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Solvability of a model for monomer–monomer surface reactions
Nonlinear Analysis: Real World Applications 35 (2017) 211–228
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Solvability of a model for monomer–monomer surface reactions A. Ambrazeviˇcius ∗ , V. Skakauskas Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania
article
info
Article history: Received 21 June 2016 Received in revised form 25 October 2016 Accepted 1 November 2016
Keywords: Coupled parabolic systems Heterogeneous catalysis Reaction–diffusion system
abstract A mathematical model for a monomer–monomer surface reaction is considered taking into account the surface diffusion of adsorbed particles of both reactants. The model is described by a coupled system of parabolic equations where some of them are defined in a domain and the other ones have to be solved on the domain surface. The existence and uniqueness theorem of a classic solution for the time-dependent problem is proved. Non-uniqueness of solutions for the steady-state problem is established. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Coupled systems of nonlinear parabolic and ordinary differential equations determined in the same domain have been extensively studied in literature (see [1–5] and references therein). In modelling of chemical surface reactions there arise coupled systems of parabolic and ordinary differential equations where some of them are solved in a domain and the other ones are defined on the domain surface (see [6,7] and references therein). In order the surface reaction to occur reactants must diffuse from domain toward the catalyst surface, adsorb on them, and their adsorbates have to react forming a product. The bulk diffusion of reactants toward the surface is described by parabolic PDEs that have to be solved in a domain. After adsorption reactant particles can stay or diffuse on the surface. The surface diffusion of adsorbed particles is described by parabolic PDEs that are determined on the domain surface. In [8] a unimolecular surface reactions model is studied which consists of two parabolic equations. One of them is solved in a domain while the other one is determined on the domain surface. A mathematical model for surface reactions between two monomers (unimolecular reactants) competing for the same adsorption site of composite catalyst is studied numerically in [9]. In the present paper we consider a mathematical model for surface reactions between two monomers that adsorb on different sites of a homogeneous surface and prove the existence and uniqueness theorem of a classical solution. ∗ Corresponding author. E-mail addresses: [email protected] (A. Ambrazeviˇ cius), [email protected] (V. Skakauskas).
http://dx.doi.org/10.1016/j.nonrwa.2016.11.001 1468-1218/© 2016 Elsevier Ltd. All rights reserved.
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The paper is organized as follows. In Sections 2 and 3, we describe the model and formulate the solvability theorem for the time-dependent problem, respectively. In Section 4, we prove a priori estimates. Section 5 is devoted to the uniqueness of the classical solution. In Sections 6 and 7, we construct the upper and lower solutions and prove the existence of a classical solution to problem (1)–(3), respectively. At last in Section 8, we consider the corresponding steady-state problem. 2. Formulation of the problem We consider the case where particles of reactants A1 and A2 do not compete for the adsorption sites and suppose that reactants A1 and A2 adsorb on the sites P1 and P2 of the catalyst surface, respectively. Suppose, that reactant Ai , i = 1, 2, occupies a bounded domain Ω ⊂ Rn , n ≥ 3, ai = ai (x, t) is the concentration of Ai at the point x ∈ Ω at time t, S := ∂Ω is (n − 1) dimension surface in Rn , S2 ⊂ S is a domain in Rn−1 on the plane xn = 0 (surface of the adsorbent), S1 = S \ S2 , ∂S2 = ∂S1 := Σ is n − 2 dimension surface (contour, if n = 3) in Rn−1 , ρi = ρi (x) is a concentration of adsorption sites Pi , i = 1, 2, at point x ∈ S, ρi (x) = 0 for x ∈ S1 , ρi (x) = ρi (x′ , 0) > 0 for x ∈ S2 , x′ = (x1 , . . . , xn−1 ), θi = θi (x′ , t) is a fraction of ρi occupied by molecules of the adsorbate at point x ∈ S2 at time t, 1 − θi is a free fraction of ρi , ρi θi is a concentration of the adsorbate, b = b(x, t) is a concentration of product B at point x ∈ Ω at time t. According to Langmuir and Hinshelwood, adsorption, desorption and reaction rates can be written as κi ρi (1 − θi )ai , κii ρi θi , and κ12 ρ1 ρ2 θ1 θ2 , i = 1, 2, where κi , κii , and κ12 are the adsorption, desorption, and reaction rate constants. The diffusion of reactants Ai , i = 1, 2, can be described by the system ′ ai − ki ∆ai = 0 in Ω × (0, T ], k ∂a /∂n = 0 on S × (0, T ], i i 1 (1) ki ∂a on S2 × (0, T ], i /∂n + κi ρi (1 − θi )ai = κii ρi θi a in Ω i t=0 = ai0 where a′i = ∂ai /∂t, ki = const > 0 is a diffusion coefficient, ∂ai /∂n is the outward normal derivative to S, n ∆ai = i=1 aixi xi , ai0 = ai0 (x) is the initial concentrations of Ai at point x ∈ Ω . For θi , i = 1, 2, we have the problem ′ θi − pi ∆θi = κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 , j ̸= i, in S2 × (0, T ], (2) pi ∂θi /∂ν = 0 on Σ × (0, T ], θ S = θ in i t=0 i0 2 where θi′ = ∂θi /∂t, pi = const > 0 is a diffusion coefficient, ∂θi /∂ν is the outward normal derivative to Σ , n−1 ∆θi = i=1 θixi xi , θi0 = θi0 (x′ ) is a fraction of ρi which is occupied by molecules of the adsorbate at initial value t = 0 at point x′ ∈ S2 . The diffusion of the product B is described by the system b′ − l∆b = 0 in Ω × (0, T ], l∂b/∂n = 0 on S × (0, T ], 1 (3) l∂b/∂n = κ ρ ρ θ on S2 × (0, T ], 12 1 2 1 θ2 b = b0 in Ω t=0 where b′i = ∂bi /∂t, l = const > 0 is a diffusion coefficient of the product B, ∂b/∂n is the outward normal n derivative to S, ∆b = i=1 bxi xi , b0 = b0 (x) is the initial concentrations of B at point x ∈ Ω . All constants are assumed to be positive. Coupled system (1), (2), and (3) forms the nonlinear mathematical model for surface reactions between two monomers.
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3. Assumptions and solvability theorem for the time-dependent problem Assumption 3.1. Suppose, that: 1. S is a surface of class C 1+α , α ∈ (0, 1). 2. Σ is a surface of class C 1+β , β ∈ (0, 1). Assumption 3.2. Let initial functions ai0 , bi0 and θi0 satisfy the following conditions: 1. ai0 , bi0 are continuous, nonnegative functions in a closed domain Ω , and in any neighbourhood of the surface S are zero, 2. θi0 is a continuous in a closed domain S2 function such that 0 ≤ θi0 (x′ ) < 1 for all x′ ∈ S 2 and in any neighbourhood of the surface Σ is zero. Assumption 3.3. Let ai0 , bi0 , θi0 and ρi satisfy the following smoothness conditions: 1. ai0 , bi0 ∈ C λ (Ω ) with any λ ∈ (0, 1), 2. θi0 , ρi ∈ C λ (S2 ) with any λ ∈ (0, 1). Definition 3.1. Solution (a1 , a2 , θ1 , θ2 , b) of problem (1), (2), and (3) forms a classical solution if ai and b ∈ C 2,1 (Ω × (0, T ]) ∩ C (Ω × [0, T ]), ∂ai /∂n is continuous on S × [0, T ], ∂b/∂n is continuous on S × [0, T ], θi ∈ C 2,1 (S2 × (0, T ]) ∩ C (S2 × [0, T ]), and ∂θi /∂ν is continuous on Σ × [0, T ], i = 1, 2. We formulate the main result as Theorem 3.1. Let the surfaces S and Σ satisfy the conditions of Assumption 3.1 and the known functions ai0 , b0 , θi0 and ρi satisfy the conditions of Assumptions 3.2 and 3.3. Then problem (1)–(3) has a unique classical solution. To prove this theorem we need a priori estimates of solution to Eqs. (1), (2) which are formulated as Lemma 3.1. Let functions ai , θi , i = 1, 2, be a classical solution of problem (1), (2) and let a∗i0 = max ai0 (x), x∈Ω
∗ θi0 = max θi0 (x′ ) < 1. x′ ∈S 2
Then ∗ κii θi0 0 ≤ ai (x, t) ≤ max a∗i0 , := mi , ∀x ∈ Ω , t ∈ [0, T ], ∗ κi 1 − θi0 κi a∗i0 ∗ 0 ≤ θi (x′ , t) ≤ max θi0 , := m′i < 1, ∀x′ ∈ S2 , t ∈ [0, T ]. κi a∗i0 + κii
(4) (5)
4. Proof of Lemma 3.1 We multiply Eq. (1) by a piecewise smooth function η and integrate the result over cylinder Ω × (0, t), t ∈ (0, T ], getting an identity which, by using the formula of integration by parts and taking into account the boundary condition of problem (1), can be written as follows: t t t aiτ η dxdτ + ki aix ηx dxdτ = ρi κii θi − κi (1 − θi )ai η dx′ dτ (6) 0
where aix ηx =
Ω
n
k=1 aixk ηxk .
0
Ω
0
S2
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We first prove that ai and θi , i = 1, 2, are nonnegative. Inserting a(x, t), if a(x, t) < 0, − η = ai := 0, if a(x, t) ≥ 0 into (6) we get t t τ =t 1 a2i dxτ =0 + ki a2ix dxdτ = ρi κii θi − κi (1 − θi )ai ai dx′ dτ − − 2 Ωi,τ 0 Ωi,τ 0 Sa−i ,τ n − where Ωi,τ = {x ∈ Ω : ai (x, τ ) < 0}, Sa−i ,τ = {x′ ∈ S2 : ai (x′ , 0, τ ) < 0}, a2ix = k=1 a2ixk . Similarly, from Eq. (2) it follows that t t 1 ′ 2 2 ′ τ =t κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 θi dx′ dτ θix′ dx dτ = θ dx τ =0 + pi − − 2 Sθ− ,τ i Sθ ,τ 0 Sθ ,τ 0
(8)
i
i
i
(7)
n−1 2 2 where Sθ−i ,τ = {x′ ∈ S2 : θi (x′ , τ ) < 0}, θix ′ = k=1 θixk . ′ Using assumptions ai0 (x, t) ≥ 0 and θi0 (x , t) ≥ 0 from Eqs. (7) and (8) we derive the estimate t t 1 1 2 ′ a2i dx + θi2 dx′ + ki a2ix dxdτ + pi θix ′ dx dτ − − 2 Ωi,t 2 Sθ− ,t 0 Ωi,τ 0 Sθ− ,τ i i t t ′ ≤ ρi κii θi ai + κi (1 − θi )ai θi dx dτ + ρi κi θi a2i dx′ dτ Sa−i ,τ ∩Sθ− ,τ
0
0
i
t
′
− 0
t
Sθ− ,τ
κ12 ρj θ1 θ2 θi dx dτ ≤ ci1
Sa−i ,τ
0
i
a2i
Sa−i ,τ
t
′
dx dτ + ci2 0
Sθ− ,τ
θi2 dx′ dτ,
where constants ci1 , ci2 depend only on κii , κi , κ12 , m′i and ρi0 = maxx′ ∈S2 ρi (x′ ). It is known (cf. [10]) that 2 2 a dS ≤ ε ax dx + Cε a2 dx S
Ω
(10)
Ω
for all ε > 0. Therefore, we have inequality − 2 2 ′ ′ 2 ai dx = (ai ) dx ≤ (a− i ) dS Sa−i ,τ
S2
S
2 (a− i )x dx + Cε
≤ε Ω
2 (a− i ) dx = ε
Ω
− Ωi,τ
a2ix dx + Cε
which with ε = ki /2ci1 reduces inequality (9) to 1 1 ki t θi2 dx′ + a2i dx + a2ix dxdτ − − 2 Ωi,t 2 Sθ− ,t 2 0 Ωi,τ i t t 2 ′ 2 + pi θix′ dx dτ ≤ c ai dx + 0
Sθ− ,τ
0
i
− Ωi,τ
Sθ− ,τ i
Let Φ(t) =
t 0
− Ωi,τ
a2i dx +
Sθ− ,τ
θi2 dx′ dτ.
i
Then Φ ′ (t) ≤ cΦ(t) =⇒ Φ(t) ≤ 0
(9)
i
− Ωi,τ
θi2′ dx′ dτ.
a2i dx
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and − Ωi,t
a2i
dx + Sθ− ,t
θi2 dx′ ≤ 0.
i
′
Therefore ai (x, t) ≥ 0, ∀x ∈ Ω , t ∈ [0, T ], θi (x , t) ≥ 0, ∀x ∈ S2 , t ∈ [0, T ]. Next we prove the boundedness of ai and θi . Inserting ai (x, τ ) − mi , if a(x, τ ) > mi , + η := (ai (x, τ ) − mi ) = 0, if a(x, τ ) ≤ mi into Eq. (6) we get 1 2
+ Ωi,τ (mi )
τ =t (ai − mi )2 dxτ =0 + ki
t = 0
Sa+i ,τ (mi )
t = 0
Sa+i ,τ (mi )
t 0
+ Ωi,τ (mi )
a2ix dxdτ
ρi κii θi − κi (1 − θi )ai (ai − mi ) dx′ dτ ρi κii (θi − m′i ) + κii m′i − κi (1 − m′i )mi
− κi (1 − θi )(ai − mi ) + κi (θi − m′i )mi (ai − mi ) dx′ dτ
(11)
where + Ωi,τ (mi ) = {x ∈ Ω : ai (x, τ ) > mi },
Sa+i ,τ (mi ) = {x′ ∈ S2 : ai (x′ , 0, τ ) > mi }.
Similarly, from Eq. (2) it follows that t τ =t 1 2 ′ (θi − m′i )2 dx′ τ =0 + pi θix ′ dx dτ 2 Sθ+ ,τ (mi ) 0 Sθ+ ,τ (mi ) i i t × κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 (θi − m′i ) dx′ dτ Sθ+ ,τ (mi )
0
i
t = 0
Sθ+ ,τ (mi ) i
κi (1 − m′i )(ai − mi ) + κi (1 − m′i )mi − κii m′i
− κi (θi − m′i )ai − κii (θi − m′i ) − κ12 ρj θ1 θ2 (θi − m′i ) dx′ dτ where Sθ+i ,τ (mi ) = {x′ ∈ S2 : θi (x′ , τ ) > m′i }. Notice, that if a∗i0 ≥
∗ κii θi0 ∗ κi 1 − θi0
∗ θi0 ≤
κi a∗i0 κi a∗i0 + κii
then
and vice versa. Thus, if mi = a∗i0 then m′i =
κi mi κi a∗i0 = . κi a∗i0 + κii κi mi + κii
Similarly, if ∗ θi0 ≥
κi a∗i0 κi a∗i0 + κii
(12)
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then a∗i0 ≤
∗ κii θi0 ∗ κi 1 − θi0
∗ and vice versa. Thus if m′i = θi0 then
mi =
∗ κii θi0 κii m′i = . ∗ κi 1 − θi0 κi 1 − m′i
In both cases we have κi (1 − m′i )mi − κii m′i = 0.
(13)
Using (13) from equalities (11) and (12) we get t 1 1 (ai − mi )2 dx + (θi − m′i )2 dx′ + ki a2ix dxdτ + + 2 Ωi,t 2 Sθ+ ,τ (m′i ) (mi ) 0 Ωi,t (mi ) i t t 2 ′ + pi θix (ai − mi )(θi − m′i ) dx′ dτ ′ dx dτ ≤ ci Sθ+ ,τ (m′i )
0
+ ci1
0
i
t 0
Sa+i ,τ (mi )
Sθ+ ,τ (m′i )∩Sa+i ,τ (mi ) i
t
(ai − mi )2 dx′ dτ +
0
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ dτ
(14)
i
where constants ci and ci1 depend only on κi , κii , mi , ρi0 . Then, by estimate (10) we get the equality (ai − mi )2 dx′ ≤ (ai − mi )2+ dS ≤ ε a2ix dx + Cε (ai − mi )2 dx Sai ,τ (mi )
which reduces estimate (14) to (ai − mi )2 dx + + Ωi,τ (mi )
Sθ+ ,τ (m′i ) i
t + 2pi 0
+ Ωi,τ (mi )
S
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ + ki
2 ′ θix ′ dx dτ ≤ Ci
0
i
where Ci = ci max{Cε , 1}. Let t Φi (t) = 0
+ Ωi,τ (mi )
+ Ωi,τ (mi )
′
2
+ Ωi,τ (mi )
a2ix dxdτ
(ai − mi )2 dx′ +
(ai − mi ) dx +
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ dτ,
i
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ dτ.
i
Then Φi′ (t) ≤ Ci Φi (t) =⇒ Φi (t) ≤ 0 and (ai − mi )2 dx + + Ωi,τ (mi )
t 0
t
+ Ωi,τ (mi )
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ ≤ 0.
i
Therefore ai (x, t) ≤ mi for all (x, t) ∈ Ω × [0, T ] and θi (x′ , t) ≤ m′i for all (x′ , t) ∈ S2 × [0, T ]. The proof is complete. Remark 1. Let θi , i = 1, 2, in system (3) be nonnegative continuous function on S 2 × [0, T ], functions b0 and ρi satisfy the conditions of Assumption 3.2 and b be a classical solution of this problem. Then by a similar argument we can prove that b is nonnegative and bounded from above for all (x, t) ∈ Ω × [0, T ]. Remark 2. Let (a1 , a2 , θ1 , θ2 , b) be a classical solution of problem (1)–(3). Then the following mass conservation laws are true: ai (x, t) + b(x, t) dx + ρi (x)θi (x, t) dS = ai0 (x) + b0 (x) dx + ρi (x)θi0 (x) dS, i = 1, 2. Ω
S2
Ω
S2
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To prove these laws it is sufficient to add Eqs. (1) and (3), then integrate over cylinder Qt = Ω × (0, t), apply the formula of integration by parts, and use Eq. (2), boundary and initial conditions. 5. Uniqueness of the classical solution In this section, we prove that problem (1)–(3) has at most one classical solution. Let a ˜i , θ˜i , ˜b and a ˆi , θˆi , ˆb, i = 1, 2, be two classical solutions to problem (1)–(3) and set ai = a ˜i − a ˆi , θi = θ˜i − θˆi , b = ˜b − ˆb. Then from Eqs. (1) and (2) it follows that a′i − ki ∆ai = 0
in Ω × (0, T ],
on S1 × (0, T ], ki ∂ai /∂n + κi ρi (1 − θ˜i )ai = (κii + κi a ˆi )ρi θi ai = 0 in Ω ki ∂ai /∂n = 0
on S2 × (0, T ],
t=0
and 1 2
a2i
a2ix
dx + ki
Ω
t
ρi (κii + κi a ˆi )θi − κi ai (1 − θ˜i ) ai dx′ dτ.
dxdτ =
Qt
0
(15)
S2
Moreover, θi′ − pi ∆θi = κi ai (1 − θ˜i ) − (κii + κi a ˆi )θi − κ12 ρj (θ1 θ˜2 + θˆ1 θ2 )
in S2 × (0, T ),
∂θi /∂ν = 0 on Σ × (0, T ), θi t=0 = 0 in S2 and, hence,
1 2
θi2
t
′
dx + pi
S2
=
t
′
(κii + κi a ˆi )θi2 dx′ dτ
dx dτ +
S2
0
t
2 θix ′
0
S2
κi ai (1 − θ˜i ) − κ12 ρj (θ1 θ˜2 + θˆ1 θ2 ) θi dx′ dτ.
(16)
S2
0
Then from Eqs. (15) and (16) we get t t 1 a2i dx + ki a2ix dxdτ + κi ρi (1 − θ˜i )a2i dx′ dτ ≤ ci |ai ||θi | dx′ dτ, 2 Ω Qt 0 S2 0 S2 t t 1 2 ′ × θi2 dx′ + pi θix dx dτ + (κii + κi a ˆi )θi2 dx′ dτ ′ 2 S2 0 S2 0 S2 t t ≤ κi |ai ||θi | dx′ dτ + κ12 ρj |θ1 | + |θ2 | |θi | dx′ dτ, 0
S2
0
S2
where ci = max{(κii + κi m ˆ ′i )ρi0 }, m ˆ ′i = maxx′ ∈S2 ,t∈[0,T ] a ˆi (x′ , 0, t). We add up these inequalities, then summarize by i getting t 1 1 (a21 + a22 ) dx + (θ12 + θ22 ) dx′ + (k1 a21x + k2 a22x ) dxdτ 2 Ω 2 S2 0 Ω t t 2 2 ′ (p1 θ1x (κ11 θ12 + κ22 θ22 ) dx′ dτ + ′ + p2 θ2x′ ) dx dτ + S2
0
t ≤ 0
S2
c+κ 2 (a1 + a22 ) dx′ dτ + 2
0
t 0
S2
S2
c+κ ¯ 2 (θ1 + θ22 ) dx′ dτ 2
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where c = max{c1 , c2 }, κ = max{κ1 , κ2 }, κ ¯ = max{κ1 + 4κ12 ρ10 , κ2 + 4κ12 ρ20 }. We apply inequality (10) to a1 and a2 , then take ε = k1 /(c + κ) and ε = k2 /(c + κ) to get 1 t 1 (k1 a21x + k2 a22x ) dxdτ (a21 + a22 ) dx + (θ12 + θ22 ) dx′ + 2 Ω 2 0 Ω S2 t ≤ c1 ¯ ). (a21 + a22 ) dx′ + (θ12 + θ22 ) dx′ dτ, c1 = c1 (k1 , k2 , c, κ, κ11 , κ22 , κ 0
Ω
S2
Set Φ(t) =
t 0
(a21 + a22 ) dx′ +
Ω
(θ12 + θ22 ) dx′ dτ.
S2
Hence, ′ Φ ′ (t) ≤ 2c1 Φ(t) ⇒ Φ(t)e−2c1 t ≤ 0 ⇒ Φ(t)e−2c1 t ≤ Φ(0) = 0 which shows that a1 ≡ 0, a2 ≡ 0, and θ1 ≡ 0, θ2 ≡ 0, and thus b ≡ 0. Therefore, we have proved Theorem 5.1. Problem (1)–(3) has at most one classical solution. 6. Upper and lower sequences The construction of upper and lower solutions is based on the Positivity Lemma (see [2], Chapter 1, Lemma 4.1, p. 54). Lemma 6.1 (Positivity Lemma). Let w ∈ C (DT ) ∩ C 1,2 (DT ) satisfy the relations wt − D∇2 w + cw ≥ 0 α0 ∂w/∂n + β0 w ≥ 0 w(0, x) ≥ 0
in on in
DT = Ω × (0, T ], ST = S × (0, T ], Ω
where α0 ≥ 0, β0 ≥ 0, α0 + β0 > 0 on ST , and c = c(t, x) is a bounded function in DT , then w ≥ 0 in DT . For the sake of convenience we rewrite Eqs. (1) and (2) as follows: Li ai = 0 in Ω × (0, T ], Bi ai = gi (ai , θi , ρi ) on S × (0, T ],i = 1, 2, a in Ω i t=0 = ai0 where Li = ∂/∂t − ki ∆, Bi = ki ∂/∂n + κi ρ, 0, if x ∈ S1 , t ∈ (, T ], gi (ai , θi , ρi ) = (κii + κi ai )ρi θi , if x ∈ S2 , t ∈ (0, T ], Li θi = fi (ai , θ1 , θ2 , ρj ) in Ω × (0, T ], Bθi = 0 on Σ × (0, T ], θ in S2 i t=0 = θi0
(17)
(18)
where Li = ∂/∂t − pi ∆, B = ∂/∂ν, fi (ai , θ1 , θ2 , ρj ) = κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 ,
i, j = 1, 2, j ̸= i.
We construct the upper and lower solutions to Eqs. (17) and (18) by using a technique given in [2]. We observe, by definition and Lemma 3.1, that gi is nondecreasing in variables ai and θi , fi is nondecreasing in ai and nonincreasing in θ1 , θ2 .
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Definition 6.1. A pair of functions u ˜ := (˜ a1 , a ˜2 , θ˜1 , θ˜2 ) and u ˆ := (ˆ a1 , a ˆ2 , θˆ1 , θˆ2 ) such that a ˜i , a ˆi ∈ C (QT ) ∩ 2,1 2,1 C (QT ), θ˜i , θˆi ∈ C ([0, T ] × S2 ) ∩ C ((0, T ] × S2 ), i = 1, 2, are called upper and lower solutions of problem (1), (2) if u ˜≥u ˆ, (i.e. a ˜i ≥ a ˆi , and θ˜i ≥ θˆi , i = 1, 2) and if they satisfy the inequalities ˜i ≥ 0 ≥ Li a ˆi in Ω × (0, T ], Li a (19) Bi a ˜ i − gi (˜ ai ,θ˜i , ρi ) ≥ 0 ≥ Bi a ˆi − gi (ˆ ai , θˆi , ρi ) on S × (0, T ], a ˜i0 t=0 ≥ ai0 t=0 ≥ a ˆi0 t=0 in Ω , i = 1, 2, L1 θ˜1 − f1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) ≥ 0 ≥ L1 θˆ1 − f1 (ˆ a1 , θˆ1 , θ˜2 , ρ2 ) in Ω × (0, T ], L θ˜ − f (˜ ˆ ˜ ˆ a2 , θ˜1 , θˆ2 , ρ1 ) in Ω × (0, T ], 2 2 2 a2 , θ1 , θ2 , ρ1 ) ≥ 0 ≥ L2 θ2 − f2 (ˆ (20) ˜ ˆ Bθi ≥ 0 ≥ Bθi on Σ × (0, T ], ˜ θi t=0 ≥ θi0 ≥ θˆi t=0 in S2 , i = 1, 2. We note that f1 (a1 , θ¯1 , θ2 , ρ2 ) − f1 (a1 , θ1 , θ2 , ρ2 ) = −(κ1 a1 + κ11 + κ12 ρ2 θ2 )(θ¯1 − θ1 ) ≥ −c1 (θ¯1 − θ1 )
if θ¯1 ≥ θ1 (21)
f2 (a2 , θ1 , θ¯2 , ρ1 ) − f2 (a2 , θ1 , θ2 , ρ1 ) = −(κ2 a2 + κ22 + κ12 ρ1 θ1 )(θ¯2 − θ2 ) ≥ −c2 (θ¯2 − θ2 )
if θ¯2 ≥ θ2
and
where ci = κi mi + κii + κ12 ρj0 , j ̸= i. Here mi is defined in Lemma 3.1 and ρj0 = maxx∈S2 ρj (x). Let us consider the equivalent (1), (2) problem Li ai = 0 in Ω × (0, T ], Bi ai = Gi (ai , θi , ρi ) on S × (0, T ], a in Ω , i t=0 = ai0 Li θi = Fi (ai , θ1 , θ2 , ρj ) in Ω × (0, T ], Bθi = 0 on Σ × (0, T ], θ in S2 i t=0 = θi0
(22)
(23)
where Bi ai := Bi ai + ρi κi ai , Gi (ai , θi , ρi ) = gi (ai , θi , ρi ) + ρi κi ai , Li θi := Li θi + ci θi , Fi (ai , θ1 , θ2 , ρj ) = fi (ai , θ1 , θ2 , ρj ) + ci θi , i, j = 1, 2, j ̸= i. From definition it follows that: Gi (ai , θi , ρi ) is nondecreasing in ai and θi , F1 (a1 , θ1 , θ2 , ρ2 ) is nondecreasing in a1 and θ1 , F2 (a2 , θ1 , θ2 , ρ1 ) is nondecreasing in a2 and θ2 , but F1 (a1 , θ1 , θ2 , ρ2 ) and F2 (a2 , θ1 , θ2 , ρ1 ) are nonincreasing in θ2 and θ1 , respectively. (0) (0) (0) (0) Starting from a suitable initial iteration u0 = (ai , a2 , θ1 , θ2 ) we construct a sequence {u(k) } = (k) (k) (k) (k) {(ai , a2 , θ1 , θ2 )} from the iteration process (k) Li ai = 0 in Ω × (0, T ], (k) (k−1) (k−1) (24) Bi ai = Gi (ai , θi , ρi ) on S × (0, T ],i = 1, 2, (k) ai t=0 = ai0 in Ω , (k) (k−1) (k−1) (k−1) , θ1 , θ2 , ρj ) in Ω × (0, T ], Li θi = Fi (ai (k) (25) Bθi = 0 on Σ × (0, T ], (k) θ = θi0 in S2 , i
where i, j = 1, 2, j ̸= i.
t=0
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
220
It is evident that a ˆi = 0, θˆi = 0, i = 1, 2, is a lower solution and Eq. (13) and inequalities (19), (20) show ˜ that a ˜i = mi and θi = m′i , i = 1, 2, is an upper solution. We take (0)
(0)
(ai , θi ) = (˜ ai , θ˜i )
(0)
(0)
(ai , θi ) = (ˆ ai , θˆi ),
and
i = 1, 2,
in Eqs. (24) and (0) (0) (0) (a1 , θ1 , θ2 ) = (˜ a1 , θ˜1 , θˆ2 ), (0) (0) (0) (a , θ , θ ) = (˜ a2 , θˆ1 , θ˜2 ), 2
1
(0) (0) (0) (a1 , θ1 , θ2 ) = (ˆ a1 , θˆ1 , θ˜2 ), (0) (0) (0) (a , θ , θ ) = (ˆ a2 , θ˜1 , θˆ2 )
2
2
1
2
in Eqs. (25) getting sequences (k)
(k)
(k)
(k)
(k)
(k)
{ai }, {ai }, {θ1 }, {θ1 }, {θ2 }, {θ2 },
i = 1, 2.
These sequences form the other two sequences (k)
(k)
(k)
(k)
(k)
{u(k) } = {(a1 , a2 , θ1 , θ2 )}
(k)
(k)
(k)
and {u(k) } = {(a1 , a2 , θ1 , θ2 )}.
Now we prove that u ˆ ≤ u(1) ≤ · · · ≤ u(k) ≤ u(k+1) ≤ · · · ≤ u(k+1) ≤ u(k) ≤ · · · ≤ u(1) ≤ u ˜. (0)
Let wi
(0)
= ai
(1)
− ai
(26)
(1)
=a ˜i − ai . By Eqs. (24) and (19),
(0)
Li wi
in Ω × (0, T ],
=0
(0) Bi wi
(1)
(0)
(0)
= Bi a ˜i + ρi κi a ˜i − Bi ai = Bi a ˜i + ρi κi a ˜i − Gi (ai , θi , ρ) = Bi a ˜i − gi (˜ ai , θ˜i , ρi ) ≥ 0 on S × (0, T ], (0) wi t=0 = a ˜i t=0 − ai0 ≥ 0 in Ω , i = 1, 2. (1)
(0)
In view of Lemma 6.1, wi (x, t) ≥ 0 which shows that a ˜i (x, t) ≥ ai (x, t). Similarly, we prove that (1) a ˆi (x, t) ≤ ai (x, t). (0) (1) (1) (0) Let v = θ − θ = θ˜i − θ . By Eqs. (25) and (20), i
i
i
i
(0)
L1 v1
(0)
Bvi
(0) = L1 θ˜1 − L1 θ1 = L1 θ˜1 + c1 θ˜1 − F1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) = L1 θ˜1 − f1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) ≥ 0 in Ω × (0, T ],
on Σ × (0, T ], ˜ = θi t=0 − θi0 ≥ 0
=0
(0) v1 t=0
in S2 .
(1) (1) (0) In view of Lemma 6.1, v1 (x, t) ≥ 0. Hence, θ˜1 (x, t) ≥ θi (x, t). Similarly, we get that θ˜2 (x, t) ≤ θ2 (x, t) (1) and θˆi (x, t) ≤ θi (x, t), i = 1, 2. (1) (1) (1) Let wi = ai − ai . Then, by Eqs. (24) and since Gi (ai , θi , ρi ) is nondecreasing in ai and θi , (1)
Li wi
(1)
(1)
= Li ai
(1)
(1)
− Li ai
(1)
=0
in Ω × (0, T ], (0)
(0)
(0)
(0)
Bi wi = Bi ai − Bi ai = Gi (ai , θi , ρi ) − Gi (ai , θi , ρi ) ≥ 0 (1) wi t=0 = ai0 − ai0 = 0 in Ω , i = 1, 2. (1)
(1)
(1)
In view of Lemma 6.1, wi (x, t) ≥ 0. Hence a1 (x, t) ≥ ai (x, t).
on S × (0, T ],
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
(1)
Let vi
(1)
(1)
− θi . Then by Eqs. (25) and monotone property of Fi we have
= θi
(1) (1) a1 , θ˜1 , θˆ2 , ρ2 ) − F1 (ˆ a1 , θˆ1 , θ˜2 , ρ2 ) = L1 θ1 − L1 θ1 = F1 (˜ = F1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) − F1 (ˆ a1 , θ˜1 , θˆ2 , ρ2 ) + F1 (ˆ a1 , θ˜1 , θˆ2 , ρ2 ) − F1 (ˆ a1 , θˆ1 , θˆ2 , ρ2 ) ˜ ˆ ˆ ˜ + F1 (ˆ a1 , θ1 , θ2 , ρ2 ) − F1 (ˆ a1 , θ1 , θ2 , ρ2 ) ≥ 0 in Ω × (0, T ],
(1)
L1 v1
(1)
Bvi
221
on Σ × (0, T ],
=0
(1) v1 t=0
in S2 .
=0
(1)
(1)
(1)
(1)
(1)
In view of Lemma 6.1, v1 (x, t) ≥ 0. Hence, θ1 (x, t) ≥ θ1 (x, t). Similarly, θ2 (x, t) ≥ θ2 (x, t). Thus, we have proved that u ˆ = u(0) ≤ u(1) ≤ u(1) ≤ u(0) = u ˜. (k−2)
Let ai
(k−1)
− ai
(k−1)
≥ 0. Then by Eq. (24) and monotone property of Gi (k)
Li (ai
− ai ) = 0
(k−1) Bi (ai (k−1) (ai −
(k) (k−2) (k−2) − ai ) = Gi (ai , θi , ρi ) (k) ai ) t=0 = 0 in Ω , i = 1, 2.
in Ω × (0, T ],
(k−1)
In view of Lemma 6.1 we have ai (k−1)
ai
(k−1)
− Gi (ai
(k−1)
, θi
, ρi ) ≥ 0
on S × (0, T ],
(k)
(x, t) ≥ ai (x, t). Similarly, we prove that (k−1)
(k)
(x, t) ≤ ai (x, t),
θi
(k)
(x, t) ≥ θi (x, t),
(k−1)
θi
(k)
(x, t) ≤ θi (x, t),
i = 1, 2. Hence we have proved inequalities (26). Now we prove that u(k) and u(k) are upper and lower solutions. From (24) and (25) by (21) and the monotone property of gi and f1 we get (k)
Li ai
(k)
Bi ai
(k)
i
t=0
(k−1)
(k−1)
(k) (k−1) , ρi ) = −ρi κi ai + ρi κi ai (k−1) (k−1) (k−i) (k) (k−1) (k−1) , θi , ρi ) = ρi κi (ai − ai ) + gi (ai , θi , ρi ) + gi (ai (k) (k−1) (k) (k−1) (k) (k) (k) (k) − gi (ai , θi , ρi ) + gi (ai , θi , ρi ) − gi (ai , θi , ρi ) + gi (ai , θi , ρi ) (k) (k) gi (ai , θi , ρi ) on S × (0, T ],
= −ρi κi ai
≥ (k) a
in Ω × (0, T ],
=0
= ai0
+ Gi (ai
, θi
in Ω
and (k)
L1 θ1
(k)
(k−1)
= −c1 θ1 + F1 (a1 =
(k−1)
, θ1
(k−1)
, θ2
, ρ2 ) (k−1) (k) (k−1) (k−1) (k−1) c1 (θ1 − θ1 ) + f1 (a1 , θ1 , θ2 , ρ2 ) (k) (k−1) (k−1) (k) (k−1) (k−1) − f1 (a1 , θ1 , θ2 , ρ2 ) + f1 (a1 , θ1 , θ2 , ρ2 ) (k) (k) (k−1) (k) (k) (k−1) − f1 (a1 , θ1 , θ2 , ρ2 ) + f1 (a1 , θ1 , θ2 , ρ2 ) (k) (k) (k) (k) (k) (k) − f1 (a1 , θ1 , θ2 , ρ2 ) + f1 (a1 , θ1 , θ2 , ρ2 ) ≥ 0 in Ω
× (0, T ],
(k)
Bθ1 = 0 on Σ × (0, T ], (k) θ1 t=0 = θ10 in S2 . (k)
(k)
(k)
Similarly, we get that θ2 satisfies inequalities (20), and ai , θi satisfy relations (19) and (20), respectively. Hence, u(k) and u(k) are the upper and lower solutions and satisfy inequalities (26).
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
222
Since {u(k) } is nondecreasing and bounded from above, and {u(k) } is nonincreasing and bounded from below, they converge monotonically to some limits lim u(k) = u,
lim u(k) = u
k→∞
k→∞
which satisfy relations u ˆ ≤ u(1) ≤ · · · ≤ u(k) ≤ u(k+1) ≤ · · · ≤ u ≤ u ≤ · · · ≤ u(k+1) ≤ u(k) ≤ · · · ≤ u(1) ≤ u ˜. In next session we prove that u and u are solutions of system (1), (2). 7. Existence of the solution (0) (0) (0) (0) Problem (24) starting from (ai , θi ) = (˜ ai , θ˜i ) or (ai , θi ) = (ˆ ai , θˆi ) has a unique classical solution (k) (k) 2,1 2,1 ai ∈ C (Ω × (0, T ]) ∩ C (Ω × [0, T ]) or ai ∈ C (Ω × (0, T ]) ∩ C (Ω × [0, T ]), respectively, for any k = 1, 2, . . ., which can be represented by the formula (see. [11,12]) (k) ai (x, t)
t Γi (x − ξ, t −
= 0 (k)
Moreover, the function ai
(k) τ )ϕi (ξ, τ ) dSξ dτ
Γi (x − y, t)ai0 (y) dy.
+
S
(27)
Ω
is H¨ older continuous in x uniformly in Ω × [0; T ] (cf. [2]). Here Γi (x, t) =
1 4πki t
n/2 e (k)
is a fundamental solution of Eq. (1). Function ϕi t
(k)
|x|2 it
− 4k
,
x ∈ Rn , t > 0
is a solution on S × [0, T ] of the equation
(k)
ϕi (η, t) =
(k−1)
Qi1 (η, t, ξ, τ )ϕi (ξ, τ ) dSξ dτ + qi (ai 0
(k−1)
, θi
, ρi )(η, t)
(28)
S
where ∂Γ (η − ξ, t − τ ) i Qi1 (η, t, ξ, τ ) = −2 ki + κi ρi (η)Γi (η − ξ, t − τ ) , ∂nη (k−1) (k−1) (k−i) (k−i) qi (ai , θi , ρi )(η, t) = 2Gi (ai , θi , ρi )(η, t) + Qi1 (η, t, y, 0)ai0 (y) dy. Ω
Eq. (28) is the Volterra integral equation with a weak singularity. Its solution can be represented by the formula (see. [11,12]) ∞ t (k) (k−1) (k−1) (k−1) (k−1) ϕi (η, t) = qi (ai , θi , ρi )(η, t) + Qij (η, t, ξ, τ )qi (ai , θi , ρi )(ξ, τ ) dSξ dτ, j=1
0
S
t Qij (η, t, ξ, τ ) =
Qi1 (η, t, ζ, s)Qij−1 (ζ, s, ξ, τ ) dSζ ds, τ
j = 2, 3, . . . .
S
Problem (25), starting from (0) (0) (0) (a1 , θ1 , θ2 ) = (˜ a1 , θ˜1 , θˆ2 )
or
(0) (0) (0) (a1 , θ1 , θ2 ) = (ˆ a1 , θˆ1 , θ˜2 )
or
(a2 , θ1 , θ2 ) = (ˆ a2 , θ˜1 , θˆ2 )
for i = 1 and (0)
(0)
(0)
(a2 , θ1 , θ2 ) = (˜ a2 , θˆ1 , θ˜2 )
(0)
(0)
(0)
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
(k)
223
(k)
for i = 2, has a unique solution θi ∈ C 2,1 (S2 × (0, T ]) ∪ C (S2 × [0, T ]) and θi ∈ C 2,1 (S2 × (0, T ]) ∪ C (S2 × [0, T ]) for any k = 1, 2, . . .. This solution can be represented by the formula (see [11,12]) t (k−1) (k−1) (k−1) (k) Zi (x′ − y ′ , t − τ )Fi (ai , θ1 θi (x′ , t) = , θ2 , ρj )(y ′ , τ ) dy ′ dτ S2
0
t
(k)
Zi (x′ − η ′ , t − τ )µi (η ′ , τ ) dΣη′ dτ +
+ Σ
0 (k)
Moreover, the function θi
Zi (x′ − y ′ , t)θi0 (y ′ ) dy ′ .
(29)
S2
is H¨ older continuous in x′ uniformly in S2 × [0; T ] (cf. [2]). Here
Zi (x′ , t) =
|x′ |2 1 − −ci t , (n−1)/2 e 4pi t e 4πpi t
x′ ∈ Rn−1 , t > 0 (k)
is a fundamental solution of the equation θi′ − pi ∆θi + ci θi = 0. The function µi solution of the integral equation t (k) (k) µi (ξ ′ , t) = Ri1 (ξ ′ , t, η ′ , τ )µi (η ′ , τ ) dΣη′ dτ
is continuous and bounded
Σ
0
(k−1)
+ ri (ai
(k−1)
, θ1
(k−1)
, θ2
ρj )(ξ ′ , t),
ξ ′ ∈ Σ , t ∈ ×[0, T ]
(30)
and can be represented by the formula (see [11,12]) (k)
(k−1)
(k−1)
(k−1)
µi (ξ ′ , t) = ri (ai , θ1 , θ2 ρj )(ξ ′ , t) ∞ t (k−1) (k−1) (k−1) + Rij (ξ ′ , t, η ′ , τ )ri (ai , θ1 , θ2 ρj )(ξ ′ , t) dΣη′ dτ, j=1 (k−1)
ri (ai
(k−1)
, θ1
0
(k−1)
, θ2
Σ
ρj )(ξ ′ , t) =
Ri1 (ξ ′ , t, y ′ , τ )θi0 (y ′ ) dy ′ S2 t (k−1) (k−1) (k−1) + Ri1 (ξ ′ , t, y ′ , τ )Fi (ai , θ1 , θ2 , ρj )(y ′ , τ ) dy ′ dτ, 0
D
∂Zi (ξ ′ − η ′ , t − τ ) Ri1 (ξ ′ , t, η ′ , τ ) = −2 , ∂νξ′ t Rij+1 (ξ ′ , t, η ′ , τ ) = Ri1 (ξ ′ , t, ζ ′ , s)Rij (ζ ′ , s, η ′ , τ ) dΣζ ′ ds, τ
j = 1, 2, . . . .
Σ
Because of the continuity and boundedness of Fi , assumptions on θi0 , and by dominated convergence theorem, (k−1)
ri (ai
(k−1)
, θ1
(k−1)
, θ2
k→∞
, ρj )(x, t) −→ ri (ai , θ1 , θ2 , ρj )(x, t).
Moreover, ri (ai , θ1 , θ2 , ρj ) is a continuous function of (x, t). Similarly, due to the continuity and boundedness of Gi , assumptions on ai0 , and by dominated convergence theorem, (k−1)
qi (ai
(k−1)
, θi
k→∞
, ρi )(x, t) −→ qi (ai , θi , ρi )(x, t), (k)
where qi (ai , θi , ρi ) is a continuous function of (x, t). Then, by [2, Lemma 2.2, p. 56], function ϕj (x, t) (k)
defined by Eq. (28) and function µj (x, t) defined by Eq. (30) converge to continuous solutions ϕj (x, t) and µj (x, t) that satisfy the integral equations t ϕi (η, t) = 2 Qi1 (η, t, ξ, τ )ϕi (ξ, τ ) dSξ dτ + qi (ai , θi , ρi )(η, t) 0
S
and µi (ξ ′ , t) =
t 0
Σ
Ri1 (ξ ′ , t, η ′ , τ )µi (η ′ , τ ) dΣη′ dτ + ri (ai , θ1 , θ2 ρj )(ξ ′ , t),
ξ ′ ∈ Σ , t ∈ ×[0, T ],
224
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(k)
respectively. Letting k → ∞ in (27) and (29) we get that the limit function ai of sequences {ai } is defined by the formula t ai (x, t) = Γi (x − ξ, t − τ )ϕi (ξ, τ ) dSξ dτ + Γi (x − y, t)ai0 (y) dy, S
0
Ω (k)
while the limit function θi of sequences {θi } satisfy the integral equation t θi (x′ , t) = Zi (x′ − y ′ , t − τ )Fi (ai , θ1 , θ2 , ρj )(y ′ , τ ) dy ′ dτ S2
0
t
′
′
′
Zi (x − η , t − τ )µi (η , τ ) dΣη′ dτ +
+ 0
Σ
Zi (x′ − y ′ , t)θi0 (y ′ ) dy ′ .
S2
These functions represent ai , θi and ai , θi , i = 1, 2. Thus we have proved that the functions u = (a1 , a2 , θ1 , θ2 ) and u = (a1 , a2 , θ1 , θ2 ) are solutions of problem (1), (2). By the uniqueness of the solution u = u. Obviously, system (3) with given continuous functions θ1 and θ2 has a unique classical solution. The proof of Theorem 3.1 is complete. 8. The long-time behaviour of the solutions In this section we consider the problem of the long-time behaviour of the solutions to problem (1), (2) and assume that S ∈ C 2+α ,
Σ ∈ C 2+α ,
ρi ∈ C 1+α ,
i = 1, 2.
(31)
We need two theorems and one lemma used in [2]: Theorem 8.1 (See [2, Chapter 2, Theorem 1.3, p. 53]). Let β0 ≥ 0, c ≥ 0, and either β0 or c be strictly positive. If q(t, x), h(t, x) converge to zero uniformly in Ω and ∂Ω , respectively, as t → ∞, then for any initial function u0 the solution u(t, x) of problem n n u − a (t, x)u + bj (t, x)uxj + c(t, x)u = q(t, x) in DT , t ij x x i j i,j=1
j=1
α0 (t, x)∂u/∂ν + β0 (t, x)u = h(t, x) u(0, x) = u0 (x)
on ST , in Ω
converges uniformly in Ω to zero as t → ∞. The convergence of u to zero is in the L2 (Ω )-space if q(t, ·), h(t, ·) converge to zero in L2 (Ω ) and L2 (∂Ω ), respectively, as t → ∞. Lemma 8.1 (See [2, Chapter 3, Lemma 1.2, p. 95]). For any p > n the Sobolev space Wpm (Ω ) is continuously embedded in C m−1+µ (Ω ) with µ = 1−n/p. Specifically, there exists a constant K such that for all u ∈ Wpm (Ω ) |u|Ω m−1+µ ≤ K∥u∥Wpm (Ω) . Theorem 8.2 (See [2, Chapter 3, Theorem 1.3, p. 98]). Let q ∈ C α (Ω ), h ∈ C 1+α (∂Ω ) when α0 > 0, and h ∈ C 2+α (∂Ω ) when α0 = 0, and let c ≥ 0 and not be identically zero when β0 ≡ 0. Then there exists a unique solution u to problem n n − aij (x)uxi xj + bj (x)uxj + c(x)u = q(x) in Ω , i,j=1 j=1 α0 (x)∂u/∂ν + β0 (x)u = h(x) on ∂Ω
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and u ∈ C 2+α (Ω ). Moreover u satisfies the Schauder estimate Ω ∂Ω |u|Ω 2+α ≤ K |q|α + |h|1+α where K is a constant independent of u, q, and h. To study the long-time behaviour of the solutions we also need Lemma 8.2. Let (¯ ai , θ¯i ) and (ai , θi ) be the solutions of time-dependent system (1), (2) with ai0 = a ˜i , θi0 = θ˜i ˆ ˜ ˆ and ai0 = a ˆi = 0, θi0 = θi = 0, respectively, where (˜ ai , θi ) and (ˆ ai , θi ), i = 1, 2, are the same upper and lower solutions of the steady-state and time-dependent problems (34), (35) and (1), (2), respectively. Let (a∗i , θi∗ ), i = 1, 2, be a classical solution of time-dependent system (1), (2) with ai0 ∈ [ˆ ai , a ˜i ], θi0 ∈ [θˆi , θ˜i ], i = 1, 2. Then: (i) a ¯′i ≤ 0, a′i ≥ 0, a ¯i ≥ ai in Ω × [0, ∞), ′ ′ ¯ θi ≤ 0, θi ≥ 0, θ¯i ≥ θi in S2 × [0, ∞); (ii) limt→∞ a ¯i = a ¯is , limt→∞ θ¯i = θ¯is , limt→∞ ai = ais , limt→∞ θi = θis exist and a ¯is ≥ ais , θ¯is ≥ θis ; (iii) ai ≤ a∗i ≤ a ¯i , θi ≤ θi∗ ≤ θ¯i . Proof. We first prove propositions (i) and (ii). Let a ¯δi = a ¯i (x, t+δ), θ¯iδ = θ¯i (x, t+δ), wi = a ¯i −¯ aδi , vi = θ¯i −θ¯iδ where δ ≥ 0 is small constant. Then from Eqs. (1), (2) with initial functions (a10 , θi0 ) = (˜ ai , θ˜i ) and ˆ (a10 , θi0 ) = (ˆ ai , θi ) we derive the systems wi′ − ki ∆wi = 0 in Ω × (0, T ], k ∂w /∂n = 0 on S1 × (0, T ], i i (32) ¯ ki ∂w ¯δi )vi on S2 × (0, T ], i /∂n + κi ρi 1 − θi wi = ρi (κii + κi a w ai − a ¯δi )|t=0 ≥ 0, in Ω , i t=0 = (˜ ′ vi − pi ∆vi = κi (1 − θ¯i )wi −(κ + κ a δ ¯ ¯δ in S2 × (0, T ], ii i ¯i + κ12 ρj θj )vi − κ12 ρj θi vj (33) p ∂v /∂ν = 0 on Σ × (0, T ], i i v ˜ ¯δ in S2 , i t=0 = (θi − θi )|t=0 ≥ 0 i = 1, 2, i ̸= j, j = 1, 2. It is evident that (w ˆi , vˆi ) = (0, 0), i = 1, 2, is a lower solution of system (1), (2). In view of proof of Theorem 3.1, this system has a unique nonnegative solution if there exists a positive upper solution (w ˜i , v˜i ). Let ri ∈ C 2 (Ω ) be a nonnegative function such that ∂ri /∂n ≥ 1 on S, i = 1, 2, and let 2 qi ∈ C (S2 ) be a nonnegative function such that ∂qi /∂ν ≥ 0 on Σ , i = 1, 2. Define w ˜i = λi exp{µt + µi ri },
v˜i = βi exp{µt + qi },
where λi , βi , µi , i = 1, 2, and µ are some positive constants to be chosen. Then (w ˜i , v˜i ), i = 1, 2, is a positive upper solution of problem (32), (33) if the inequalities w ˜i′ − ki ∆w ˜i = λi µ − ki µi ∆ri + µi |∇ri |2 exp{µt + µi ri } ≥ 0, in Ω × (0, T ], ki ∂ w ˜i /∂n ≥ 0 on S1 × (0, T ], δ ¯ ki ∂ w ˜i /∂n + κi ρi 1 − θi w ˜i − ρi (κii + κi a ¯i )˜ vi = λi ki µi + κi ρi (1 − θ¯i ) exp{µt + µi ri } −(κi aδi + κii )ρi βi exp{µt + qi } ≥ 0 on S2 × (0, T ], δ δ w ˜i t=0 − (˜ ai − ai )|t=0 = λi exp{µi ri } − (˜ ai − ai )|t=0 ≥ 0 in Ω
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and v˜i′ − pi ∆˜ vi − κi 1 − θ¯i w ˜i + (κii + κi a ¯δ + κ12 ρj θ¯j )˜ v i i δ +κ12 ρj θ¯i vˆj = βi exp{µt + qi } µ − pi ∆qi + |∇qi |2 −κi 1 − θ¯i exp{µi ri − qi }λi /βi + κii + κi a ¯δi + κ12 ρj θ¯j ≥ 0 in S2 × (0, T ], ∂˜ vi /∂ν = βi exp{µt + qi }∂qi /∂ν ≥ 0 on Σ × (0, T ], v˜i t=0 − (θ˜i − θ¯iδ )|t=0 = βi exp{qi } − (θ˜i − θ¯iδ )|t=0 ≥ 0 in S2 are satisfied. It is evident that: (a) initial inequalities are satisfied by choosing sufficiently large λi and βi ; (b) boundary inequality for w ˜i is satisfied with sufficiently large µi ; (c) differential equations for w ˜i and v˜i can be satisfied by choosing sufficiently large µ. Thus w ˜i and v˜i form the upper solution of system (32), (33). Then by Theorem 3.1, wi ≥ 0 and vi ≥ 0. This shows that a ¯′i ≤ 0 and θ¯i′ ≤ 0. The proof of a′i ≥ 0 and θ′i ≥ 0 is similar. Functions a ¯i and θ¯i are non-increasing and bounded from below. Functions ai and θi are non-decreasing and bounded from above. Therefore, the limits limt→∞ a ¯i = a ¯is , limt→∞ θ¯i = θ¯is , limt→∞ ai = ais , limt→∞ θi = θis exist. Now we prove that a ¯is ≥ ais and θ¯is ≥ θis . Let wi = a ¯i − ai , vi = θ¯i − θi . Then by the argument above we derive the system for wi and vi , wi′ − ki ∆wi = 0 in Ω × (0, T ], k ∂w /∂n = 0 on S1 × (0, T ], i i ¯i wi = ρi (κii + κi a )vi on S2 × (0, T ], ki ∂w /∂n + κ ρ 1 − θ i i i i w ai − ai )|t=0 ≥ 0, in Ω , i t=0 = (˜ ′ ¯ vi − pi ∆vi = κi (1 − θi )wi −(κ + κ a + κ ρ θ¯ )v − κ ρ θ v in S × (0, T ], ii i 12 j j i 12 j j 2 i
pi ∂vi /∂ν = 0 v ˜ i t=0 = (θi − θ i )|t=0 ≥ 0
i
on Σ × (0, T ], in S2 ,
i = 1, 2, i ̸= j, j = 1, 2. This system is of type (32), (33) and therefore wi ≥ 0 and vi ≥ 0. Hence, a ¯i ≥ ai , θ¯i ≥ θ and a ¯is ≥ a , θ¯is ≥ θ . i
is
is
It remains to prove proposition (iii). Let (a∗i , θi∗ ), i = 1, 2, be a solution of system (1), (2) with ai0 ∈ [ˆ ai , a ˜i ], ∗ ∗ ˆ ˜ ˆ θi0 ∈ [θi , θi ]. If we consider pairs the (ˆ ai , θi ) and (ai , θi ) as the lower and upper solutions of system (1), (2), then we get that ai ≤ a∗i , θi ≤ θi∗ . Similarly, if we take the pairs (a∗i , θi∗ ), (˜ ai , θ˜i ) as the lower and upper ∗ ∗ solutions, then we get that ai ≤ a ¯i , θi ≤ θ¯i . The proof of lemma is complete. The steady-state system corresponding to time-depending problem (1), (2) is −ki ∆ai = 0 in Ω , ki ∂ai /∂n = 0 on S1 , k ∂a /∂n + κ ρ (1 − θ )a = κ ρ θ on S , i i i i i i ii i i 2 −pi ∆θi = κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 , in S2 , pi ∂θi /∂ν = 0 on Σ , i = 1, 2, j ̸= i, j = 1, 2.
(34)
(35)
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Now we prove that a ¯is , θ¯is and ais , θis , i = 1, 2, are classical solutions of system (34), (35). Let wi ∈ 2 vi ∈ Wp (S2 ) be a generalized solution of problem ¯is in Ω , −ki ∆wi + Ci wi = Ci a (36) ki ∂wi /∂n = 0 on S1 , k ∂w /∂n + κ ρ (1 − θ¯ )¯ ¯ on S2 , i i i i is ais = κii ρi θis −pi ∆vi + ci vi = ci θ¯is + κi (1 − θ¯is )¯ ais − κii θ¯is − κ12 ρj θ¯1s θ¯2s in S2 (37) pi ∂θi /∂ν = 0 on Σ ,
Wp2 (Ω ),
i = 1, 2, j ̸= i, j = 1, 2. From systems (1), (2) and (36), (37) we derive equations for Wi = a ¯i − wi , Vi = θ¯i − vi , Wi′ − ki ∆wi + Ci Wi = Ci (¯ ai − a ¯is ) in Ω × (0, T ], k ∂W /∂n = 0 on S × (0, T ], i i 1 k ∂W /∂n + κ ρ (1 − θ¯i )¯ ai = κii ρi θ¯i + κi ρi (1 − θ¯is )¯ ais − κii ρi θ¯is on S2 × (0, T ], i i i i W | ˜i − wi in Ω , i t=0 = a Vi′ − pi ∆Vi + ci Vi = ci (θ¯i − θ¯is ) + κi (1 − θ¯i )¯ ai − κii θ¯i −κ ρ θ¯ θ¯ − κ (1 − θ¯ )¯ ¯ ¯ ¯ in S2 × (0, T ] 12 j 1 2 i is ais + κii θis + κ12 ρj θ1s θ2s pi ∂Vi /∂ν = 0 on Σ × (0, T ], V | ˜ in S2 . i t=0 = θi − vi Theorem 8.1 shows that
a ¯is − wi
2
dx = 0,
Ω
θ¯is − vi
2
dx = 0
Ω
since the right-hand sides of equations of these systems converge to zero as t → ∞. This shows that wi = a ¯is 1+α ¯ in L2 (Ω ) and vi = θis in L2 (S2 ). By choosing p > n the embedding Lemma 8.1 implies that a ¯is ∈ C (Ω ), θ¯is ∈ C 1+α (S2 ). Then Theorem 8.2 shows that a ¯is ∈ C 2+α (Ω ), θ¯is ∈ C 2+α (S2 ). Thus a ¯is , θ¯is , i = 1, 2, is a classical solution of steady-state problem (34), (35). By the same argument we can prove that ais , θis , i = 1, 2, is a classical solution of the same problem. We observe that the steady-state problem has not a unique solution, since functions κ2 c a1 = 0, θ1 = 0, a2 = c, θ2 = , κ2 c + κ22 and κ1 c , a2 = 0, θ2 = 0 a1 = c, θ1 = κ1 c + κ11 where c = const ≥ 0 satisfy this system. But there exists a domain of initial functions such that each solution of the time-dependent problem with (ai0 , θi0 ), i = 1, 2, from this domain monotonically converges to corresponding solution of the steady-state problem as t → ∞. Theorem 8.3. Let (ai , θi ) be a classical solution of system (1), (2) with initial functions (ai0 , θi0 ) ∈ [¯ ais , a ˜i ] × [θ¯is , θ˜i ] ∪ [ˆ ai , ais ] × [θˆi , θis ],
i = 1, 2.
Then limits lim ai = a ¯is ,
t→∞
lim θi = θ¯is
with (ai0 , θi0 ) ∈ [¯ ais , a ˜i ] × [θ¯is , θ˜i ]
lim θi = θis
with (ai0 , θi0 ) ∈ [ˆ ai , ais ] × [θˆi , θis ],
t→∞
and lim ai = ais ,
t→∞
i = 1, 2, exist.
t→∞
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To prove this theorem we observe that (¯ ais , θ¯is ) and (ais , θis ), i = 1, 2, are also solutions of system (1), ¯ ais , θ¯is ) and (2) provided that ai0 = a ¯is , θi0 = θis and ai0 = ais , θi0 = θis , respectively. Then by considering (¯ ˜ (˜ ais , θis ), i = 1, 2, as lower and upper solutions of problem (1) and (2), Theorem 3.1 implies that a ¯is ≤ ai ≤ a ¯i ,
θ¯is ≤ θi ≤ θ¯i
if ai0 ∈ [¯ ais , a ˜is ], θi0 ∈ [θ¯i , θ˜i ], i = 1, 2.
a ˆi ≤ ai ≤ ais ,
θˆi ≤ θi ≤ θis
if ai0 ∈ [ˆ ai , ais ], θi0 ∈ [θˆis , θis ], i = 1, 2.
Similarly,
Corollary 8.1. Under assumption (31) all classical solutions of problem (1), (2) corresponding to initial functions (ai0 , θi0 ) ∈ [¯ ais , a ˜i ] × [θ¯is , θ˜i ] and (ai0 , θi0 ) ∈ [ˆ ai , ais ] × [θˆi , θis ] monotonically converge to classical ¯ solutions, (¯ ais , θis ) and (ais , θis ), of the steady-state problem, respectively, as t → ∞. Remark 8.1. In the case where (ai0 , θi0 ) ∈ (ais , a ¯is ) × (θis , θ¯is ), the existence of limits lim ai ,
t→∞
lim θi
t→∞
is an open problem. If these limits exist, then under assumption (31) they are classical solutions of the steady-state problem. Each solution of the steady-state problem is also solution of the time-dependent problem if (ai0 , θi0 ), i = 1, 2, is a solution of steady-state problem. References [1] C.V. Pao, W.H. Ruan, Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions, J. Math. Anal. Appl. 333 (2007) 472–499. [2] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, New York, 1992. [3] J.R. Anderson, K. Deng, Global existence for nonlinear diffusion equations, J. Math. Anal. Appl. 196 (1995) 479–501. [4] S. Carl, Ch. Grossmann, C.V. Pao, Existence and monotone iterations for parabolic differential inclusions, Commun. Appl. Nonlinear Anal. 3 (1996) 1–24. [5] P.E. Sacks, Continuity of solutions of a singular parabolic equation, J. Nonlinear Anal. 7 (1983) 384–407. [6] A. Ambrazeviˇ cius, Solvability of a coupled system of parabolic and ordinary differential equations, Cent. Eur. J. Math. 8 (3) (2010) 537–547. [7] V. Skakauskas, P. Katauskis, Modelling dimer-dimer reactions on supported catalysts, J. Math. Chem. 53 (2015) 604–617. [8] A. Ambrazeviˇ cius, Solvability theorem for a model of a unimolecular heterogeheous reaction with adsorbate diffusion, J. Math. Sci. (N. Y.) 184 (4) (2012) 383–398. (Translated from Probl. Mat. Anal. 65 (2012) 13–26). [9] V. Skakauskas, P. Katauskis, Spillover in monomer-monomer reactions on supported catalysts: dynamic mean-feld study, JOMC 52 (5) (2014) 1350–1363. [10] D.K. Fadeev, B.Z. Vulix, N.N. Ural’ceva, Selected Chapters of Analysis and Algebra, LGU, 1981, p. 200 (in Russian). [11] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Clifs, NJ, 1964. [12] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasi-linear Equation of Parabolic Type, Am. Math. Soc. Transl., Providence, RI, 1968 (English transl.).
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Solvability of a model for monomer–monomer surface reactions A. Ambrazeviˇcius ∗ , V. Skakauskas Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania
article
info
Article history: Received 21 June 2016 Received in revised form 25 October 2016 Accepted 1 November 2016
Keywords: Coupled parabolic systems Heterogeneous catalysis Reaction–diffusion system
abstract A mathematical model for a monomer–monomer surface reaction is considered taking into account the surface diffusion of adsorbed particles of both reactants. The model is described by a coupled system of parabolic equations where some of them are defined in a domain and the other ones have to be solved on the domain surface. The existence and uniqueness theorem of a classic solution for the time-dependent problem is proved. Non-uniqueness of solutions for the steady-state problem is established. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Coupled systems of nonlinear parabolic and ordinary differential equations determined in the same domain have been extensively studied in literature (see [1–5] and references therein). In modelling of chemical surface reactions there arise coupled systems of parabolic and ordinary differential equations where some of them are solved in a domain and the other ones are defined on the domain surface (see [6,7] and references therein). In order the surface reaction to occur reactants must diffuse from domain toward the catalyst surface, adsorb on them, and their adsorbates have to react forming a product. The bulk diffusion of reactants toward the surface is described by parabolic PDEs that have to be solved in a domain. After adsorption reactant particles can stay or diffuse on the surface. The surface diffusion of adsorbed particles is described by parabolic PDEs that are determined on the domain surface. In [8] a unimolecular surface reactions model is studied which consists of two parabolic equations. One of them is solved in a domain while the other one is determined on the domain surface. A mathematical model for surface reactions between two monomers (unimolecular reactants) competing for the same adsorption site of composite catalyst is studied numerically in [9]. In the present paper we consider a mathematical model for surface reactions between two monomers that adsorb on different sites of a homogeneous surface and prove the existence and uniqueness theorem of a classical solution. ∗ Corresponding author. E-mail addresses: [email protected] (A. Ambrazeviˇ cius), [email protected] (V. Skakauskas).
http://dx.doi.org/10.1016/j.nonrwa.2016.11.001 1468-1218/© 2016 Elsevier Ltd. All rights reserved.
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The paper is organized as follows. In Sections 2 and 3, we describe the model and formulate the solvability theorem for the time-dependent problem, respectively. In Section 4, we prove a priori estimates. Section 5 is devoted to the uniqueness of the classical solution. In Sections 6 and 7, we construct the upper and lower solutions and prove the existence of a classical solution to problem (1)–(3), respectively. At last in Section 8, we consider the corresponding steady-state problem. 2. Formulation of the problem We consider the case where particles of reactants A1 and A2 do not compete for the adsorption sites and suppose that reactants A1 and A2 adsorb on the sites P1 and P2 of the catalyst surface, respectively. Suppose, that reactant Ai , i = 1, 2, occupies a bounded domain Ω ⊂ Rn , n ≥ 3, ai = ai (x, t) is the concentration of Ai at the point x ∈ Ω at time t, S := ∂Ω is (n − 1) dimension surface in Rn , S2 ⊂ S is a domain in Rn−1 on the plane xn = 0 (surface of the adsorbent), S1 = S \ S2 , ∂S2 = ∂S1 := Σ is n − 2 dimension surface (contour, if n = 3) in Rn−1 , ρi = ρi (x) is a concentration of adsorption sites Pi , i = 1, 2, at point x ∈ S, ρi (x) = 0 for x ∈ S1 , ρi (x) = ρi (x′ , 0) > 0 for x ∈ S2 , x′ = (x1 , . . . , xn−1 ), θi = θi (x′ , t) is a fraction of ρi occupied by molecules of the adsorbate at point x ∈ S2 at time t, 1 − θi is a free fraction of ρi , ρi θi is a concentration of the adsorbate, b = b(x, t) is a concentration of product B at point x ∈ Ω at time t. According to Langmuir and Hinshelwood, adsorption, desorption and reaction rates can be written as κi ρi (1 − θi )ai , κii ρi θi , and κ12 ρ1 ρ2 θ1 θ2 , i = 1, 2, where κi , κii , and κ12 are the adsorption, desorption, and reaction rate constants. The diffusion of reactants Ai , i = 1, 2, can be described by the system ′ ai − ki ∆ai = 0 in Ω × (0, T ], k ∂a /∂n = 0 on S × (0, T ], i i 1 (1) ki ∂a on S2 × (0, T ], i /∂n + κi ρi (1 − θi )ai = κii ρi θi a in Ω i t=0 = ai0 where a′i = ∂ai /∂t, ki = const > 0 is a diffusion coefficient, ∂ai /∂n is the outward normal derivative to S, n ∆ai = i=1 aixi xi , ai0 = ai0 (x) is the initial concentrations of Ai at point x ∈ Ω . For θi , i = 1, 2, we have the problem ′ θi − pi ∆θi = κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 , j ̸= i, in S2 × (0, T ], (2) pi ∂θi /∂ν = 0 on Σ × (0, T ], θ S = θ in i t=0 i0 2 where θi′ = ∂θi /∂t, pi = const > 0 is a diffusion coefficient, ∂θi /∂ν is the outward normal derivative to Σ , n−1 ∆θi = i=1 θixi xi , θi0 = θi0 (x′ ) is a fraction of ρi which is occupied by molecules of the adsorbate at initial value t = 0 at point x′ ∈ S2 . The diffusion of the product B is described by the system b′ − l∆b = 0 in Ω × (0, T ], l∂b/∂n = 0 on S × (0, T ], 1 (3) l∂b/∂n = κ ρ ρ θ on S2 × (0, T ], 12 1 2 1 θ2 b = b0 in Ω t=0 where b′i = ∂bi /∂t, l = const > 0 is a diffusion coefficient of the product B, ∂b/∂n is the outward normal n derivative to S, ∆b = i=1 bxi xi , b0 = b0 (x) is the initial concentrations of B at point x ∈ Ω . All constants are assumed to be positive. Coupled system (1), (2), and (3) forms the nonlinear mathematical model for surface reactions between two monomers.
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3. Assumptions and solvability theorem for the time-dependent problem Assumption 3.1. Suppose, that: 1. S is a surface of class C 1+α , α ∈ (0, 1). 2. Σ is a surface of class C 1+β , β ∈ (0, 1). Assumption 3.2. Let initial functions ai0 , bi0 and θi0 satisfy the following conditions: 1. ai0 , bi0 are continuous, nonnegative functions in a closed domain Ω , and in any neighbourhood of the surface S are zero, 2. θi0 is a continuous in a closed domain S2 function such that 0 ≤ θi0 (x′ ) < 1 for all x′ ∈ S 2 and in any neighbourhood of the surface Σ is zero. Assumption 3.3. Let ai0 , bi0 , θi0 and ρi satisfy the following smoothness conditions: 1. ai0 , bi0 ∈ C λ (Ω ) with any λ ∈ (0, 1), 2. θi0 , ρi ∈ C λ (S2 ) with any λ ∈ (0, 1). Definition 3.1. Solution (a1 , a2 , θ1 , θ2 , b) of problem (1), (2), and (3) forms a classical solution if ai and b ∈ C 2,1 (Ω × (0, T ]) ∩ C (Ω × [0, T ]), ∂ai /∂n is continuous on S × [0, T ], ∂b/∂n is continuous on S × [0, T ], θi ∈ C 2,1 (S2 × (0, T ]) ∩ C (S2 × [0, T ]), and ∂θi /∂ν is continuous on Σ × [0, T ], i = 1, 2. We formulate the main result as Theorem 3.1. Let the surfaces S and Σ satisfy the conditions of Assumption 3.1 and the known functions ai0 , b0 , θi0 and ρi satisfy the conditions of Assumptions 3.2 and 3.3. Then problem (1)–(3) has a unique classical solution. To prove this theorem we need a priori estimates of solution to Eqs. (1), (2) which are formulated as Lemma 3.1. Let functions ai , θi , i = 1, 2, be a classical solution of problem (1), (2) and let a∗i0 = max ai0 (x), x∈Ω
∗ θi0 = max θi0 (x′ ) < 1. x′ ∈S 2
Then ∗ κii θi0 0 ≤ ai (x, t) ≤ max a∗i0 , := mi , ∀x ∈ Ω , t ∈ [0, T ], ∗ κi 1 − θi0 κi a∗i0 ∗ 0 ≤ θi (x′ , t) ≤ max θi0 , := m′i < 1, ∀x′ ∈ S2 , t ∈ [0, T ]. κi a∗i0 + κii
(4) (5)
4. Proof of Lemma 3.1 We multiply Eq. (1) by a piecewise smooth function η and integrate the result over cylinder Ω × (0, t), t ∈ (0, T ], getting an identity which, by using the formula of integration by parts and taking into account the boundary condition of problem (1), can be written as follows: t t t aiτ η dxdτ + ki aix ηx dxdτ = ρi κii θi − κi (1 − θi )ai η dx′ dτ (6) 0
where aix ηx =
Ω
n
k=1 aixk ηxk .
0
Ω
0
S2
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We first prove that ai and θi , i = 1, 2, are nonnegative. Inserting a(x, t), if a(x, t) < 0, − η = ai := 0, if a(x, t) ≥ 0 into (6) we get t t τ =t 1 a2i dxτ =0 + ki a2ix dxdτ = ρi κii θi − κi (1 − θi )ai ai dx′ dτ − − 2 Ωi,τ 0 Ωi,τ 0 Sa−i ,τ n − where Ωi,τ = {x ∈ Ω : ai (x, τ ) < 0}, Sa−i ,τ = {x′ ∈ S2 : ai (x′ , 0, τ ) < 0}, a2ix = k=1 a2ixk . Similarly, from Eq. (2) it follows that t t 1 ′ 2 2 ′ τ =t κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 θi dx′ dτ θix′ dx dτ = θ dx τ =0 + pi − − 2 Sθ− ,τ i Sθ ,τ 0 Sθ ,τ 0
(8)
i
i
i
(7)
n−1 2 2 where Sθ−i ,τ = {x′ ∈ S2 : θi (x′ , τ ) < 0}, θix ′ = k=1 θixk . ′ Using assumptions ai0 (x, t) ≥ 0 and θi0 (x , t) ≥ 0 from Eqs. (7) and (8) we derive the estimate t t 1 1 2 ′ a2i dx + θi2 dx′ + ki a2ix dxdτ + pi θix ′ dx dτ − − 2 Ωi,t 2 Sθ− ,t 0 Ωi,τ 0 Sθ− ,τ i i t t ′ ≤ ρi κii θi ai + κi (1 − θi )ai θi dx dτ + ρi κi θi a2i dx′ dτ Sa−i ,τ ∩Sθ− ,τ
0
0
i
t
′
− 0
t
Sθ− ,τ
κ12 ρj θ1 θ2 θi dx dτ ≤ ci1
Sa−i ,τ
0
i
a2i
Sa−i ,τ
t
′
dx dτ + ci2 0
Sθ− ,τ
θi2 dx′ dτ,
where constants ci1 , ci2 depend only on κii , κi , κ12 , m′i and ρi0 = maxx′ ∈S2 ρi (x′ ). It is known (cf. [10]) that 2 2 a dS ≤ ε ax dx + Cε a2 dx S
Ω
(10)
Ω
for all ε > 0. Therefore, we have inequality − 2 2 ′ ′ 2 ai dx = (ai ) dx ≤ (a− i ) dS Sa−i ,τ
S2
S
2 (a− i )x dx + Cε
≤ε Ω
2 (a− i ) dx = ε
Ω
− Ωi,τ
a2ix dx + Cε
which with ε = ki /2ci1 reduces inequality (9) to 1 1 ki t θi2 dx′ + a2i dx + a2ix dxdτ − − 2 Ωi,t 2 Sθ− ,t 2 0 Ωi,τ i t t 2 ′ 2 + pi θix′ dx dτ ≤ c ai dx + 0
Sθ− ,τ
0
i
− Ωi,τ
Sθ− ,τ i
Let Φ(t) =
t 0
− Ωi,τ
a2i dx +
Sθ− ,τ
θi2 dx′ dτ.
i
Then Φ ′ (t) ≤ cΦ(t) =⇒ Φ(t) ≤ 0
(9)
i
− Ωi,τ
θi2′ dx′ dτ.
a2i dx
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
215
and − Ωi,t
a2i
dx + Sθ− ,t
θi2 dx′ ≤ 0.
i
′
Therefore ai (x, t) ≥ 0, ∀x ∈ Ω , t ∈ [0, T ], θi (x , t) ≥ 0, ∀x ∈ S2 , t ∈ [0, T ]. Next we prove the boundedness of ai and θi . Inserting ai (x, τ ) − mi , if a(x, τ ) > mi , + η := (ai (x, τ ) − mi ) = 0, if a(x, τ ) ≤ mi into Eq. (6) we get 1 2
+ Ωi,τ (mi )
τ =t (ai − mi )2 dxτ =0 + ki
t = 0
Sa+i ,τ (mi )
t = 0
Sa+i ,τ (mi )
t 0
+ Ωi,τ (mi )
a2ix dxdτ
ρi κii θi − κi (1 − θi )ai (ai − mi ) dx′ dτ ρi κii (θi − m′i ) + κii m′i − κi (1 − m′i )mi
− κi (1 − θi )(ai − mi ) + κi (θi − m′i )mi (ai − mi ) dx′ dτ
(11)
where + Ωi,τ (mi ) = {x ∈ Ω : ai (x, τ ) > mi },
Sa+i ,τ (mi ) = {x′ ∈ S2 : ai (x′ , 0, τ ) > mi }.
Similarly, from Eq. (2) it follows that t τ =t 1 2 ′ (θi − m′i )2 dx′ τ =0 + pi θix ′ dx dτ 2 Sθ+ ,τ (mi ) 0 Sθ+ ,τ (mi ) i i t × κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 (θi − m′i ) dx′ dτ Sθ+ ,τ (mi )
0
i
t = 0
Sθ+ ,τ (mi ) i
κi (1 − m′i )(ai − mi ) + κi (1 − m′i )mi − κii m′i
− κi (θi − m′i )ai − κii (θi − m′i ) − κ12 ρj θ1 θ2 (θi − m′i ) dx′ dτ where Sθ+i ,τ (mi ) = {x′ ∈ S2 : θi (x′ , τ ) > m′i }. Notice, that if a∗i0 ≥
∗ κii θi0 ∗ κi 1 − θi0
∗ θi0 ≤
κi a∗i0 κi a∗i0 + κii
then
and vice versa. Thus, if mi = a∗i0 then m′i =
κi mi κi a∗i0 = . κi a∗i0 + κii κi mi + κii
Similarly, if ∗ θi0 ≥
κi a∗i0 κi a∗i0 + κii
(12)
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
216
then a∗i0 ≤
∗ κii θi0 ∗ κi 1 − θi0
∗ and vice versa. Thus if m′i = θi0 then
mi =
∗ κii θi0 κii m′i = . ∗ κi 1 − θi0 κi 1 − m′i
In both cases we have κi (1 − m′i )mi − κii m′i = 0.
(13)
Using (13) from equalities (11) and (12) we get t 1 1 (ai − mi )2 dx + (θi − m′i )2 dx′ + ki a2ix dxdτ + + 2 Ωi,t 2 Sθ+ ,τ (m′i ) (mi ) 0 Ωi,t (mi ) i t t 2 ′ + pi θix (ai − mi )(θi − m′i ) dx′ dτ ′ dx dτ ≤ ci Sθ+ ,τ (m′i )
0
+ ci1
0
i
t 0
Sa+i ,τ (mi )
Sθ+ ,τ (m′i )∩Sa+i ,τ (mi ) i
t
(ai − mi )2 dx′ dτ +
0
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ dτ
(14)
i
where constants ci and ci1 depend only on κi , κii , mi , ρi0 . Then, by estimate (10) we get the equality (ai − mi )2 dx′ ≤ (ai − mi )2+ dS ≤ ε a2ix dx + Cε (ai − mi )2 dx Sai ,τ (mi )
which reduces estimate (14) to (ai − mi )2 dx + + Ωi,τ (mi )
Sθ+ ,τ (m′i ) i
t + 2pi 0
+ Ωi,τ (mi )
S
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ + ki
2 ′ θix ′ dx dτ ≤ Ci
0
i
where Ci = ci max{Cε , 1}. Let t Φi (t) = 0
+ Ωi,τ (mi )
+ Ωi,τ (mi )
′
2
+ Ωi,τ (mi )
a2ix dxdτ
(ai − mi )2 dx′ +
(ai − mi ) dx +
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ dτ,
i
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ dτ.
i
Then Φi′ (t) ≤ Ci Φi (t) =⇒ Φi (t) ≤ 0 and (ai − mi )2 dx + + Ωi,τ (mi )
t 0
t
+ Ωi,τ (mi )
Sθ+ ,τ (m′i )
(θi − m′i )2 dx′ ≤ 0.
i
Therefore ai (x, t) ≤ mi for all (x, t) ∈ Ω × [0, T ] and θi (x′ , t) ≤ m′i for all (x′ , t) ∈ S2 × [0, T ]. The proof is complete. Remark 1. Let θi , i = 1, 2, in system (3) be nonnegative continuous function on S 2 × [0, T ], functions b0 and ρi satisfy the conditions of Assumption 3.2 and b be a classical solution of this problem. Then by a similar argument we can prove that b is nonnegative and bounded from above for all (x, t) ∈ Ω × [0, T ]. Remark 2. Let (a1 , a2 , θ1 , θ2 , b) be a classical solution of problem (1)–(3). Then the following mass conservation laws are true: ai (x, t) + b(x, t) dx + ρi (x)θi (x, t) dS = ai0 (x) + b0 (x) dx + ρi (x)θi0 (x) dS, i = 1, 2. Ω
S2
Ω
S2
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
217
To prove these laws it is sufficient to add Eqs. (1) and (3), then integrate over cylinder Qt = Ω × (0, t), apply the formula of integration by parts, and use Eq. (2), boundary and initial conditions. 5. Uniqueness of the classical solution In this section, we prove that problem (1)–(3) has at most one classical solution. Let a ˜i , θ˜i , ˜b and a ˆi , θˆi , ˆb, i = 1, 2, be two classical solutions to problem (1)–(3) and set ai = a ˜i − a ˆi , θi = θ˜i − θˆi , b = ˜b − ˆb. Then from Eqs. (1) and (2) it follows that a′i − ki ∆ai = 0
in Ω × (0, T ],
on S1 × (0, T ], ki ∂ai /∂n + κi ρi (1 − θ˜i )ai = (κii + κi a ˆi )ρi θi ai = 0 in Ω ki ∂ai /∂n = 0
on S2 × (0, T ],
t=0
and 1 2
a2i
a2ix
dx + ki
Ω
t
ρi (κii + κi a ˆi )θi − κi ai (1 − θ˜i ) ai dx′ dτ.
dxdτ =
Qt
0
(15)
S2
Moreover, θi′ − pi ∆θi = κi ai (1 − θ˜i ) − (κii + κi a ˆi )θi − κ12 ρj (θ1 θ˜2 + θˆ1 θ2 )
in S2 × (0, T ),
∂θi /∂ν = 0 on Σ × (0, T ), θi t=0 = 0 in S2 and, hence,
1 2
θi2
t
′
dx + pi
S2
=
t
′
(κii + κi a ˆi )θi2 dx′ dτ
dx dτ +
S2
0
t
2 θix ′
0
S2
κi ai (1 − θ˜i ) − κ12 ρj (θ1 θ˜2 + θˆ1 θ2 ) θi dx′ dτ.
(16)
S2
0
Then from Eqs. (15) and (16) we get t t 1 a2i dx + ki a2ix dxdτ + κi ρi (1 − θ˜i )a2i dx′ dτ ≤ ci |ai ||θi | dx′ dτ, 2 Ω Qt 0 S2 0 S2 t t 1 2 ′ × θi2 dx′ + pi θix dx dτ + (κii + κi a ˆi )θi2 dx′ dτ ′ 2 S2 0 S2 0 S2 t t ≤ κi |ai ||θi | dx′ dτ + κ12 ρj |θ1 | + |θ2 | |θi | dx′ dτ, 0
S2
0
S2
where ci = max{(κii + κi m ˆ ′i )ρi0 }, m ˆ ′i = maxx′ ∈S2 ,t∈[0,T ] a ˆi (x′ , 0, t). We add up these inequalities, then summarize by i getting t 1 1 (a21 + a22 ) dx + (θ12 + θ22 ) dx′ + (k1 a21x + k2 a22x ) dxdτ 2 Ω 2 S2 0 Ω t t 2 2 ′ (p1 θ1x (κ11 θ12 + κ22 θ22 ) dx′ dτ + ′ + p2 θ2x′ ) dx dτ + S2
0
t ≤ 0
S2
c+κ 2 (a1 + a22 ) dx′ dτ + 2
0
t 0
S2
S2
c+κ ¯ 2 (θ1 + θ22 ) dx′ dτ 2
218
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
where c = max{c1 , c2 }, κ = max{κ1 , κ2 }, κ ¯ = max{κ1 + 4κ12 ρ10 , κ2 + 4κ12 ρ20 }. We apply inequality (10) to a1 and a2 , then take ε = k1 /(c + κ) and ε = k2 /(c + κ) to get 1 t 1 (k1 a21x + k2 a22x ) dxdτ (a21 + a22 ) dx + (θ12 + θ22 ) dx′ + 2 Ω 2 0 Ω S2 t ≤ c1 ¯ ). (a21 + a22 ) dx′ + (θ12 + θ22 ) dx′ dτ, c1 = c1 (k1 , k2 , c, κ, κ11 , κ22 , κ 0
Ω
S2
Set Φ(t) =
t 0
(a21 + a22 ) dx′ +
Ω
(θ12 + θ22 ) dx′ dτ.
S2
Hence, ′ Φ ′ (t) ≤ 2c1 Φ(t) ⇒ Φ(t)e−2c1 t ≤ 0 ⇒ Φ(t)e−2c1 t ≤ Φ(0) = 0 which shows that a1 ≡ 0, a2 ≡ 0, and θ1 ≡ 0, θ2 ≡ 0, and thus b ≡ 0. Therefore, we have proved Theorem 5.1. Problem (1)–(3) has at most one classical solution. 6. Upper and lower sequences The construction of upper and lower solutions is based on the Positivity Lemma (see [2], Chapter 1, Lemma 4.1, p. 54). Lemma 6.1 (Positivity Lemma). Let w ∈ C (DT ) ∩ C 1,2 (DT ) satisfy the relations wt − D∇2 w + cw ≥ 0 α0 ∂w/∂n + β0 w ≥ 0 w(0, x) ≥ 0
in on in
DT = Ω × (0, T ], ST = S × (0, T ], Ω
where α0 ≥ 0, β0 ≥ 0, α0 + β0 > 0 on ST , and c = c(t, x) is a bounded function in DT , then w ≥ 0 in DT . For the sake of convenience we rewrite Eqs. (1) and (2) as follows: Li ai = 0 in Ω × (0, T ], Bi ai = gi (ai , θi , ρi ) on S × (0, T ],i = 1, 2, a in Ω i t=0 = ai0 where Li = ∂/∂t − ki ∆, Bi = ki ∂/∂n + κi ρ, 0, if x ∈ S1 , t ∈ (, T ], gi (ai , θi , ρi ) = (κii + κi ai )ρi θi , if x ∈ S2 , t ∈ (0, T ], Li θi = fi (ai , θ1 , θ2 , ρj ) in Ω × (0, T ], Bθi = 0 on Σ × (0, T ], θ in S2 i t=0 = θi0
(17)
(18)
where Li = ∂/∂t − pi ∆, B = ∂/∂ν, fi (ai , θ1 , θ2 , ρj ) = κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 ,
i, j = 1, 2, j ̸= i.
We construct the upper and lower solutions to Eqs. (17) and (18) by using a technique given in [2]. We observe, by definition and Lemma 3.1, that gi is nondecreasing in variables ai and θi , fi is nondecreasing in ai and nonincreasing in θ1 , θ2 .
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
219
Definition 6.1. A pair of functions u ˜ := (˜ a1 , a ˜2 , θ˜1 , θ˜2 ) and u ˆ := (ˆ a1 , a ˆ2 , θˆ1 , θˆ2 ) such that a ˜i , a ˆi ∈ C (QT ) ∩ 2,1 2,1 C (QT ), θ˜i , θˆi ∈ C ([0, T ] × S2 ) ∩ C ((0, T ] × S2 ), i = 1, 2, are called upper and lower solutions of problem (1), (2) if u ˜≥u ˆ, (i.e. a ˜i ≥ a ˆi , and θ˜i ≥ θˆi , i = 1, 2) and if they satisfy the inequalities ˜i ≥ 0 ≥ Li a ˆi in Ω × (0, T ], Li a (19) Bi a ˜ i − gi (˜ ai ,θ˜i , ρi ) ≥ 0 ≥ Bi a ˆi − gi (ˆ ai , θˆi , ρi ) on S × (0, T ], a ˜i0 t=0 ≥ ai0 t=0 ≥ a ˆi0 t=0 in Ω , i = 1, 2, L1 θ˜1 − f1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) ≥ 0 ≥ L1 θˆ1 − f1 (ˆ a1 , θˆ1 , θ˜2 , ρ2 ) in Ω × (0, T ], L θ˜ − f (˜ ˆ ˜ ˆ a2 , θ˜1 , θˆ2 , ρ1 ) in Ω × (0, T ], 2 2 2 a2 , θ1 , θ2 , ρ1 ) ≥ 0 ≥ L2 θ2 − f2 (ˆ (20) ˜ ˆ Bθi ≥ 0 ≥ Bθi on Σ × (0, T ], ˜ θi t=0 ≥ θi0 ≥ θˆi t=0 in S2 , i = 1, 2. We note that f1 (a1 , θ¯1 , θ2 , ρ2 ) − f1 (a1 , θ1 , θ2 , ρ2 ) = −(κ1 a1 + κ11 + κ12 ρ2 θ2 )(θ¯1 − θ1 ) ≥ −c1 (θ¯1 − θ1 )
if θ¯1 ≥ θ1 (21)
f2 (a2 , θ1 , θ¯2 , ρ1 ) − f2 (a2 , θ1 , θ2 , ρ1 ) = −(κ2 a2 + κ22 + κ12 ρ1 θ1 )(θ¯2 − θ2 ) ≥ −c2 (θ¯2 − θ2 )
if θ¯2 ≥ θ2
and
where ci = κi mi + κii + κ12 ρj0 , j ̸= i. Here mi is defined in Lemma 3.1 and ρj0 = maxx∈S2 ρj (x). Let us consider the equivalent (1), (2) problem Li ai = 0 in Ω × (0, T ], Bi ai = Gi (ai , θi , ρi ) on S × (0, T ], a in Ω , i t=0 = ai0 Li θi = Fi (ai , θ1 , θ2 , ρj ) in Ω × (0, T ], Bθi = 0 on Σ × (0, T ], θ in S2 i t=0 = θi0
(22)
(23)
where Bi ai := Bi ai + ρi κi ai , Gi (ai , θi , ρi ) = gi (ai , θi , ρi ) + ρi κi ai , Li θi := Li θi + ci θi , Fi (ai , θ1 , θ2 , ρj ) = fi (ai , θ1 , θ2 , ρj ) + ci θi , i, j = 1, 2, j ̸= i. From definition it follows that: Gi (ai , θi , ρi ) is nondecreasing in ai and θi , F1 (a1 , θ1 , θ2 , ρ2 ) is nondecreasing in a1 and θ1 , F2 (a2 , θ1 , θ2 , ρ1 ) is nondecreasing in a2 and θ2 , but F1 (a1 , θ1 , θ2 , ρ2 ) and F2 (a2 , θ1 , θ2 , ρ1 ) are nonincreasing in θ2 and θ1 , respectively. (0) (0) (0) (0) Starting from a suitable initial iteration u0 = (ai , a2 , θ1 , θ2 ) we construct a sequence {u(k) } = (k) (k) (k) (k) {(ai , a2 , θ1 , θ2 )} from the iteration process (k) Li ai = 0 in Ω × (0, T ], (k) (k−1) (k−1) (24) Bi ai = Gi (ai , θi , ρi ) on S × (0, T ],i = 1, 2, (k) ai t=0 = ai0 in Ω , (k) (k−1) (k−1) (k−1) , θ1 , θ2 , ρj ) in Ω × (0, T ], Li θi = Fi (ai (k) (25) Bθi = 0 on Σ × (0, T ], (k) θ = θi0 in S2 , i
where i, j = 1, 2, j ̸= i.
t=0
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
220
It is evident that a ˆi = 0, θˆi = 0, i = 1, 2, is a lower solution and Eq. (13) and inequalities (19), (20) show ˜ that a ˜i = mi and θi = m′i , i = 1, 2, is an upper solution. We take (0)
(0)
(ai , θi ) = (˜ ai , θ˜i )
(0)
(0)
(ai , θi ) = (ˆ ai , θˆi ),
and
i = 1, 2,
in Eqs. (24) and (0) (0) (0) (a1 , θ1 , θ2 ) = (˜ a1 , θ˜1 , θˆ2 ), (0) (0) (0) (a , θ , θ ) = (˜ a2 , θˆ1 , θ˜2 ), 2
1
(0) (0) (0) (a1 , θ1 , θ2 ) = (ˆ a1 , θˆ1 , θ˜2 ), (0) (0) (0) (a , θ , θ ) = (ˆ a2 , θ˜1 , θˆ2 )
2
2
1
2
in Eqs. (25) getting sequences (k)
(k)
(k)
(k)
(k)
(k)
{ai }, {ai }, {θ1 }, {θ1 }, {θ2 }, {θ2 },
i = 1, 2.
These sequences form the other two sequences (k)
(k)
(k)
(k)
(k)
{u(k) } = {(a1 , a2 , θ1 , θ2 )}
(k)
(k)
(k)
and {u(k) } = {(a1 , a2 , θ1 , θ2 )}.
Now we prove that u ˆ ≤ u(1) ≤ · · · ≤ u(k) ≤ u(k+1) ≤ · · · ≤ u(k+1) ≤ u(k) ≤ · · · ≤ u(1) ≤ u ˜. (0)
Let wi
(0)
= ai
(1)
− ai
(26)
(1)
=a ˜i − ai . By Eqs. (24) and (19),
(0)
Li wi
in Ω × (0, T ],
=0
(0) Bi wi
(1)
(0)
(0)
= Bi a ˜i + ρi κi a ˜i − Bi ai = Bi a ˜i + ρi κi a ˜i − Gi (ai , θi , ρ) = Bi a ˜i − gi (˜ ai , θ˜i , ρi ) ≥ 0 on S × (0, T ], (0) wi t=0 = a ˜i t=0 − ai0 ≥ 0 in Ω , i = 1, 2. (1)
(0)
In view of Lemma 6.1, wi (x, t) ≥ 0 which shows that a ˜i (x, t) ≥ ai (x, t). Similarly, we prove that (1) a ˆi (x, t) ≤ ai (x, t). (0) (1) (1) (0) Let v = θ − θ = θ˜i − θ . By Eqs. (25) and (20), i
i
i
i
(0)
L1 v1
(0)
Bvi
(0) = L1 θ˜1 − L1 θ1 = L1 θ˜1 + c1 θ˜1 − F1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) = L1 θ˜1 − f1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) ≥ 0 in Ω × (0, T ],
on Σ × (0, T ], ˜ = θi t=0 − θi0 ≥ 0
=0
(0) v1 t=0
in S2 .
(1) (1) (0) In view of Lemma 6.1, v1 (x, t) ≥ 0. Hence, θ˜1 (x, t) ≥ θi (x, t). Similarly, we get that θ˜2 (x, t) ≤ θ2 (x, t) (1) and θˆi (x, t) ≤ θi (x, t), i = 1, 2. (1) (1) (1) Let wi = ai − ai . Then, by Eqs. (24) and since Gi (ai , θi , ρi ) is nondecreasing in ai and θi , (1)
Li wi
(1)
(1)
= Li ai
(1)
(1)
− Li ai
(1)
=0
in Ω × (0, T ], (0)
(0)
(0)
(0)
Bi wi = Bi ai − Bi ai = Gi (ai , θi , ρi ) − Gi (ai , θi , ρi ) ≥ 0 (1) wi t=0 = ai0 − ai0 = 0 in Ω , i = 1, 2. (1)
(1)
(1)
In view of Lemma 6.1, wi (x, t) ≥ 0. Hence a1 (x, t) ≥ ai (x, t).
on S × (0, T ],
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
(1)
Let vi
(1)
(1)
− θi . Then by Eqs. (25) and monotone property of Fi we have
= θi
(1) (1) a1 , θ˜1 , θˆ2 , ρ2 ) − F1 (ˆ a1 , θˆ1 , θ˜2 , ρ2 ) = L1 θ1 − L1 θ1 = F1 (˜ = F1 (˜ a1 , θ˜1 , θˆ2 , ρ2 ) − F1 (ˆ a1 , θ˜1 , θˆ2 , ρ2 ) + F1 (ˆ a1 , θ˜1 , θˆ2 , ρ2 ) − F1 (ˆ a1 , θˆ1 , θˆ2 , ρ2 ) ˜ ˆ ˆ ˜ + F1 (ˆ a1 , θ1 , θ2 , ρ2 ) − F1 (ˆ a1 , θ1 , θ2 , ρ2 ) ≥ 0 in Ω × (0, T ],
(1)
L1 v1
(1)
Bvi
221
on Σ × (0, T ],
=0
(1) v1 t=0
in S2 .
=0
(1)
(1)
(1)
(1)
(1)
In view of Lemma 6.1, v1 (x, t) ≥ 0. Hence, θ1 (x, t) ≥ θ1 (x, t). Similarly, θ2 (x, t) ≥ θ2 (x, t). Thus, we have proved that u ˆ = u(0) ≤ u(1) ≤ u(1) ≤ u(0) = u ˜. (k−2)
Let ai
(k−1)
− ai
(k−1)
≥ 0. Then by Eq. (24) and monotone property of Gi (k)
Li (ai
− ai ) = 0
(k−1) Bi (ai (k−1) (ai −
(k) (k−2) (k−2) − ai ) = Gi (ai , θi , ρi ) (k) ai ) t=0 = 0 in Ω , i = 1, 2.
in Ω × (0, T ],
(k−1)
In view of Lemma 6.1 we have ai (k−1)
ai
(k−1)
− Gi (ai
(k−1)
, θi
, ρi ) ≥ 0
on S × (0, T ],
(k)
(x, t) ≥ ai (x, t). Similarly, we prove that (k−1)
(k)
(x, t) ≤ ai (x, t),
θi
(k)
(x, t) ≥ θi (x, t),
(k−1)
θi
(k)
(x, t) ≤ θi (x, t),
i = 1, 2. Hence we have proved inequalities (26). Now we prove that u(k) and u(k) are upper and lower solutions. From (24) and (25) by (21) and the monotone property of gi and f1 we get (k)
Li ai
(k)
Bi ai
(k)
i
t=0
(k−1)
(k−1)
(k) (k−1) , ρi ) = −ρi κi ai + ρi κi ai (k−1) (k−1) (k−i) (k) (k−1) (k−1) , θi , ρi ) = ρi κi (ai − ai ) + gi (ai , θi , ρi ) + gi (ai (k) (k−1) (k) (k−1) (k) (k) (k) (k) − gi (ai , θi , ρi ) + gi (ai , θi , ρi ) − gi (ai , θi , ρi ) + gi (ai , θi , ρi ) (k) (k) gi (ai , θi , ρi ) on S × (0, T ],
= −ρi κi ai
≥ (k) a
in Ω × (0, T ],
=0
= ai0
+ Gi (ai
, θi
in Ω
and (k)
L1 θ1
(k)
(k−1)
= −c1 θ1 + F1 (a1 =
(k−1)
, θ1
(k−1)
, θ2
, ρ2 ) (k−1) (k) (k−1) (k−1) (k−1) c1 (θ1 − θ1 ) + f1 (a1 , θ1 , θ2 , ρ2 ) (k) (k−1) (k−1) (k) (k−1) (k−1) − f1 (a1 , θ1 , θ2 , ρ2 ) + f1 (a1 , θ1 , θ2 , ρ2 ) (k) (k) (k−1) (k) (k) (k−1) − f1 (a1 , θ1 , θ2 , ρ2 ) + f1 (a1 , θ1 , θ2 , ρ2 ) (k) (k) (k) (k) (k) (k) − f1 (a1 , θ1 , θ2 , ρ2 ) + f1 (a1 , θ1 , θ2 , ρ2 ) ≥ 0 in Ω
× (0, T ],
(k)
Bθ1 = 0 on Σ × (0, T ], (k) θ1 t=0 = θ10 in S2 . (k)
(k)
(k)
Similarly, we get that θ2 satisfies inequalities (20), and ai , θi satisfy relations (19) and (20), respectively. Hence, u(k) and u(k) are the upper and lower solutions and satisfy inequalities (26).
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
222
Since {u(k) } is nondecreasing and bounded from above, and {u(k) } is nonincreasing and bounded from below, they converge monotonically to some limits lim u(k) = u,
lim u(k) = u
k→∞
k→∞
which satisfy relations u ˆ ≤ u(1) ≤ · · · ≤ u(k) ≤ u(k+1) ≤ · · · ≤ u ≤ u ≤ · · · ≤ u(k+1) ≤ u(k) ≤ · · · ≤ u(1) ≤ u ˜. In next session we prove that u and u are solutions of system (1), (2). 7. Existence of the solution (0) (0) (0) (0) Problem (24) starting from (ai , θi ) = (˜ ai , θ˜i ) or (ai , θi ) = (ˆ ai , θˆi ) has a unique classical solution (k) (k) 2,1 2,1 ai ∈ C (Ω × (0, T ]) ∩ C (Ω × [0, T ]) or ai ∈ C (Ω × (0, T ]) ∩ C (Ω × [0, T ]), respectively, for any k = 1, 2, . . ., which can be represented by the formula (see. [11,12]) (k) ai (x, t)
t Γi (x − ξ, t −
= 0 (k)
Moreover, the function ai
(k) τ )ϕi (ξ, τ ) dSξ dτ
Γi (x − y, t)ai0 (y) dy.
+
S
(27)
Ω
is H¨ older continuous in x uniformly in Ω × [0; T ] (cf. [2]). Here Γi (x, t) =
1 4πki t
n/2 e (k)
is a fundamental solution of Eq. (1). Function ϕi t
(k)
|x|2 it
− 4k
,
x ∈ Rn , t > 0
is a solution on S × [0, T ] of the equation
(k)
ϕi (η, t) =
(k−1)
Qi1 (η, t, ξ, τ )ϕi (ξ, τ ) dSξ dτ + qi (ai 0
(k−1)
, θi
, ρi )(η, t)
(28)
S
where ∂Γ (η − ξ, t − τ ) i Qi1 (η, t, ξ, τ ) = −2 ki + κi ρi (η)Γi (η − ξ, t − τ ) , ∂nη (k−1) (k−1) (k−i) (k−i) qi (ai , θi , ρi )(η, t) = 2Gi (ai , θi , ρi )(η, t) + Qi1 (η, t, y, 0)ai0 (y) dy. Ω
Eq. (28) is the Volterra integral equation with a weak singularity. Its solution can be represented by the formula (see. [11,12]) ∞ t (k) (k−1) (k−1) (k−1) (k−1) ϕi (η, t) = qi (ai , θi , ρi )(η, t) + Qij (η, t, ξ, τ )qi (ai , θi , ρi )(ξ, τ ) dSξ dτ, j=1
0
S
t Qij (η, t, ξ, τ ) =
Qi1 (η, t, ζ, s)Qij−1 (ζ, s, ξ, τ ) dSζ ds, τ
j = 2, 3, . . . .
S
Problem (25), starting from (0) (0) (0) (a1 , θ1 , θ2 ) = (˜ a1 , θ˜1 , θˆ2 )
or
(0) (0) (0) (a1 , θ1 , θ2 ) = (ˆ a1 , θˆ1 , θ˜2 )
or
(a2 , θ1 , θ2 ) = (ˆ a2 , θ˜1 , θˆ2 )
for i = 1 and (0)
(0)
(0)
(a2 , θ1 , θ2 ) = (˜ a2 , θˆ1 , θ˜2 )
(0)
(0)
(0)
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
(k)
223
(k)
for i = 2, has a unique solution θi ∈ C 2,1 (S2 × (0, T ]) ∪ C (S2 × [0, T ]) and θi ∈ C 2,1 (S2 × (0, T ]) ∪ C (S2 × [0, T ]) for any k = 1, 2, . . .. This solution can be represented by the formula (see [11,12]) t (k−1) (k−1) (k−1) (k) Zi (x′ − y ′ , t − τ )Fi (ai , θ1 θi (x′ , t) = , θ2 , ρj )(y ′ , τ ) dy ′ dτ S2
0
t
(k)
Zi (x′ − η ′ , t − τ )µi (η ′ , τ ) dΣη′ dτ +
+ Σ
0 (k)
Moreover, the function θi
Zi (x′ − y ′ , t)θi0 (y ′ ) dy ′ .
(29)
S2
is H¨ older continuous in x′ uniformly in S2 × [0; T ] (cf. [2]). Here
Zi (x′ , t) =
|x′ |2 1 − −ci t , (n−1)/2 e 4pi t e 4πpi t
x′ ∈ Rn−1 , t > 0 (k)
is a fundamental solution of the equation θi′ − pi ∆θi + ci θi = 0. The function µi solution of the integral equation t (k) (k) µi (ξ ′ , t) = Ri1 (ξ ′ , t, η ′ , τ )µi (η ′ , τ ) dΣη′ dτ
is continuous and bounded
Σ
0
(k−1)
+ ri (ai
(k−1)
, θ1
(k−1)
, θ2
ρj )(ξ ′ , t),
ξ ′ ∈ Σ , t ∈ ×[0, T ]
(30)
and can be represented by the formula (see [11,12]) (k)
(k−1)
(k−1)
(k−1)
µi (ξ ′ , t) = ri (ai , θ1 , θ2 ρj )(ξ ′ , t) ∞ t (k−1) (k−1) (k−1) + Rij (ξ ′ , t, η ′ , τ )ri (ai , θ1 , θ2 ρj )(ξ ′ , t) dΣη′ dτ, j=1 (k−1)
ri (ai
(k−1)
, θ1
0
(k−1)
, θ2
Σ
ρj )(ξ ′ , t) =
Ri1 (ξ ′ , t, y ′ , τ )θi0 (y ′ ) dy ′ S2 t (k−1) (k−1) (k−1) + Ri1 (ξ ′ , t, y ′ , τ )Fi (ai , θ1 , θ2 , ρj )(y ′ , τ ) dy ′ dτ, 0
D
∂Zi (ξ ′ − η ′ , t − τ ) Ri1 (ξ ′ , t, η ′ , τ ) = −2 , ∂νξ′ t Rij+1 (ξ ′ , t, η ′ , τ ) = Ri1 (ξ ′ , t, ζ ′ , s)Rij (ζ ′ , s, η ′ , τ ) dΣζ ′ ds, τ
j = 1, 2, . . . .
Σ
Because of the continuity and boundedness of Fi , assumptions on θi0 , and by dominated convergence theorem, (k−1)
ri (ai
(k−1)
, θ1
(k−1)
, θ2
k→∞
, ρj )(x, t) −→ ri (ai , θ1 , θ2 , ρj )(x, t).
Moreover, ri (ai , θ1 , θ2 , ρj ) is a continuous function of (x, t). Similarly, due to the continuity and boundedness of Gi , assumptions on ai0 , and by dominated convergence theorem, (k−1)
qi (ai
(k−1)
, θi
k→∞
, ρi )(x, t) −→ qi (ai , θi , ρi )(x, t), (k)
where qi (ai , θi , ρi ) is a continuous function of (x, t). Then, by [2, Lemma 2.2, p. 56], function ϕj (x, t) (k)
defined by Eq. (28) and function µj (x, t) defined by Eq. (30) converge to continuous solutions ϕj (x, t) and µj (x, t) that satisfy the integral equations t ϕi (η, t) = 2 Qi1 (η, t, ξ, τ )ϕi (ξ, τ ) dSξ dτ + qi (ai , θi , ρi )(η, t) 0
S
and µi (ξ ′ , t) =
t 0
Σ
Ri1 (ξ ′ , t, η ′ , τ )µi (η ′ , τ ) dΣη′ dτ + ri (ai , θ1 , θ2 ρj )(ξ ′ , t),
ξ ′ ∈ Σ , t ∈ ×[0, T ],
224
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(k)
respectively. Letting k → ∞ in (27) and (29) we get that the limit function ai of sequences {ai } is defined by the formula t ai (x, t) = Γi (x − ξ, t − τ )ϕi (ξ, τ ) dSξ dτ + Γi (x − y, t)ai0 (y) dy, S
0
Ω (k)
while the limit function θi of sequences {θi } satisfy the integral equation t θi (x′ , t) = Zi (x′ − y ′ , t − τ )Fi (ai , θ1 , θ2 , ρj )(y ′ , τ ) dy ′ dτ S2
0
t
′
′
′
Zi (x − η , t − τ )µi (η , τ ) dΣη′ dτ +
+ 0
Σ
Zi (x′ − y ′ , t)θi0 (y ′ ) dy ′ .
S2
These functions represent ai , θi and ai , θi , i = 1, 2. Thus we have proved that the functions u = (a1 , a2 , θ1 , θ2 ) and u = (a1 , a2 , θ1 , θ2 ) are solutions of problem (1), (2). By the uniqueness of the solution u = u. Obviously, system (3) with given continuous functions θ1 and θ2 has a unique classical solution. The proof of Theorem 3.1 is complete. 8. The long-time behaviour of the solutions In this section we consider the problem of the long-time behaviour of the solutions to problem (1), (2) and assume that S ∈ C 2+α ,
Σ ∈ C 2+α ,
ρi ∈ C 1+α ,
i = 1, 2.
(31)
We need two theorems and one lemma used in [2]: Theorem 8.1 (See [2, Chapter 2, Theorem 1.3, p. 53]). Let β0 ≥ 0, c ≥ 0, and either β0 or c be strictly positive. If q(t, x), h(t, x) converge to zero uniformly in Ω and ∂Ω , respectively, as t → ∞, then for any initial function u0 the solution u(t, x) of problem n n u − a (t, x)u + bj (t, x)uxj + c(t, x)u = q(t, x) in DT , t ij x x i j i,j=1
j=1
α0 (t, x)∂u/∂ν + β0 (t, x)u = h(t, x) u(0, x) = u0 (x)
on ST , in Ω
converges uniformly in Ω to zero as t → ∞. The convergence of u to zero is in the L2 (Ω )-space if q(t, ·), h(t, ·) converge to zero in L2 (Ω ) and L2 (∂Ω ), respectively, as t → ∞. Lemma 8.1 (See [2, Chapter 3, Lemma 1.2, p. 95]). For any p > n the Sobolev space Wpm (Ω ) is continuously embedded in C m−1+µ (Ω ) with µ = 1−n/p. Specifically, there exists a constant K such that for all u ∈ Wpm (Ω ) |u|Ω m−1+µ ≤ K∥u∥Wpm (Ω) . Theorem 8.2 (See [2, Chapter 3, Theorem 1.3, p. 98]). Let q ∈ C α (Ω ), h ∈ C 1+α (∂Ω ) when α0 > 0, and h ∈ C 2+α (∂Ω ) when α0 = 0, and let c ≥ 0 and not be identically zero when β0 ≡ 0. Then there exists a unique solution u to problem n n − aij (x)uxi xj + bj (x)uxj + c(x)u = q(x) in Ω , i,j=1 j=1 α0 (x)∂u/∂ν + β0 (x)u = h(x) on ∂Ω
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
225
and u ∈ C 2+α (Ω ). Moreover u satisfies the Schauder estimate Ω ∂Ω |u|Ω 2+α ≤ K |q|α + |h|1+α where K is a constant independent of u, q, and h. To study the long-time behaviour of the solutions we also need Lemma 8.2. Let (¯ ai , θ¯i ) and (ai , θi ) be the solutions of time-dependent system (1), (2) with ai0 = a ˜i , θi0 = θ˜i ˆ ˜ ˆ and ai0 = a ˆi = 0, θi0 = θi = 0, respectively, where (˜ ai , θi ) and (ˆ ai , θi ), i = 1, 2, are the same upper and lower solutions of the steady-state and time-dependent problems (34), (35) and (1), (2), respectively. Let (a∗i , θi∗ ), i = 1, 2, be a classical solution of time-dependent system (1), (2) with ai0 ∈ [ˆ ai , a ˜i ], θi0 ∈ [θˆi , θ˜i ], i = 1, 2. Then: (i) a ¯′i ≤ 0, a′i ≥ 0, a ¯i ≥ ai in Ω × [0, ∞), ′ ′ ¯ θi ≤ 0, θi ≥ 0, θ¯i ≥ θi in S2 × [0, ∞); (ii) limt→∞ a ¯i = a ¯is , limt→∞ θ¯i = θ¯is , limt→∞ ai = ais , limt→∞ θi = θis exist and a ¯is ≥ ais , θ¯is ≥ θis ; (iii) ai ≤ a∗i ≤ a ¯i , θi ≤ θi∗ ≤ θ¯i . Proof. We first prove propositions (i) and (ii). Let a ¯δi = a ¯i (x, t+δ), θ¯iδ = θ¯i (x, t+δ), wi = a ¯i −¯ aδi , vi = θ¯i −θ¯iδ where δ ≥ 0 is small constant. Then from Eqs. (1), (2) with initial functions (a10 , θi0 ) = (˜ ai , θ˜i ) and ˆ (a10 , θi0 ) = (ˆ ai , θi ) we derive the systems wi′ − ki ∆wi = 0 in Ω × (0, T ], k ∂w /∂n = 0 on S1 × (0, T ], i i (32) ¯ ki ∂w ¯δi )vi on S2 × (0, T ], i /∂n + κi ρi 1 − θi wi = ρi (κii + κi a w ai − a ¯δi )|t=0 ≥ 0, in Ω , i t=0 = (˜ ′ vi − pi ∆vi = κi (1 − θ¯i )wi −(κ + κ a δ ¯ ¯δ in S2 × (0, T ], ii i ¯i + κ12 ρj θj )vi − κ12 ρj θi vj (33) p ∂v /∂ν = 0 on Σ × (0, T ], i i v ˜ ¯δ in S2 , i t=0 = (θi − θi )|t=0 ≥ 0 i = 1, 2, i ̸= j, j = 1, 2. It is evident that (w ˆi , vˆi ) = (0, 0), i = 1, 2, is a lower solution of system (1), (2). In view of proof of Theorem 3.1, this system has a unique nonnegative solution if there exists a positive upper solution (w ˜i , v˜i ). Let ri ∈ C 2 (Ω ) be a nonnegative function such that ∂ri /∂n ≥ 1 on S, i = 1, 2, and let 2 qi ∈ C (S2 ) be a nonnegative function such that ∂qi /∂ν ≥ 0 on Σ , i = 1, 2. Define w ˜i = λi exp{µt + µi ri },
v˜i = βi exp{µt + qi },
where λi , βi , µi , i = 1, 2, and µ are some positive constants to be chosen. Then (w ˜i , v˜i ), i = 1, 2, is a positive upper solution of problem (32), (33) if the inequalities w ˜i′ − ki ∆w ˜i = λi µ − ki µi ∆ri + µi |∇ri |2 exp{µt + µi ri } ≥ 0, in Ω × (0, T ], ki ∂ w ˜i /∂n ≥ 0 on S1 × (0, T ], δ ¯ ki ∂ w ˜i /∂n + κi ρi 1 − θi w ˜i − ρi (κii + κi a ¯i )˜ vi = λi ki µi + κi ρi (1 − θ¯i ) exp{µt + µi ri } −(κi aδi + κii )ρi βi exp{µt + qi } ≥ 0 on S2 × (0, T ], δ δ w ˜i t=0 − (˜ ai − ai )|t=0 = λi exp{µi ri } − (˜ ai − ai )|t=0 ≥ 0 in Ω
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
226
and v˜i′ − pi ∆˜ vi − κi 1 − θ¯i w ˜i + (κii + κi a ¯δ + κ12 ρj θ¯j )˜ v i i δ +κ12 ρj θ¯i vˆj = βi exp{µt + qi } µ − pi ∆qi + |∇qi |2 −κi 1 − θ¯i exp{µi ri − qi }λi /βi + κii + κi a ¯δi + κ12 ρj θ¯j ≥ 0 in S2 × (0, T ], ∂˜ vi /∂ν = βi exp{µt + qi }∂qi /∂ν ≥ 0 on Σ × (0, T ], v˜i t=0 − (θ˜i − θ¯iδ )|t=0 = βi exp{qi } − (θ˜i − θ¯iδ )|t=0 ≥ 0 in S2 are satisfied. It is evident that: (a) initial inequalities are satisfied by choosing sufficiently large λi and βi ; (b) boundary inequality for w ˜i is satisfied with sufficiently large µi ; (c) differential equations for w ˜i and v˜i can be satisfied by choosing sufficiently large µ. Thus w ˜i and v˜i form the upper solution of system (32), (33). Then by Theorem 3.1, wi ≥ 0 and vi ≥ 0. This shows that a ¯′i ≤ 0 and θ¯i′ ≤ 0. The proof of a′i ≥ 0 and θ′i ≥ 0 is similar. Functions a ¯i and θ¯i are non-increasing and bounded from below. Functions ai and θi are non-decreasing and bounded from above. Therefore, the limits limt→∞ a ¯i = a ¯is , limt→∞ θ¯i = θ¯is , limt→∞ ai = ais , limt→∞ θi = θis exist. Now we prove that a ¯is ≥ ais and θ¯is ≥ θis . Let wi = a ¯i − ai , vi = θ¯i − θi . Then by the argument above we derive the system for wi and vi , wi′ − ki ∆wi = 0 in Ω × (0, T ], k ∂w /∂n = 0 on S1 × (0, T ], i i ¯i wi = ρi (κii + κi a )vi on S2 × (0, T ], ki ∂w /∂n + κ ρ 1 − θ i i i i w ai − ai )|t=0 ≥ 0, in Ω , i t=0 = (˜ ′ ¯ vi − pi ∆vi = κi (1 − θi )wi −(κ + κ a + κ ρ θ¯ )v − κ ρ θ v in S × (0, T ], ii i 12 j j i 12 j j 2 i
pi ∂vi /∂ν = 0 v ˜ i t=0 = (θi − θ i )|t=0 ≥ 0
i
on Σ × (0, T ], in S2 ,
i = 1, 2, i ̸= j, j = 1, 2. This system is of type (32), (33) and therefore wi ≥ 0 and vi ≥ 0. Hence, a ¯i ≥ ai , θ¯i ≥ θ and a ¯is ≥ a , θ¯is ≥ θ . i
is
is
It remains to prove proposition (iii). Let (a∗i , θi∗ ), i = 1, 2, be a solution of system (1), (2) with ai0 ∈ [ˆ ai , a ˜i ], ∗ ∗ ˆ ˜ ˆ θi0 ∈ [θi , θi ]. If we consider pairs the (ˆ ai , θi ) and (ai , θi ) as the lower and upper solutions of system (1), (2), then we get that ai ≤ a∗i , θi ≤ θi∗ . Similarly, if we take the pairs (a∗i , θi∗ ), (˜ ai , θ˜i ) as the lower and upper ∗ ∗ solutions, then we get that ai ≤ a ¯i , θi ≤ θ¯i . The proof of lemma is complete. The steady-state system corresponding to time-depending problem (1), (2) is −ki ∆ai = 0 in Ω , ki ∂ai /∂n = 0 on S1 , k ∂a /∂n + κ ρ (1 − θ )a = κ ρ θ on S , i i i i i i ii i i 2 −pi ∆θi = κi (1 − θi )ai − κii θi − κ12 ρj θ1 θ2 , in S2 , pi ∂θi /∂ν = 0 on Σ , i = 1, 2, j ̸= i, j = 1, 2.
(34)
(35)
A. Ambrazeviˇ cius, V. Skakauskas / Nonlinear Analysis: Real World Applications 35 (2017) 211–228
227
Now we prove that a ¯is , θ¯is and ais , θis , i = 1, 2, are classical solutions of system (34), (35). Let wi ∈ 2 vi ∈ Wp (S2 ) be a generalized solution of problem ¯is in Ω , −ki ∆wi + Ci wi = Ci a (36) ki ∂wi /∂n = 0 on S1 , k ∂w /∂n + κ ρ (1 − θ¯ )¯ ¯ on S2 , i i i i is ais = κii ρi θis −pi ∆vi + ci vi = ci θ¯is + κi (1 − θ¯is )¯ ais − κii θ¯is − κ12 ρj θ¯1s θ¯2s in S2 (37) pi ∂θi /∂ν = 0 on Σ ,
Wp2 (Ω ),
i = 1, 2, j ̸= i, j = 1, 2. From systems (1), (2) and (36), (37) we derive equations for Wi = a ¯i − wi , Vi = θ¯i − vi , Wi′ − ki ∆wi + Ci Wi = Ci (¯ ai − a ¯is ) in Ω × (0, T ], k ∂W /∂n = 0 on S × (0, T ], i i 1 k ∂W /∂n + κ ρ (1 − θ¯i )¯ ai = κii ρi θ¯i + κi ρi (1 − θ¯is )¯ ais − κii ρi θ¯is on S2 × (0, T ], i i i i W | ˜i − wi in Ω , i t=0 = a Vi′ − pi ∆Vi + ci Vi = ci (θ¯i − θ¯is ) + κi (1 − θ¯i )¯ ai − κii θ¯i −κ ρ θ¯ θ¯ − κ (1 − θ¯ )¯ ¯ ¯ ¯ in S2 × (0, T ] 12 j 1 2 i is ais + κii θis + κ12 ρj θ1s θ2s pi ∂Vi /∂ν = 0 on Σ × (0, T ], V | ˜ in S2 . i t=0 = θi − vi Theorem 8.1 shows that
a ¯is − wi
2
dx = 0,
Ω
θ¯is − vi
2
dx = 0
Ω
since the right-hand sides of equations of these systems converge to zero as t → ∞. This shows that wi = a ¯is 1+α ¯ in L2 (Ω ) and vi = θis in L2 (S2 ). By choosing p > n the embedding Lemma 8.1 implies that a ¯is ∈ C (Ω ), θ¯is ∈ C 1+α (S2 ). Then Theorem 8.2 shows that a ¯is ∈ C 2+α (Ω ), θ¯is ∈ C 2+α (S2 ). Thus a ¯is , θ¯is , i = 1, 2, is a classical solution of steady-state problem (34), (35). By the same argument we can prove that ais , θis , i = 1, 2, is a classical solution of the same problem. We observe that the steady-state problem has not a unique solution, since functions κ2 c a1 = 0, θ1 = 0, a2 = c, θ2 = , κ2 c + κ22 and κ1 c , a2 = 0, θ2 = 0 a1 = c, θ1 = κ1 c + κ11 where c = const ≥ 0 satisfy this system. But there exists a domain of initial functions such that each solution of the time-dependent problem with (ai0 , θi0 ), i = 1, 2, from this domain monotonically converges to corresponding solution of the steady-state problem as t → ∞. Theorem 8.3. Let (ai , θi ) be a classical solution of system (1), (2) with initial functions (ai0 , θi0 ) ∈ [¯ ais , a ˜i ] × [θ¯is , θ˜i ] ∪ [ˆ ai , ais ] × [θˆi , θis ],
i = 1, 2.
Then limits lim ai = a ¯is ,
t→∞
lim θi = θ¯is
with (ai0 , θi0 ) ∈ [¯ ais , a ˜i ] × [θ¯is , θ˜i ]
lim θi = θis
with (ai0 , θi0 ) ∈ [ˆ ai , ais ] × [θˆi , θis ],
t→∞
and lim ai = ais ,
t→∞
i = 1, 2, exist.
t→∞
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To prove this theorem we observe that (¯ ais , θ¯is ) and (ais , θis ), i = 1, 2, are also solutions of system (1), ¯ ais , θ¯is ) and (2) provided that ai0 = a ¯is , θi0 = θis and ai0 = ais , θi0 = θis , respectively. Then by considering (¯ ˜ (˜ ais , θis ), i = 1, 2, as lower and upper solutions of problem (1) and (2), Theorem 3.1 implies that a ¯is ≤ ai ≤ a ¯i ,
θ¯is ≤ θi ≤ θ¯i
if ai0 ∈ [¯ ais , a ˜is ], θi0 ∈ [θ¯i , θ˜i ], i = 1, 2.
a ˆi ≤ ai ≤ ais ,
θˆi ≤ θi ≤ θis
if ai0 ∈ [ˆ ai , ais ], θi0 ∈ [θˆis , θis ], i = 1, 2.
Similarly,
Corollary 8.1. Under assumption (31) all classical solutions of problem (1), (2) corresponding to initial functions (ai0 , θi0 ) ∈ [¯ ais , a ˜i ] × [θ¯is , θ˜i ] and (ai0 , θi0 ) ∈ [ˆ ai , ais ] × [θˆi , θis ] monotonically converge to classical ¯ solutions, (¯ ais , θis ) and (ais , θis ), of the steady-state problem, respectively, as t → ∞. Remark 8.1. In the case where (ai0 , θi0 ) ∈ (ais , a ¯is ) × (θis , θ¯is ), the existence of limits lim ai ,
t→∞
lim θi
t→∞
is an open problem. If these limits exist, then under assumption (31) they are classical solutions of the steady-state problem. Each solution of the steady-state problem is also solution of the time-dependent problem if (ai0 , θi0 ), i = 1, 2, is a solution of steady-state problem. References [1] C.V. Pao, W.H. Ruan, Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions, J. Math. Anal. Appl. 333 (2007) 472–499. [2] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, New York, 1992. [3] J.R. Anderson, K. Deng, Global existence for nonlinear diffusion equations, J. Math. Anal. Appl. 196 (1995) 479–501. [4] S. Carl, Ch. Grossmann, C.V. Pao, Existence and monotone iterations for parabolic differential inclusions, Commun. Appl. Nonlinear Anal. 3 (1996) 1–24. [5] P.E. Sacks, Continuity of solutions of a singular parabolic equation, J. Nonlinear Anal. 7 (1983) 384–407. [6] A. Ambrazeviˇ cius, Solvability of a coupled system of parabolic and ordinary differential equations, Cent. Eur. J. Math. 8 (3) (2010) 537–547. [7] V. Skakauskas, P. Katauskis, Modelling dimer-dimer reactions on supported catalysts, J. Math. Chem. 53 (2015) 604–617. [8] A. Ambrazeviˇ cius, Solvability theorem for a model of a unimolecular heterogeheous reaction with adsorbate diffusion, J. Math. Sci. (N. Y.) 184 (4) (2012) 383–398. (Translated from Probl. Mat. Anal. 65 (2012) 13–26). [9] V. Skakauskas, P. Katauskis, Spillover in monomer-monomer reactions on supported catalysts: dynamic mean-feld study, JOMC 52 (5) (2014) 1350–1363. [10] D.K. Fadeev, B.Z. Vulix, N.N. Ural’ceva, Selected Chapters of Analysis and Algebra, LGU, 1981, p. 200 (in Russian). [11] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Clifs, NJ, 1964. [12] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasi-linear Equation of Parabolic Type, Am. Math. Soc. Transl., Providence, RI, 1968 (English transl.).