)|2)|˜u(t)|2H+(12a2|˜u2y(t)|2H+12η0(2la3)2|˜u2y(t)|2H+12˜u22(t,1)+2)|s(t)|2+12a|st(t)|2. | (4.13) |
Now, we put the summation of all coefficients of |˜u(t)|2H by L(2)s(t) for t∈[0,T0] and L(3)s2(t)=|˜u2y(t)|2H/2a2+(4l2|˜u2y(t)|2H)/2η0a6+˜u22(t,1)/2+2. Then, we have
12ddt|˜u(t)|2H+121s21(t)|˜uy(τ)|2H≤L(2)s(t)|˜u(t)|2H+L(3)s2(t)|s(t)|2+12a|st(t)|2 for t∈[0,T0]. | (4.14) |
By Eq (3.24) and Eq (4.1) we note that ˜u2i(⋅,1),˜u4i(⋅,1)∈L1(0,T0) for i=1,2. From this and the fact that si∈MK(T0) for i=1,2, we see that L(2)s∈L1(0,T0) and L(3)s2∈L1(0,T0). Also, it holds that
L(3)s2(t)|s(t)|2≤L(3)s2(t)T0|st|2L2(0,t) for t∈[0,T0]. |
Therefore, Gronwall's inequality guarantees that
12|˜u(t)|2H+121l2∫t0|˜uy(τ)|2Hdτ≤[(2|L(3)s2|L1(0,T0)T0+1a)|st|2L2(0,t)]e2∫t0L(2)s(τ)dτ for t∈[0,T0]. | (4.15) |
By using Eq (4.15) we show that there exists T∗≤T0 such that ΓT∗ is a contraction mapping on the closed subset of MK(T∗). To do so, from the subtraction of the time derivatives of ΓT0(s1) and ΓT0(s2) and relying on Eq (3.24) and Eq (4.15), we have for T1≤T0 the following estimate:
|(ΓT1(s1))t−(ΓT1(s2))t|L2(0,T1)≤a0(|σ(˜u1(⋅,1))−σ(˜u2(⋅,1))|L2(0,T1)+α|s1(t)−s2(t)|L2(0,T1))≤a0√Ce(∫T10(|˜uy(t)|H|˜u(t)|H+|˜u(t)|2H)dt)1/2+a0αT1|s|W1,2(0,T1)≤a0√Ce(|˜u|12L∞(0,T1;H)(∫T10|˜uy(t)|Hdt)12+√T1|˜u|L∞(0,T1;H))+a0αT1|s|W1,2(0,T1). | (4.16) |
Using Eq (4.15) and Eq (4.16), we obtain
|ΓT1(s1)−ΓT1(s2)|L2(0,T1)≤T1C6(T141|s|W1,2(0,T1)+√T1|s|W1,2(0,T1)+T1|s|W1,2(0,T1)), | (4.17) |
where C6 is a positive constant obtained by Eq (4.15). Therefore, by Eq (4.16) and Eq (4.17) we see that there exists T∗≤T0 such that ΓT∗ is a contraction mapping on a closed subset of MK(T∗).
From Lemma 4.1, by applying Banach's fixed point theorem, there exists s∈MK(T∗), where T∗ is the same as in Lemma 4.1 such that ΓT∗(s)=s. This implies that (PC)(˜u0,s0,b) has a unique solution (s,˜u) on [0,T∗].
At the end of this section, we show the boundedness of a solution ˜u to (PC)(˜u0,s0,b) which completes Theorem 2.2. By (S2) it holds that
⟨˜ut,z⟩X+∫101s2˜uyzydy+1sσ(˜u(⋅,1))stz(1)−1sβ(b(⋅)−γ˜u(⋅,0))z(0)=∫10ysts˜uyzdy for z∈X a.e. on [0,T]. | (4.18) |
First, we note that the solution ˜u of (PC)(˜u0,s0,b) is non-negative.
Lemma 4.2. Let T>0 and (s,˜u) be a solution of (PC)(˜u0,s0,b) on [0,T]. Then, ˜u(t)≥0 on [0,1] for t∈[0,T].
Proof. Lemma 4.2 is proved by taking z=−[−˜u]+ in Eq (4.18) and using the argument of the proof of Lemma 3.7. Here, by st(t)=a0(σ(˜u(t,1))−αs(t)) we note that it holds that
∫10yst(t)s(t)[−˜u(t)]+y[−˜u(t)]+dy=st(t)s(t)∫1012(ddy(y|[−˜u(t)]+|2)−|[−˜u(t)]+|2)=st(t)2s(t)|[−˜u(t,1)]+|2−st(t)2s(t)|[−˜u(t)]+|2H=a0(σ(˜u(t,1))−αs(t))2s(t)|[−˜u(t,1)]+|2−a0(σ(˜u(t,1))−αs(t))2s(t)|[−˜u(t)]+|2H≤a0α2|[−˜u(t)]+|2H. |
From this, we derive
12ddt|[−˜u(t)]+|2H+12s(t)∫10|[−˜u(t)]+y|2dy≤a0α2|[−˜u(t)]+|2H for a.e. t∈[0,T]. |
This implies that ˜u(t)≥0 on [0,1] for t∈[0,T].
Next, we show the boundedness of the solution (s,˜u) of (PC)(˜u0,s0,b).
Lemma 4.3. Let T>0 and (s,˜u) be a solution of (PC)(˜u0,s0,b) on [0,T]. Then, it holds that
(i)s(t)≤Mfort∈[0,T],(ii)0≤˜u(t)≤u∗:=max{αM,b∗γ}on[0,1]fort∈[0,T], |
where M is a positive constant which depends on β, γ, α, b∗, |˜u0|H, s0, |bt|L2(0,T) and |bt|L1(0,T).
Proof. First, we prove (i). By taking z=s(˜u−bγ) in Eq (4.18) it holds that:
s(t)2ddt|˜u(t)−b(t)γ|2H+(bt(t)γ,s(t)(˜u(t)−b(t)γ))H+1s(t)∫10|˜uy(t)|2dy+σ(˜u(t,1))st(t)(˜u(t,1)−b(t)γ)−β(b(t)−γ˜u(t,0))(˜u(t,0)−b(t)γ)=∫10yst(t)˜uy(t)(˜u(t)−b(t)γ)dy for a.e. t∈[0,T]. | (4.19) |
The second term of the left-hand side of Eq (4.19) is follows:
|(bt(t)γ,s(t)(˜u(t)−b(t)γ))H|=|bt(t)γs(t)∫10(˜u(t,y)−˜u(t,0)+˜u(t,0)−b(t)γ)dy|≤1γ|bt(t)||s(t)|(|˜uy(t)|H+|˜u(t,0)−b(t)γ|). | (4.20) |
Also, by Lemma 4.2 we note that σ(˜u(t,1))st(t)=˜u(t,1)st(t). Moreover, we observe that
∫10yst(t)˜uy(t)(˜u(t)−b(t)γ)dy=st(t)∫1012(∂∂y(y(˜u(t)−b(t)γ)2)−(˜u(t)−b(t)γ)2)dy=st(t)2(˜u(t,1)−b(t)γ)2−st(t)2∫10(˜u(t)−b(t)γ)2dy. | (4.21) |
Hence, by Eq (4.20) and Eq (4.21) we have
12ddt(s(t)|˜u(t)−b(t)γ|2H)+1s(t)∫10|˜uy(t)|2dy+˜u(t,1)st(t)(˜u(t,1)−b(t)γ)−st(t)2(˜u(t,1)−b(t)γ)2+βγ|˜u(t,0)−b(t)γ|2≤1γ|bt(t)||s(t)|(|˜uy(t)|H+|˜u(t,0)−b(t)γ|) for a.e. t∈[0,T]. | (4.22) |
Here, the third and forth terms of the left-hand side of Eq (4.22) is as follows:
\begin{align} & \tilde{u}(t, 1)s_t(t) \biggl (\tilde{u}(t, 1)-\frac{b(t)}{\gamma} \biggr) -\frac{s_t(t)}{2}\biggl (\tilde{u}(t, 1)-\frac{b(t)}{\gamma}\biggr)^2 \\ = & s_t(t)\tilde{u}^2(t, 1)-s_t(t)\tilde{u}(t, 1)\frac{b(t)}{\gamma}-\frac{s_t(t)}{2} \biggl (\tilde{u}^2(t, 1)-2\tilde{u}(t, 1)\frac{b(t)}{\gamma} + \biggl (\frac{b(t)}{\gamma} \biggr)^2 \biggr) \\ = & \frac{s_t(t)}{2} \tilde{u}^2(t, 1)-\frac{s_t(t)}{2} \biggl (\frac{b(t)}{\gamma} \biggr)^2. \end{align} | (4.23) |
Since s_t(t) = a_0(\tilde{u}(t, 1)-\alpha s(t)) , it holds that
\begin{align} & \frac{s_t(t)}{2} \tilde{u}^2(t, 1) = \frac{\tilde{u}(t, 1)}{2} \left(\frac{|s_t(t)|^2}{a_0} + \alpha s(t) s_t(t)\right) \\ & = \frac{1}{2a_0} \tilde{u}(t, 1)|s_t(t)|^2 + \frac{\alpha s(t)}{2} \left(\frac{|s_t(t)|^2}{a_0} + \alpha s(t) s_t(t)\right) \\ & = \frac{1}{2a_0} \tilde{u}(t, 1)|s_t(t)|^2 + \frac{\alpha s(t)}{2} \frac{|s_t(t)|^2}{a_0} + \frac{\alpha^2}{6} \frac{d}{dt}s^3(t). \end{align} | (4.24) |
Note that the first and second terms in the last of Eq (4.24) are non-negative. Accordingly, by Eq (4.23) and Eq (4.24) we have that
\begin{align} & \tilde{u}(t, 1)s_t(t) \biggl (\tilde{u}(t, 1)-\frac{b(t)}{\gamma} \biggr) -\frac{s_t(t)}{2}\biggl (\tilde{u}(t, 1)-\frac{b(t)}{\gamma}\biggr)^2 \\ \geq & \frac{\alpha^2}{6} \frac{d}{dt}s^3(t)-\frac{s_t(t)}{2} \biggl (\frac{b(t)}{\gamma} \biggr)^2 \\ = & \frac{\alpha^2}{6} \frac{d}{dt}s^3(t)-\frac{1}{2\gamma^2} \biggl (\frac{d}{dt} (s(t)b^2(t))-2s(t)b(t)b_t(t) \biggr). \end{align} | (4.25) |
Also, for the right-hand side of Eq (4.22) we have
\begin{align} & \frac{1}{\gamma}|b_t(t)||s(t)| \biggl (|\tilde{u}_y(t)|_H +|\tilde{u}(t, 0)-\frac{b(t)}{\gamma}| \biggr) \\ \leq & \frac{1}{2s(t)}|\tilde{u}_y(t)|^2_H + \frac{1}{2} s^3(t) \biggl (\frac{|b_t(t)|}{\gamma} \biggr)^2+ \frac{\beta \gamma}{2}\biggl |\tilde{u}(t, 0)-\frac{b(t)}{\gamma} \biggr|^2 + \frac{1}{2\beta \gamma} |s(t)|^2 \biggl (\frac{|b_t(t)|}{\gamma} \biggr)^2. \end{align} | (4.26) |
By combining Eq (4.25) and Eq (4.26) with Eq (4.22) we have
\begin{align} & \frac{1}{2} \frac{d}{dt} \biggl (s(t) \biggl |\tilde{u}(t)-\frac{b(t)}{\gamma} \biggr|^2_{H} \biggr) + \frac{1}{2s(t)} \int_0^1 |\tilde{u}_y(t)|^2 dy \\ & +\frac{\alpha^2}{6}\frac{d}{dt} s^3(t) - \frac{1}{2\gamma^2} \frac{d}{dt} (s(t)b^2(t)) + \frac{\beta \gamma}{2}|\tilde{u}(t, 0)-\frac{b(t)}{\gamma}|^2 \\ \leq & \frac{1}{\gamma^2} s(t)b(t)|b_t(t)| + \frac{1}{2} s^3(t) \biggl (\frac{|b_t(t)|}{\gamma} \biggr)^2 + \frac{1}{2\beta \gamma} |s(t)|^2 \biggl (\frac{|b_t(t)|}{\gamma} \biggr)^2 \mbox{ for a.e. }t\in [0, T]. \end{align} | (4.27) |
Then, by integrating Eq (4.27) over [0, t] for t\in [0, T] we obtain that
\begin{align} & \frac{1}{2}s(t) \biggl |\tilde{u}(t)-\frac{b(t)}{\gamma} \biggr|^2_{H} + \int_0^t \frac{1}{2s(\tau)} |\tilde{u}_y(\tau)|^2_H d\tau + \frac{\alpha^2}{6} s^3(t) - \frac{1}{2\gamma^2} (s(t)b^2(t)) \\ \leq & \frac{1}{2}s_0 \biggl |\tilde{u}_0-\frac{b(0)}{\gamma} \biggr|^2_{H} + \frac{\alpha^2}{6} s^3_0 +\frac{1}{2\gamma^2} \int_0^t s^3(\tau) |b_t(\tau)|^2d\tau \\ & + \frac{b^*}{\gamma^2} \int_0^t s(\tau)|b_t(\tau)|d\tau + \frac{1}{2\beta \gamma^3} \int_0^t |s(\tau)|^2 |b_t(\tau)|^2 d\tau \mbox{ for }t\in [0, T]. \end{align} | (4.28) |
Here, by Young's inequality it follows that
\begin{align} & \frac{\alpha^2}{6} s^3(t) -\frac{1}{2\gamma^2} (s(t)b^2(t)) \\ \geq & \frac{\alpha^2}{6} s^3(t)-\biggl( \frac{\eta^3}{3} s^3(t) + \frac{2}{3\eta^{3/2}} \biggl(\frac{1}{2\gamma^2} (b^*)^2\biggr)^{3/2} \biggr), \end{align} | (4.29) |
where \eta is an arbitrary positive number. Therefore, by Eq (4.28) and Eq (4.29) we obtain
\begin{align} & \biggl( \frac{\alpha^2}{6}-\frac{\eta^3}{3}\biggr) s^3(t) \leq \frac{2}{3\eta^{3/2}} \biggl(\frac{1}{2\gamma^2} (b^*)^2\biggr)^{3/2} \\ & + \frac{1}{2}s_0 \biggl |\tilde{u}_0-\frac{b(0)}{\gamma} \biggr|^2_{H} + \frac{\alpha^2}{6} s^3_0 + \frac{1}{2\gamma^2} \int_0^t s^3(\tau) |b_t(\tau)|^2d\tau \\ & + \frac{b^*}{\gamma^2} \int_0^t s(\tau)|b_t(\tau)|d\tau + \frac{1}{2\beta \gamma^3} \int_0^t |s(\tau)|^2 |b_t(\tau)|^2 d\tau \mbox{ for }t\in [0, T]. \end{align} | (4.30) |
We put M(\eta) = (s_0|\tilde{u}_0-\frac{b(0)}{\gamma}|^2_H)/2 + (\alpha^2s^3_0)/6+(2(\frac{1}{2\gamma^2} (b^*)^2)^{3/2}))/3\eta^{3/2} . Then, by taking a suitable \eta = \eta_0 and J_1(t) = \int_0^t s^3(\tau)|b_t(\tau)|^2 d\tau , J_2(t) = \int_0^t s(\tau)|b_t(\tau)|d\tau and J_3(t) = \int_0^t |s(\tau)|^2 |b_t(\tau)|^2 d\tau for t\in [0, T] we derive that
\begin{align} \frac{\alpha^2}{12 }s^3(t) \leq M(\eta_0) + \frac{1}{2\gamma^2} J_1(t) + \frac{b^*}{\gamma^2} J_2(t) + \frac{1}{2\beta \gamma^3} J_3(t) \mbox{ for }t\in [0, T]. \end{align} | (4.31) |
Then, we have that
\begin{align*} J'_1(t) & \leq \frac{12M(\eta_0)}{\alpha^2}|b_t(t)|^2 + \frac{6}{(\gamma \alpha)^2}|b_t(t)|^2 J_1(t) \nonumber \\ & +\frac{12}{\alpha^2}\biggl (\frac{b^*}{\gamma^2} J_2(t)+ \frac{1}{2\beta \gamma^3} J_3(t) \biggr)|b_t(t)|^2 \mbox{ for }t\in [0, T]. \end{align*} |
Hence, by Gronwall's inequality we obtain that
\begin{align} J_1(t)& \leq \biggl[ \frac{12M(\eta_0)}{\alpha^2}|b_t|^2_{L^2(0, T)} \\ & + \frac{12}{\alpha^2}\biggl (\frac{b^*}{\gamma^2} J_2(t) + \frac{1}{2\beta \gamma^3} J_3(t) \biggr)|b_t|^2_{L^2(0, T)} \biggr] e^{\frac{6}{(\gamma \alpha)^2}|b_t|^2_{L^2(0, T)}} \mbox{ for }t\in [0, T]. \end{align} | (4.32) |
Here, we put N(T) = \frac{12}{\alpha^2}|b_t|^2_{L^2(0, T)}e^{\frac{6}{(\gamma \alpha)^2}|b_t|^2_{L^2(0, T)}} . Then, by Eq (4.31) and Eq (4.32) we obtain that
\begin{align} & \frac{\alpha^2}{12} s^3(t) \leq M(\eta_0) + \frac{1}{2\gamma^2} M(\eta_0)N(T) \\ & + \frac{b^*}{\gamma^2} (1+\frac{N(T)}{2\gamma^2})J_2(t) + \frac{1}{2\beta \gamma^3} (1+\frac{N(T)}{2\gamma^2} )J_3(t) \mbox{ for }t\in [0, T]. \end{align} | (4.33) |
Now, we put l(T) = \max_{0\leq t\leq T}|s(t)| . Then, we have
\begin{align*} & \frac{b^*}{\gamma^2} (1+\frac{N(T)}{2\gamma^2})J_2(t) \leq \frac{b^*}{\gamma^2} (1+\frac{N(T)}{2\gamma^2})l(T)|b_t|_{L^1(0, T)} \\ \leq & \frac{\eta^3}{3}l^3(T) + \frac{2}{3\eta^{3/2}} \biggl( \frac{b^*}{\gamma^2} (1+\frac{N(T)}{2\gamma^2}) |b_t|_{L^1(0, T)} \biggr)^{3/2}, \end{align*} |
and
\begin{align*} & \frac{1}{2\beta \gamma^3} (1+\frac{N(T)}{2\gamma^2} )J_3(t) \leq \frac{1}{2\beta \gamma^3} (1+\frac{N(T)}{2\gamma^2}) l^2(T) |b_t|^2_{L^2(0, T)} \\ \leq & \frac{2\eta^{3/2}}{3}l^3(T) + \frac{1}{3\eta^3} \biggl( \frac{1}{2\beta \gamma^3} (1+\frac{N(T)}{2\gamma^2} ) |b_t|^2_{L^2(0, T)} \biggr)^{3}. \end{align*} |
Hence, by adding these estimates to Eq (4.33) and taking a suitable \eta = \eta_0 we see that there exists a positive constant M which depends on \beta , \gamma , \alpha , b^* , s_0 , |b_t|_{L^2(0, T)} and |b_t|_{L^1(0, T)} such that s(t)\leq M for t \in [0, T] .
Next, we show (ii). Put U(t, y) = [\tilde{u}(t, y)-u^*]^+ for y \in [0, 1] and t\in [0, T] , and then take z = U(t, y) in Eq (4.18). Here, by \tilde{u}(t)\geq 0 on [0, 1] for t\in [0, T] we note that s_t(t) = a_0(\sigma(\tilde{u}(t, 1))-\alpha s(t)) = a_0(\tilde{u}(t, 1)-\alpha s(t)) . Then, we observe that:
\begin{align*} & \int_0^1\frac{y s_t(t)}{s(t)} \tilde{u}_y(t)U(t) dy = \frac{s_t(t)}{2s(t)} \int_0^1 \biggl (\frac{d}{dy}(yU^2(t))-U^2(t) \biggr) dy \\ & = \frac{s_t(t)}{2s(t)}|U(t, 1)|^2 -\frac{s_t(t)}{2s(t)}|U(t)|^2_H \leq \frac{s_t(t)}{2s(t)}|U(t, 1)|^2 + \frac{a_0\alpha }{2}|U(t)|^2_H. \end{align*} |
Then, by using the argument of the proof of Lemma 3.7 we derive
\begin{align} & \frac{1}{2} \frac{d}{dt}|U(t)|^2_{H}+ \frac{1}{s^2(t)} \int_0^1 |U_y(t)|^2 dy + \frac{1}{s(t)} \tilde{u}(t, 1)s_t(t)U(t, 1)-\frac{s_t(t)}{2s(t)}|U(t, 1)|^2 \\ & \leq \frac{a_0\alpha}{2}|U(t)|^2_H \mbox{ for a.e. }t\in [0, T]. \end{align} | (4.34) |
Here, by u^*\geq \alpha M \geq \alpha s(t) for t\in [0, T] , it holds that
\begin{align*} & \frac{s_t(t)}{s(t)}\tilde{u}(t, 1)U(t, 1)-\frac{s_t(t)}{2s(t)}|U(t, 1)|^2 = \frac{s_t(t)}{2s(t)}|U(t, 1)|^2 + \frac{s_t(t)}{s(t)}u^*U(t, 1)\\ \geq & \frac{a_0(u^*-\alpha s(t))}{s(t)}(\frac{|U(t, 1)|^2}{2}+ u^*U(t, 1)) \geq 0. \end{align*} |
By applying the result to Eq (4.34) we obtain that
\begin{align*} & \frac{1}{2}\frac{d}{dt}\int_0^1|U(t)|^2 dy + \frac{1}{2s^2(t)}\int_0^1 |U_y(t)|^2 dy \leq \frac{a_0\alpha}{2}|U(t)|^2_H \mbox{ for a.e. }t\in[0, T]. \end{align*} |
This implies that \tilde{u}(t) \leq u^* on [0, 1] for t\in [0, T] . Thus, Lemma 4.3 is now proven.
Combining the statements of Lemma 4.2 and Lemma 4.3, we can conclude that Theorem 2.2 holds.
The first author is supported by Grant-in-Aid No. 20K03704, JSPS. The second author is supported by Grant-in-Aid No. 19K03572, JSPS
The authors declare there is no conflict of interest.
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