Solvability theorem for a surface reaction type model

Journal Pre-proof Solvability theorem for a surface reaction type model

A. Ambrazeviˇcius, V. Skakauskas

PII:

S0022-247X(20)30059-7

DOI:

https://doi.org/10.1016/j.jmaa.2020.123897

Reference:

YJMAA 123897

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

28 June 2019

Please cite this article as: A. Ambrazeviˇcius, V. Skakauskas, Solvability theorem for a surface reaction type model, J. Math. Anal. Appl. (2020), 123897, doi: https://doi.org/10.1016/j.jmaa.2020.123897.

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Solvability theorem for a surface reaction type model A. Ambrazeviˇciusa , V. Skakauskasa a Faculty

of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania

Abstract The purpose of this paper is to investigate the existence and uniqueness of classical solutions to a model of surface reactions between two polyatomic reactants. The model is described by a coupled system of four quasilinear parabolic equations where two of them are determined in the domain and the other two are considered on a part of its surface. The elliptic operators of the parabolic equations determined on the surface are allowed to be degenerate in the sense that the density-dependent diffusion coefficients pi (θi ) may have the property pi (1) = 0, i = 1, 2. Keywords: coupled parabolic systems; heterogeneous catalysis; reaction-diffusion system

1. Introduction Coupled systems of parabolic and elliptic equations usually arise from applied sciences. In recent years, such systems have been extensively studied in both theory and applications. In theory of differential equations, coupled systems of parabolic and ordinary differential equations are considered in the same domain (see, e.g., [8, 9, 10] and references therein). The process of the bulk diffusion of reactants and their surface reaction can be modelled by: (i) coupled systems of parabolic and ordinary differential equations with the latter considered on the boundary of the domain (see [1]), (ii) coupled systems of parabolic equations where some of them are determined in the domain while the other ones are considered on a part of its surface (see [3, 15]) or in the domain of less dimension (see [2] or [5]). In [2], [3], the elliptic operator of the parabolic equations determined on the domain boundary is uniformly elliptic. In the present paper, we study a mathematical model of a surface reaction between two reactants which is composed of a coupled system of quasilinear parabolic equations. The elliptic operator of the parabolic equations determined on the surface is allowed to be degenerate in the sense that the densitydependent diffusion coefficients pi (θi ) may have the property pi (1) = 0, i = 1, 2. Email addresses: [email protected] (A. Ambrazeviˇ cius), [email protected] (V. Skakauskas)

Preprint submitted to Elsevier

January 24, 2020

The purpose of this paper is to investigate the existence and uniqueness of classical solutions of the model. The paper is organized as follows. In Section 2, we describe the model. Main results are formulated in Section 3. In Section 4, we give a priori estimates of the solutions. Sections 5 and 6 are devoted to the uniqueness and existence of the nonnegative classical solution to problem (1) and (2). At last, in section 7, we investigate the long-time behavior of the mean values of components of the solution. 2. Formulation of the problem (2)

(1)

We study the r1 Ar2 + r2 Ar1 → r1 r2 A(1) A(2) surface reaction with positive (1) (2) integers r1 and r2 between reactants Ar1 and Ar2 . According to [6, 14] this surface reaction occurs via steps q1 κ11

q2 κ22

(1) K1 , A(1) r1 + r1 K1  r1 A

(2) A(2) K2 , r2 + r2 K2  r2 A

q1 κ1

q2 κ2

κ

A(1) K1 + A(2) K2 → P + K1 + K2 . Here P = A(1) A(2) is the reaction product, K1 and K2 are the vacant adsorption sites, qi κii and qi κi , i = 1, 2, are the adsorption and desorption rates of reactants (i) Ari , κii and κi are some positive constants, κ is the reaction between adsorbates A(1) K1 and A(2) K2 rate constant. (1) (2) Suppose, that reactants Ar1 and Ar2 occupy a bounded domain Ω ⊂ Rn , n ≥ 3; a1 = a1 (x, t) and a2 = a2 (x, t) are their concentrations at point x ∈ Ω at time t, respectively; S := ∂Ω is a surface of (n − 1) dimension 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 a surface of n − 2 dimension (contour, if n = 3) in Rn−1 , ρi = ρi (x ) is the concentration of the adsorption sites of the surface S2 at (i) point x = (x1 , . . . , xn−1 ) ∈ S2 for molecules of reactant Ari , θ1 = θ1 (x , t)   and θ2 = θ2 (x , t) are the surface coverages at point x ∈ S2 at time t by (1) (2) adsorbed molecules of reactants Ar1 and Ar2 , respectively. The bulk diffusion (i) of reactants Ari , i = 1, 2, can be described by the system ⎧ in Ω × (0, T ), ∂ai /∂t − ki Δai = 0 ⎪ ⎪ ⎪ ⎪ ⎨ ki ∂ai /∂n = 0 on S1 × (0, T ),   ⎪ ki ∂ai /∂n = qi ρri i κi θiri − κii ai (1 − θi )ri on S2 × (0, T ), ⎪ ⎪ ⎪  ⎩  ai t=0 = ai0 in Ω,

(1)

n where ki = const > 0 are the diffusion coefficients, Δai = k=1 aixk xk , ∂ai /∂n is the outward normal derivative of ai at point x ∈ S (ki ∂ai /∂n means a flux (i) of reactant Ari through surface S), qi = qi (x ), x ∈ S2 , are given functions such that qi (x ) > 0 for x ∈ S2 , ri ∈ N, where N is the set of positive integers,

2

(i)

κii = const > 0, κi = const > 0, ai0 = ai0 (x) is the initial concentration of Ari at point x ∈ Ω, and T < ∞. Applying the Langmuir and Hinshelwood reactions scheme for θi , i = 1, 2, we get the system ⎧ ρi ∂θi /∂t − div(ϕi pi ∇θi ) = ⎪ ⎪ ⎪   ⎪ ⎨ qi ri ρri κii ai (1 − θi )ri − κi θri − κρi ρj θi θj in S2 × (0, T ), i i (2) ⎪ on Σ × (0, T ), ∂θi /∂ν = 0 ⎪ ⎪ ⎪  ⎩ in S2 , θi t=0 = θi0  n−1  where j = 1, 2, j = i, div(ϕi pi ∇θi ) = k=1 ϕi pi θixk xk , pi = pi (θi ) and ϕi = ϕi (ρi ) are given functions, pi (θi ) > 0 if θi < 1, pi (1) = 0, pi ϕi stands for the diffusivity, ∂θi /∂ν is the outward normal derivative of θi on Σ, θi0 = θi0 (x ) is the initial surface coverage. The bulk diffusion of the reaction product P is described by the system ⎧ ∂b/∂t − kb Δb = 0 in Ω × (0, T ), ⎪ ⎪ ⎪ ⎪ ⎨ kb ∂b/∂n = 0 on S1 × (0, T ), (3) ⎪ kb ∂b/∂n = κρ1 ρ2 θ1 θ2 on S2 × (0, T ), ⎪ ⎪ ⎪ ⎩  in Ω, b t=0 = b0 n where kb = const > 0 is the diffusion coefficient, Δb = k=1 bxk xk , ∂b/∂n is the outward normal derivative of b on S, b0 = b0 (x) is the initial concentration of P at point x ∈ Ω. Systems (1), (2), and (3) compose a nonlinear mathematical model of the (1) (2) surface reaction proceeding between reactants Ar1 and Ar2 . Obviously, system (3) can be solved after problem (1)–(2) is solved. 3. Main results Let

a∗i0 := max ai0 (x), x∈Ω

∗ θi0 := max θi0 (x ) < 1 x ∈S2

and ri



∗ κi  θi0 (κii a∗i0 )1/ri  ∗ , m < 1, := max θ , mi := max a∗i0 , i i0 ∗ 1/r κii 1 − θi0 κi i + (κii a∗i0 )1/ri (4) Let α, β ∈ (0, 1) and α ≥ β. Suppose that the surfaces S and Σ, initial functions ai0 , θi0 , and given functions ρi , qi , ϕi , pi , i = 1, 2, satisfy the following conditions:  (i) S is a surface of class C 1+α , (A1 ) (5) (ii) Σ is a surface of class C 2+β , 3

⎧ (i) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (ii) (A2 ) ⎪ ⎪ ⎪ ⎪ ⎪ (iii) ⎪ ⎩

⎧ ⎨ (i) (A3 ) (ii) ⎩ (iii)

ai0 ∈ C α (Ω), ai0 (x) ≥ 0 for all x ∈ Ω and are continuously differentiable on any neighbourhood of the surfaceS, θi0 ∈ C 2+β (S2 ) is a nonnegative function in the closed domain S2 such that : ∂θi0 /∂ν = 0 on Σ,  ρi ∈ C 2+β (S2 ), div ϕi (ρi (x ))∇ρi (x ) ≤ 0 for all x ∈ S2 , ρi , ρ˜i ], ρˆi and ρ˜i are positive, ρi ∈ [ˆ (6) qi ∈ C β (S2),  ρi , ρ˜i ] , ϕi (ρi ) > 0, ϕi ∈ C 1+β [ˆ pi ∈ C 1+β [0, mi ] .

(7)

Definition 3.1. Collection of functions (a1 , a2 , θ1 , θ2 ) is called a classical solution of problem (1), (2) if ai ∈ C 2,1 (Ω × (0, T ]) ∩ C (Ω × [0, T ]), ∂ai /∂n is continuous on S1 × [0, T ], S2 × [0, T ], θi ∈ C 2,1 (S2 × (0, T ]) ∩ C (S2 × [0, T ]), ∂θi /∂ν, i = 1, 2, is continuous on Σ × [0, T ] and if it satisfies equations (1) – (2). The basic theorem reads as follows: Theorem 3.1. Let assumptions (A1 ), (A2 ), and (A3 ) hold. Then problem (1), (2) has a unique classical solution such that 0 ≤ ai (x, t) ≤ mi , 0 ≤ θi (x , t) ≤ mi ,

∀x ∈ Ω, t ∈ [0, T ],

(8)

∀x ∈ S2 , t ∈ [0, T ].

(9)

Remark 3.1. Let (a1 , a2 , θ1 , θ2 , b) be a classical solution of problem (1)–(3). Then the following mass conservation laws are true:     (ri ai + b) dx + ρi θi dx = (ri ai0 + b0 ) dx + ρi θi0 dx , i = 1, 2. (10) Ω

S2

Ω

S2

4. A priori estimates In parallel with problem (1), (2), we consider an auxiliary problem consisting of (1) and ⎧ ρi ∂θi /∂t − div(ϕi p∗i ∇θi ) = ⎪ ⎪ ⎪   ⎪ ⎨ qi ri ρri κii ai (1 − θi )ri − κi θri − κρi ρj θi θj i i ⎪ ∂θi /∂ν = 0 ⎪ ⎪ ⎪ ⎩  θi = θi0

in S2 × (0, T ), on Σ × (0, T ), in S2 ,

t=0

4

(11)



where p∗i (θi )

if θi ≤ mi , if θi > mi ,

pi (θi ), pi (mi ),

=

i = 1, 2. It is obvious that every solution (a1 , a2 , θ1 , θ2 ) to auxiliary problem (1), (11) satisfying the condition θi ≤ mi , i = 1, 2, is also a solution of the problem (1), (2). To prove Theorem 3.1 we need a priori estimates for solutions to Eqs. (1), (11) that are formulated as Lemma 4.1. Let (a1 , a2 , θ1 , θ2 ) be a classical solution of problem (1), (11). Then estimates (8), (9) are true. Proof. We first prove that ai and θi , i = 1, 2, are nonnegative. To do this we multiply equation (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 

ai η dxdτ

t  + ki

0 Ω

t 

  ρri i κi θiri − κii ai (1 − θi )ri qi η dx dτ.

aix ηx dxdτ = 0 Ω

0 S2

n Here and in what follows ai = ∂ai /∂t, aix ηx = k=1 aixk ηxk . Inserting  ai (x, t), if ai (x, t) < 0, η= 0, if ai (x, t) ≥ 0

(12)

into (12) we get 1 2



τ =t a2i dxτ =0 +ki

Ω− i,τ

t 

t 

0 Ω− i,τ

t

  ρri i κi θiri −κii ai (1−θi )ri qi ai dx dτ

a2ix dxdτ = 0 Sa− ,τ i



≤

ρri i κi θiri qi ai

t 



dx dτ −

0 Sa− ,τ ∩S − θ ,τ i

ρri i κii (1 − θi )ri qi a2i dx dτ

0 Sa− ,τ i

i

≤ c1i

t   0

a2i dx +

Sa−i ,τ



θi2 dx dτ,

(13)

Sθ− ,τ i

where Ω− S − = {x ∈ S2 : ai (x , 0, τ ) < 0}, i,τ = {x ∈ Ω : ai (x, τ ) < 0}, n ai ,τ 2 −   2 Sθi ,τ = {x ∈ S2 : θi (x , τ ) < 0}, aix = k=1 aixk .

5

Similarly, from Eq. (11) it follows that 1 2

 ρi θi2



τ =t dx τ =0

t 

ϕi pi |∇θi | dx dτ =

+

Sθ− ,τ

t 



2

ϕi pi

0 S− θ ,τ

i

0 Σ− θ ,τ

i

t 



+

∂θi θi dΣdτ ∂ν

i

   ri ρri i κii ai (1 − θi )ri − κi θiri qi − κρ1 ρ2 θi θj θi dx dτ,

(14)

0 S− θ ,τ i

n−1 2   where |∇θi |2 = k=1 θix , Σ− θi ,τ = {x ∈ Σ : θi (x , τ ) < 0}. k The boundary condition for θi shows that  ∂θi θi dΣ = 0. ϕi pi ∂ν Σ− θ ,τ i

For the second integral on the right hand side of Eq.(14) we have the estimate t 



   ri ρri i κii ai (1 − θi )ri − κi θiri qi − κρi ρj θ1 θ2 θi dx dτ

0 S− θ ,τ i

≤ c2i

t 

θi2 dx dτ +

t 

0 S− θ ,τ

0 Sa− ,τ i

i

It is known that





 a2 dS ≤ ε

S

for all ε > 0. Therefore 

a2i dx dτ .

a2x dx + Cε Ω

a2i dx ≤ ε

Sa−i ,τ

a2 dx

(15)

Ω



 a2ix dx + Cε

Ω− i,τ

a2i dx.

Ω− i,τ

Using this inequality with sufficiently small ε > 0, assumption (A2 ) for ai0 and ρi , ρ˜i ] from (13) and (14) we derive the estimate θi0 and that ρi ∈ [ˆ 

 a2i

Ω− i,t

θi2

dx + Sθ− ,t



dx ≤ c

t   Ω− i,τ

0

i

6

 a2i

dx + Sθ− ,τ i

θi2 dx dτ.

where the constant c depends only on κi , κ, κii , ri , ρˆi , ρ˜i and max |θi (x )|. From x ∈S2

here by the Gronwall lemma it follows that   a2i dx + θi2 dx ≤ 0. Ω− i,t

Sθ− ,t i

Therefore ai (x, t) ≥ 0, ∀x ∈ Ω, t ∈ [0, T ], θi (x , t) ≥ 0, ∀x ∈ S2 , t ∈ [0, T ]. Next, we prove that ai (x, t) ≤ mi , ∀x ∈ Ω, t ∈ [0, T ], θi (x , t) ≤ mi , ∀x ∈ S2 , t ∈ [0, T ]. Inserting  ai (x, t) − mi , if ai (x, t) > mi , η= 0, if ai (x, t) ≤ mi into (12) we get 

1 2

τ =t (ai − mi ) dxτ =0 + ki

t



2

0 Ω+ (mi ) i,τ

Ω+ i,τ (mi )

t



0

Sa+i ,τ (mi )

a2ix dxdτ =

  ρri i κi θiri − κii ai (1 − θi )ri qi (ai − mi ) dx dτ,

(16)

+   where Ω+ i,τ (mi ) = {x ∈ Ω : ai (x, τ ) > mi }, Sai ,τ (mi ) = {x ∈ S2 : ai (x , 0, τ ) > mi }.  ri i Equations (4) show that κi mr i −κii mi (1−mi ) = 0. Therefore, for ai > mi ,  ri i κi θiri − κii ai (1 − θi )ri ≤ κi mr i − κii mi (1 − mi ) = 0,

if θi (x , t) ≤ mi and κi θiri − κii ai (1 − θi )ri

   ri ri i = κi (θiri − mr i ) + κii mi (1 − mi ) − (1 − θi ) −κii (ai − mi )(1 − θi )ri ,

if θi (x , t) > mi . Hence,    ρri i κi θiri − κii ai (1 − θi )ri qi (ai − mi ) dx ≤ Sa+i ,τ (mi )



ρi (θi − mi )(ai − mi ) + (ai − mi )2 dx ≤

C1i Sa+i ,τ (mi )∩Sθ+ ,τ (mi )  C1i





i



(ai − mi )2 dx +

Sθ+ ,τ (mi )

Sa+i ,τ (mi )

i

7

(θi − mi )2 dx ,

where Sθ+i ,τ (m ) = {x ∈ S2 : θi (x , τ ) > m }. Similarly, from Eq. (11) it follows that 

1 2

ρi (θi −

mi )2



τ =t dx τ =0

t



ϕi p∗i (θi )|∇θi |2 dx dτ

+

Sθ+ ,τ (mi )

0 S + (m ) i θ ,τ

i

i

t



ϕi p∗i (θi )

=

∂θi (θi − mi ) dΣdτ ∂ν

0 Σ+ (m ) i θ ,τ i

t





+

   ri ρri i κii ai (1 − θi )ri − κi θiri qi − κρ1 ρ2 θ1 θ2 (θi − mi ) dx dτ. (17)

0 S + (m ) θ,τ

The boundary condition for θi (∂θi /∂ν = 0 on Σ) shows that  ∂θi ρi (θi − mi ) dΣ = 0. ϕi p∗i (θi ) ∂ν  Σ+ θ ,τ (mi ) i

 ri i Using relation κi mr i − κii mi (1 − mi ) = 0 we get      ri ρri i κii ai (1 − θi )ri − κi θiri qi − κρ1 ρ2 θ1 θ2 (θi − mi ) dx ≤ Sθ+ ,τ (mi ) i



   ri ρri i κii ai (1 − θi )ri − (1 − mi )ri + κii (ai − mi )(1 − mi )ri +

Sθ+ ,τ (mi ) i

 ri ri i + κ (m − θ ) qi (θi − mi ) dx κii mi (1 − mi )ri − κi mr i i i i    2  (θi − mi ) dx + ri ρri i κii (ai − mi )qi (θi − mi ) dx

≤ C2i

Sθ+ ,τ (mi ) i

 ≤ C2i





Sθ+ ,τ (mi )∩Sa+i ,τ (mi ) i

(θi − mi )2 dx +

Sθ+ ,τ (mi )

Sa+i ,τ (mi )

(ai − mi )2 dx .

Sa+i ,τ (mi )

i

By (15), 



(ai − mi )2 dx ≤ ε



 a2ix dx + Cε

Ω+ ai ,τ (mi )

8

Ω+ ai ,τ (mi )

(ai − mi )2 dx.

Using these inequalities with sufficiently small ε > 0, assumption (A2 ) for ai0 and θi0 , and that ρi ∈ [ˆ ρi , ρ˜i ], from (16) and (17) we derive the estimate   2 (ai − mi ) dx + (θi − mi )2 dx Sθ+ ,t (mi )

Ω+ i,t (mi )

≤C

t  0

i

t





(θi − mi )2 dx dτ.

(ai − mi ) dx + 2

0 S + (m ) i θ ,τ

Ω+ i,τ (mi )

i

where constant C depends only on

max x∈Ω,t∈[0,T ]

|ai (x, t)|,

max

x ∈S2 ,t∈[0,T ]

|θi (x , t)|,

κi , κ, κii , ri , ρˆi , ρ˜i . At last, the Gronwall lemma shows that   2 (ai − mi ) dx + (θi − mi )2 dx ≤ 0. Ω+ i,t

Sθ+ ,t i

Hence, ai (x, t) ≤ mi , ∀x ∈ Ω, t ∈ [0, T ] and θi (x , t) ≤ mi , ∀x ∈ S2 , t ∈ [0, T ]. The proof is complete.  In what follows we need Lemma 4.2. 1. Let functions θˆi , θ˜i for each fixed i = 1, 2 be continuous on S2 × [0, T ] and let 0 ≤ θˆi ≤ θ˜i ≤ mi . Assume that a ˜i and a ˆi are the classical solutions to problem (1) with θi = θ˜i and θi = θˆi , respectively. ˜i ≤ mi in Ω × [0, T ]. Then 0 ≤ a ˆi ≤ a ˜i , θˆi∗ , θ˜i∗ for each fixed i = 1, 2, be continuous on S2 × 2. Let functions a ˆi , a ˜i ≤ mi , 0 ≤ θˆi∗ ≤ θ˜i∗ ≤ mi . Assume that θ˜i and [0, T ] and let 0 ≤ a ˆi ≤ a ˆ ˜i , θj = θˆj∗ and θi are the classical solutions to problem (11) with ai = a ai = a ˆi , θj = θ˜j∗ , j = i, respectively. Then 0 ≤ θˆi ≤ θ˜i ≤ mi in S2 × [0, T ]. Proof. The first proposition of this lemma can be easily proved by using the Positivity Lemma (see [8, Chapter 1, Lemma 4.1, p. 19]) or by the argument used to prove Lemma 4.1. The proof of the second assertion is as follows. First, similarly as in the proof of Lemma 4.1 we prove that estimates (9) are true for the solutions θˆi , θ˜i , i = 1, 2. Next, we prove that θˆi ≤ θ˜i . Set θi = θ˜i − θˆi . Then from (2) it follows that    ˜ ri − κi θ˜ri ρi ∂θi /∂t − div ϕi (pi (θ˜i )∇θ˜i − pi (θˆi )∇θˆi ) = ri ρri qi κii a ˜i (1 − θ) i

   ˆ ri + κi θˆri − κρ1 ρ2 θ˜i θˆ∗ − θˆi θ˜∗ . −κii a ˆi (1 − θ) j j i Multiplying this equation by θi , and integrating the result, we get 1 2

 Sθ− ,t i

ρi θi2 dx +

t 

  ϕi pi (θ˜i )∇θ˜i − pi (θˆi )∇θˆi · ∇θi dx dτ

0 S− θ ,τ i

9

i

t  =

 ∂ θ˜i ∂ θˆi − pi (θˆi ) θi dΣdτ + ϕi pi (θ˜i ) ∂ν ∂ν 

0 Σ− θ ,τ

t





κii a ˜i (1 − θ˜i )ri

0 S− θ ,τ

i

i

  −κi θ˜iri − κii a ˆi (1 − θˆi )ri + κi θˆiri ri ρri i qi − κρ1 ρ2 (θ˜i θˆj∗ − θˆi θ˜j∗ θi dx dτ. It is easy to verify that   ϕi pi (θ˜i )∇θ˜i − pi (θˆi )∇θˆi · ∇θi ≥ ε|∇θi |2 − Cε θi2 ,   κii a ˜i (1 − θ˜i )ri − κi θ˜iri − κii a ˆi (1 − θˆi )ri + κi θˆiri ri ρri i qi θi ≤ 0,     −κρ1 ρ2 θ˜i θˆj∗ − θˆi θ˜j∗ ρi θi = −κρ1 ρ2 θi θˆj∗ + θˆi (θˆj∗ − θ˜j∗ ) ρi θi ≤ 0 in Sθ−i ,τ and

 ∂ θ˜i ∂ θˆi − pi (θˆi ) θi = 0 ϕi pi (θ˜i ) ∂ν ∂ν

on Σ− θi ,τ . These estimates show that  θi2 Sθ− ,t i



t 

dx ≤ C

θi2 dx dτ.

0 S− θ ,τ i

From here by the Gronwall lemma we conclude that θi ≥ 0. Therefore θˆi ≤ θ˜i . The proof is complete.  Remark 4.1. Suppose that given functions ai and θj∗ , i, j = 1, 2, i = j, are continuous on S2 × [0, T ] and satisfy inequalities (8) and (9), respectively. Let θi , i = 1, 2, be a classical solution of the problem ⎧ ρi ∂θi /∂t − λ div(ϕi (ρi )p∗i (θi )∇θi ) − (1 − λ)Δθi = ⎪ ⎪ ⎪   ⎪  ri  ⎨ in S2 × (0, T ), λ ri ρi κii ai (1 − θ)ri − κi θiri qi − κρ1 ρ2 θi θj∗ ⎪ ∂θi /∂ν = 0 ⎪ ⎪ ⎪ ⎩  θi = θi0

on Σ × (0, T ), in S2 ,

t=0

depending on the parameter λ ∈ [0, 1]. It is easy to see that for solution θi estimate (9) is also preserved. 5. Uniqueness of the classical solution In this section, we prove that problem (1) and (2) has at most one classical solution that satisfies inequalities (8) and (9). Theorem 5.1. Problem (1) and (2) has at most one classical solution satisfying estimates (8) and (9). 10

Proof. Let (˜ a1 , a ˜2 , θ˜1 , θ˜2 ) and (ˆ a1 , a ˆ2 , θˆ1 , θˆ2 ) be two classical solutions of problem (1) and (2) such that a ˜i , a ˆi ∈ [0, mi ], θ˜i , θˆi ∈ [0, mi ], i = 1, 2, and set ai = a ˜i − a ˆi , θi = θ˜i − θˆi . Then ∂ai /∂t − κi Δai = 0 κi ∂ai /∂n = 0    ˜i (1− θ˜i )ri −ˆ ai (1− θˆi )ri qi κi∂ai /∂n = ρri i κi (θ˜iri − θˆiri )−κii a ai t=0 = 0

in Ω × (0, T ), on S1 × (0, T ), on S2 × (0, T ), in Ω

and 1 2



t  a2i

t  a2ix

dx + κi

dxdτ =

Ω

0 S2

−ˆ ai (1− θˆi )ri

 qi ai dx dτ ≤ ci

  ˜i (1− θ˜i )ri ρri i κi (θ˜iri − θˆiri )−κii a

0 S2

t 

ci |ai θi | dx dτ ≤ 2 

0 S2

t 

(a2i + θi2 ) dx dτ. (18)

0 S2

Similarly,       ˜i (1 − θ˜i )ri ρi ∂θi /∂t − div ϕi pi (θˆi )∇θi + ϕi pi (θ˜i ) − pi (θˆi ) ∇θ˜i = ri ρri i κii a     −ˆ ai (1 − θˆi )ri − κi (θ˜iri − θˆiri ) qi − κρi ρj θi θ˜j + θˆi θj ∂θi /∂ν = 0  θi  =0 t=0

in

S2 × (0, T ),

on

Σ × (0, T ),

in

S2

and 1 2

 ρi θi2 S2

t  =



t 

dx +

  ϕi pi (θ˜i ) − pi (θˆi ) ∇θ˜i · ∇θi + ϕi pi (θˆi )|∇θi |2 dx dτ

0 S2

t       ∂ θ˜i ∂θi  + pi (θˆi ) ˜i (1− ϕi pi (θ˜i ) − pi (θˆi ) ri ρri i κii a θi dΣdτ + ∂ν ∂ν

0 Σ

ˆi (1 − θˆi ) θ˜i )ri − a

0 S2

 ri





− κi (θ˜iri − θˆiri ) qi − κρ1 ρ2 θi θ˜j + θˆi θj t 

≤ Ci



θi dx dτ

(θi2 + θj2 + a2i ) dx dτ.

(19)

0 S2

Assumption (A3 ) for pi implies that      ϕi pi (θ˜i ) − pi (θˆi ) ∇θ˜i · ∇θi dx ≤ ε |∇θi |2 dx + Cε θi2 dΣ. S2

S2

11

S2

Using these inequalities with sufficiently small ε > 0 from (18) and (19) we derive the estimate   2

a2i

dx +

Ω i=1

  2

θi2

dx ≤ C

S2 i=1

Set Φ(t) =

t   2 0

t   2

a2i dx +

Ω i=1

a2i dx +

Ω i=1

0

Then



  2

  2

θi2 dx dτ.

S2 i=1

ρ2i θi2 dx dτ.

S2 i=1

  Φ (t) ≤ CΦ(t) ⇒ Φ(t)e−Ct ≤ 0 ⇒ Φ(t)e−Ct ≤ Φ(0) = 0.

Hence, ai ≡ 0 and θi ≡ 0. The proof is complete.  6. Existence of a classical solution In this section, we prove that problem (1) and (2) has a classical solution. Proof. For any continuous nonnegative on S 2 × [0, T ] functions θ1 and θ2 problem (1) has a unique classical solution ai ∈ C 2,1 (Ω × (0, T ]) ∩ C (Ω × [0, T ]) which can be represented by the formula (see [4, Chapter 5, Section 3, Theorem 2]) t 

 Γi (x − ξ, t − τ )μi (ξ, τ ) dSξ dτ +

ai (x, t) = 0

S

Γi (x − y, t)ai0 (y) dy,

(20)

Ω0

where

1

Γi (x, t) = 

n/2 e

−|x|2 /4ki t

4πki t

,

x ∈ Rn , t > 0,

is a fundamental solution of equation (1), Ω0 = Ω, if ai0 = 0 in any neighbourhood of surface S, and Ω0 ⊃ Ω, if ai0 is continuously differentiable in any neighbourhood of surface S. In the first case, we extend function ai0 = 0 on Rn \ Ω. In the other case, we extend function ai0 on Ω0 \ Ω preserving the same smoothness and nonnegativity so that supp ai0 ⊂ Ω0 . In both cases, we denote the extended function by ai0 . The function μi is a solution of the Volterra integral equation 1 μi (η, t) + 2

t   ∂Γi (η − ξ, t − τ ) σi (η, t) + Γi (η − ξ, t − τ ) μi (ξ, τ ) dSξ dτ ∂nη ki 0

S

ψi (η, t) − = ki

  ∂Γi (η − y, t) σi (η, t) + Γi (η − y, t) ai0 (y) dy ∂nη ki Ω0

12

(21)

with the weak singularity    ∂Γ (η − ξ, t − τ ) σ (η, t)  C  i  i + Γi (η − ξ, t − τ ) ≤ ,    (t − τ )1−γ |η − ξ|n−1−δ ∂nη ki δ = α − 2γ > 0, 0 < γ < 1/2. Here  0, if η ∈ S1 , t ∈ [0, T ],   ri σi (η, t) = κii ρri i (η  ) 1 − θi (η  , t) qi (η  ), if η ∈ S2 , t ∈ [0, T ],  0, if η ∈ S1 , t ∈ [0, T ], ψi (η, t) = ri  ri   κi ρi (η )θ (η , t)qi (η ), if η ∈ S2 , t ∈ [0, T ]. The solution of Eq. (21) can be represented by the formula (see [4, Chapter 5, Section 3, Theorem 2]) μi (η, t) = gi (η, t) +

∞ t   k=1 0

Qik (η, t, ξ, τ )gi (ξ, τ ) dSξ dτ,

(22)

S

where gi (η, t) =      ψi (η, t) ∂Γi (η − y, t) σi (η, t) − + Γ(η − y, t) ai0 (y) dy , 2 ki ∂nη ki Ω0

 ∂Γ (η − ξ, t − τ ) σ (η, t) i i + Γi (η − ξ, t − τ ) , ∂nη ki

Qi1 (η, t, ξ, τ ) = −2

t  Qik (η, t, ξ, τ ) =

Qi1 (η, t, ζ, s)Qik−1 (ζ, s, ξ, τ ) dSζ ds, τ

k = 2, 3, . . .

S



Since

∂Γi (η − y, t) ai0 (y) dy ∂nη

Ω0

is continuous on S × [0, T ], then function gi is bounded and continuous on surfaces S1 × [0, T ] and S2 × [0, T ] as well. Function Qik (η, t, ξ, τ ) can be written in the form (1)

(2)

Qik (η, t, ξ, τ ) = Qik (η, t, ξ, τ ) + σi (η, t)Qik (η, t, ξ, τ ). By induction (see [4, Chapter 5, Section 3, Theorem 2] or [7, Chapter 4, section 11]) it is possible to show that (j)

|Qik (η, t, ξ, τ )| ≤

Cik 1 Γk (γ) , |ξ − η|n−1−kδ (t − τ )1−kγ Γ(kγ) 13

j = 1, 2,

where Γ(t) is the Gamma-function and the constant Ci is independent of θi and satisfies the inequalities 0 ≤ θi (x , t) ≤ 1 for all (x , t) ∈ S2 × [0, T ]. The (j) functions Qik , j = 1, 2, have a week singularity if k ≤ k0j and are continuous if k > k0j . Therefore, the series ∞ t   k=1 0

(j)

Qik (η, t, ξ, τ )gi (ξ, τ ) dSξ dτ,

j = 1, 2,

S

converge uniformly. Hence, the function μi is bounded,   μi (η, t) ≤ Mi ∀η ∈ S, t ∈ [0, T ], and is continuous1 on each of the surfaces S1 × [0, T ] and S2 × [0, T ]. Moreover, the constant Mi is independent of θi satisfying the condition 0 ≤ θi (x , t) ≤ 1 for all (x , t) ∈ S2 × [0, T ]. By assumption (A2 ), ai0 ∈ C α (Ω) . Therefore in formula (20), the integral  Γi (x − y, t)ai0 (y) dy Ω0

belongs to the H¨ older space C α,α/2 (Ω × [0, T ]) . In formula (20), the potential of the simple layer belongs to H¨older space C α,α/2 (Ω × [0, T ]) (see [11, Section 2, Theorems 2 and 3] or [4, Chapter 5, Section 4, Theorem 3]). Hence, ai determined by (20) belongs to C α,α/2 (Ω × [0, T ]) if θi ∈ C(S 2 × [0, T ]). Let in (11), for each fixed i = 1, 2, functions ai , θj ∈ C β,β/2 (S2 × [0, T ]), j = i, are given and satisfy the inequalities 0 ≤ ai ≤ mi , 0 ≤ θj ≤ 1. By Lema 4.2, estimates (9) are true for solutions θi of problem (11). Moreover, by Remark 4.1 such estimates hold true for a whole family of problems. Then problem (2) has a unique classical solution θi ∈ C 2+β,1+β/2 (S2 × [0, T ]) such that 0 ≤ θi ≤ mi in S2 × [0, T ] (see [7, Chapter V, Section 7, Theorem 7.4]). (0) (0) Substituting either θi = θ˜i := mi or θi = θˆi := 0 into (1) and either (0) (0) (0) (0) ˜i := mi , θj = θˆj := 0 or ai = a ˆi := 0, θj = θ˜j := mj into (2) as the ai = a (s) (s) (s) (s) (s) initial iteration we construct sequences {˜ ai }, {˜ μi }, {ˆ ai }, {ˆ μi } and {θ˜i }, (s) {θˆi }. By Lemma 4.2 and induction principle we get that these sequences are uniformly bounded and possess the monotone property (0)

0=a ˆi

(1)

≤a ˆi

(s)

≤ ··· ≤ a ˆi

(s)

≤ ··· ≤ a ˜i

(1)

≤ ··· ≤ a ˜i

(0)

≤a ˜i

= mi ,

(0) (1) (s) (s) (1) (0) 0 = θˆi ≤ θˆi ≤ · · · ≤ θˆi ≤ · · · ≤ θ˜i ≤ · · · ≤ θ˜i ≤ θ˜i = mi ,

i = 1, 2. Therefore, the pointwise limits (s) ˆ lim a s→∞ i

1 If

(s)

=a ˆi , lim a ˜i s→∞

(s) (s) =a ˜i , lim θˆi = θˆi , lim θ˜i = θ˜i . s→∞

qi = 0 on Σ then μi ∈ C (S × [0, T ]) .

14

s→∞

exist and (0)

0=a ˆi

(1)

≤a ˆi

(s)

≤ ··· ≤ a ˆi

(s)

≤a ˆi ≤ a ˜i ≤ · · · ≤ a ˜i

(1)

≤ ··· ≤ a ˜i

(0)

≤a ˜i

= mi ,

(0) (1) (s) (s) (1) (0) 0 = θˆi ≤ θˆi ≤ · · · ≤ θˆi ≤ θˆi ≤ θ˜i ≤ · · · ≤ θ˜i ≤ · · · ≤ θ˜i ≤ θ˜i = mi , (s) (s) (s) i = 1, 2. Moreover, θ˜i , θˆi ∈ C β,β/2 (S2 × [0, T ]) . Therefore, sequences {θ˜i }, (s) {θˆi } are equicontinuous and, hence, converge uniformly to θ˜i and θˆi , respectively. (s) For any s = 1, 2, . . . the function ai is defined by the formula

t  (s) a ˜i (x, t)

0 (s)

where μ ˜i

S

Γi (x − y, t)ai0 (y) dy, Ω0

is a solution of the integral equation

(s)

μ ˜i (η, t) + 2

t   (s−1) ˜ (η, t) ∂Γi (η − ξ, t − τ ) σ (s) + i Γi (η −ξ, t−τ ) μ ˜i (ξ, τ )dSξ dτ ∂nη ki 0

=

 (s)

Γi (x − ξ, t − τ )˜ μi (ξ, τ ) dSξ dτ +

=

S

  (s−1) (s−1) (η, t) (η, t) ˜ ψ˜i ∂Γi (η − y, t) σ − + i Γi (η − y, t) ai0 (y) dy ki ∂nη ki Ω0

(s−1) σ ˜i

(s−1) ψ˜i

(s−1) with and obtained from σi and ψi by replacing θi by θ˜i . In this integral equation, the term

t  0

∂Γi (η − ξ, t − τ ) (s) μ ˜i (ξ, t) dSξ dt ∂nη

S

belongs to the H¨ older space C λ,λ/2 (S × [0, T ]) with λ < 2α/3 (see [11, Section 2, Theorem 4] or [4, Chapter 5, Section 4, Theorem 4 ]). Since the sequence (s) (s) {θ˜i } converges uniformly, then sequence {˜ μi } also converges uniformly on ˜i . The potential of the surfaces S1 × [0, T ] and S2 × [0, T ] to a limit function μ simple layer t  (s) Γi (x − ξ, t − τ )˜ μi (ξ, τ ) dSξ dτ 0

S

belongs to the H¨ older space C λ,λ/2 (Ω × [0, T ]) with any λ ∈ (0, 1) (see [11, Section 2, Theorems 2 and 3] or [4, Chapter 5, Section 4, Theorem 3 ]). Therefore, (s) sequence {˜ ai } is equicontinuous and converges uniformly to a limit function a ˜i and the formula t 

 Γi (x − ξ, t − τ )˜ μi (ξ, τ ) dSξ dτ +

a ˜i (x, t) = 0

S

Γi (x − y, t)ai0 (y) dy, Ω0

15

is true. Hence, a ˜i ∈ C 2,1 (Ω × (0, T ]) ∩ C (Ω × [0, T ]) is a solution to problem ˜i ∈ (1) with a continuous function θi = θ˜i in S 2 × [0, T ]. Moreover, solution a C α,α/2 (Ω × [0, T ]) . (s) (s) Similarly, the sequences {ˆ μi } and {ˆ ai } converge uniformly to some limit ˆi and a ˆi ∈ C 2,1 (Ω × (0, T ]) ∩ C (Ω × [0, T ]) is a solution to probfunctions μ ˆi , a lem (1) with a continuous function θi = θˆi in S 2 × [0, T ]. Moreover, a ˆi ∈ C α,α/2 (Ω × [0, T ]) . Next, we show that θ˜i and θˆi are classical solutions of the problem (2) with ai = a ˜i , θj = θˆj and ai = a ˆi , θj = θ˜j , respectively. For every s function (s) (s−1) 2+β,1+β/2 ˜ (S2 × [0, T ]) is a solution of problem (2) with ai = a ˜i , θi ∈ C (s−1) (s) ˆ ˜ . The sequence {θi } is uniformly bounded and all the conditions θj = θ j of Theorem 7.2 (given in [7, Chapter V, Section 7]) are satisfied. Then the estimates (s)

θ˜ix C (S2 ×[0,T ]) ≤ Mi ,

(s)

θ˜i C 1+δi ,(1+δi )/2 (S2 ×[0,T ]) ≤ ci

(23)

are true, where the constants Mi , ci and δi > 0 are independent of s. Let (s) (s) Li (θi ) = ρi ∂θi /∂t − div(ϕi (ρi )pi (θ˜i )∇θi ) (s) ≡ ρi ∂θi /∂t − ϕi (ρi )pi (θ˜i )Δθi  (s)  (s)   (s) + pi (θ˜i )∇ϕi (ρi ) + ϕi (ρi )pi θ˜i ∇θ˜i · ∇θi .   (s) ri (s) (s−1)  (s) ri (s) (s−1) F (x , t) = ri ρri κii a 1 − θ˜ ˜ − κi θ˜ . qi − κρ1 ρ2 θ˜ θˆ i

i

i

i

i

i

j

 (s)  (s) function ui = θi is a solution of the By construction of the sequence θi (s) (s) linear equation Li (ui ) = Fi in S2 × (0, T ]. Let S2 be strictly interior subdomain of S2 . Based on the estimate (23) we can assert that all the conditions of Theorem 15 (given in [4, Chapter 3, Section 6]) are satisfied in S2 × (0, T ].  (s )   (s)  Hence, there exists a subsequence θi of sequence θi such that (s )

θi

(s )

, ∂θi

(s )

/∂xk , ∂ 2 θi

(s )

/∂xk ∂xl , ∂θi

/∂t, k, l = 1, . . . , n − 1,

are all uniformly convergent in S2 ×[0, T ] to θ˜i and its corresponding derivatives, (s) (s) the coefficients of the operator Li and function Fi converge to corresponding 2,1  ˜ limits. Then the limit functions θi ∈ C (S2 × [0, T ]) satisfy the equations     ri   ˜i 1 − θ˜i − κi θ˜iri qi − κρ1 ρ2 θ˜i θˆj , ρi ∂ θ˜i /∂t − div ϕi (ρi )pi (θ˜i )∇θ˜i = ri ρri i κii a i, j = 1, 2, j = i. Since the domain S  is arbitrary, then the limit functions θ˜i satisfy these equations in S2 × (0, T ]. (s) For every s, θ˜i (x , 0) = θi0 (x ). Therefore limit function θ˜i satisfies the initial condition θ˜i (x , 0) = θi0 (x ). It remains to prove that the limit function (s) satisfies the boundary condition. We recall that the sequence θ˜i converges 16

to the limit function θ˜i uniformly. Moreover, in view of (23), the sequence (s) of derivatives {θ˜ix } also converges uniformly. Hence, the derivative ∂ θ˜i /∂ν is continuous and satisfies the boundary condition. Similarly, we prove that the limit function θˆi is a classical solution of Eq. (2) ˆi , θj = θ˜j . Based on the proof of the uniqueness Theorem 5.1 we can with ai = a assert that a ˜i = a ˆi , θ˜i = θˆi , i = 1, 2. The proof of Theorem 3.1 is complete. 7. The long-time behaviour of the solutions of system (1)–(3) The steady-state system corresponding to time-depending problem (1), (2) is

⎧ in Ω, −ki Δai = 0 ⎪ ⎨ on S1 , ki ∂ai /∂n = 0 ⎪   ⎩ ki ∂ai /∂n = qi ρri i κi θiri − κii ai (1 − θi )ri on S2 , ⎧ − div(ϕi pi ∇(θi )) = ⎪ ⎨   ri qi ρri i κii ai (1 − θi )ri − κi θiri − κρi ρj θi θj in S2 , ⎪ ⎩ on Σ, ∂θi /∂ν = 0

(24)

(25)

i = 1, 2, j = i, j = 1, 2. For this system, the uniqueness theorem does not hold. It suffices to note that (a1 , a2 , θ1 , θ2 ) = (c1 , 0, d1 /(1+d1 ), 0) and (a1 , a2 , θ1 , θ2 ) = (0, c2 , 0, d2 /(1 + d2 ) are solutions of this system, where ci = const ≥ 0, di = (κii ci /κi )1/ri . In what follows we study the long-time behaviour of average values of ai and ρi θi . To do this we integrate system (3) getting t 



 b(x, t) dx − Ω

b0 (x) dx =

κρ1 (x )ρ2 (x )θ1 (x , τ )θ2 (x , τ ) dxdτ

0 S2

Ω

and rewrite mass conservation law (10) as follows:  ri





ai (x, t) dx + Ω

= ri



ai0 (x) dx + Ω



t 

ρi (x )θi (x , t) dx +

S2





κρ1 (x )ρ2 (x )θ1 (x , τ )θ2 (x , τ ) dxdτ

0 S2

ρi (x )θi0 (x ) dx , i = 1, 2.

S2



Let Ai = ri

 ai0 (x) dx +

Ω

ρi (x )θi0 (x ) dx .

S2

The function t  A(t) =

κρ1 (x )ρ2 (x )θ1 (x , τ )θ2 (x , τ ) dxdτ

0 S2

17

is non-decreasing and bounded from above, A(t) ≤ Ai , i = 1, 2. Therefore it has a finite limit A = lim A(t) ≤ Ai , i = 1, 2, and t→∞



 b(x, t) dx → A + Ω

b0 (x) dx Ω

as t → ∞. There are only three possible cases: A = A1 = A2 ; A = A1 < A2 ; A = A2 < A1 . Suppose that A = A1 = A2 . Then   ri ai (x, t) dx + ρi (x )θi (x , t) dx → 0, i = 1, 2, Ω

S2

as t → ∞. Since ai and θi are nonnegative, then   ai (x, t) dx → 0 and ρi (x )θi (x , t) dx → 0, Ω

i = 1, 2,

S2

as t → ∞. Moreover, if ai → 0, θi → 0 as t → ∞, then classical solution (ai , a2 , θ1 , θ2 ) of time-dependent system (1), (2) converges as t → ∞ to solution (0, 0, 0, 0) of steady-state problem (24), (25). If A1 = A, A2 > A, then   a1 (x, t) dx → 0 and ρ1 (x )θ1 (x , t) dx → 0, Ω

S2



and r2

 a2 (x, t) dx +

Ω

ρ2 (x )θ2 (x , t) dx → A2 − A > 0

S2

as t → ∞. There exists a unique pair of positive constants c2 ≤ m2 , c2 ≤ m2 satisfying the system   r  dx + c2 ρ2 (x ) dx = A2 − A, κ2 c2 2 − κ22 c2 (1 − c2 )r2 = 0. r2 c 2 Ω

S2

Then the mass conservation law shows that   r2 (a2 (x, t) − c2 ) dx + ρ2 (x )(θ2 (x , t) − c2 ) dx → 0 Ω

S2

as t → ∞. But it does not follow from here that   (a2 (x, t) − c2 ) dx → 0 and ρ2 (x )(θ2 (x , t) − c2 ) dx → 0 Ω

S2

18

as t → ∞. If a1 → 0, θ1 → 0, a2 → c2 , θ2 → c2 as t → ∞, then classical solution (ai , a2 , θ1 , θ2 ) of time-dependent system (1), (2) converges as t → ∞ to solution (0, c2 , 0, c2 ) of steady-state problem (24), (25). The case where A1 > A, A2 = A can be considered similarly and the case when A1 > A, A2 > A is physically impossible. The proof that ai , θi → 0, i = 1, 2, as t → ∞ in case A = A1 = A2 ; a1 , θ1 → 0, a2 → c2 , θ2 → c2 in case A = A1 < A2 ; and a2 , θ2 → 0, a1 → c1 , θ1 → c1 in case A = A2 < A1 is an open problem. Funding This work was supported by the Research Council of Lithuania (project No. S-MIP-17-65). References References [1] A. Ambrazeviˇcius, Solvability of a coupled system of parabolic and ordinary differential equations, Centr. Eur. J. Math. 8(3) (2010) 537–547. [2] A. Ambrazeviˇcius and V. Skakauskas, Positive solution of a nonlinear parabolic system arising in grain drying, Acta Appl. Math. 150 (2017) 123– 140. [3] A. Ambrazeviˇcius and V. Skakauskas, Solvability of a model for monomermonomer surface reactions. Nonlinear Anal.: real world applications 35 (2017) 211–228. [4] A. Friedman, Partial Differential Equations of Parabolic Type (PrenticeHall, 1964). [5] A. Friedman, A. E. Tzavaras, A Quasilinear Parabolic System Arising in Modeling of Catalytic Reactors, J. Differential Equations 70 (1987) 167-196. [6] A. Garcia, Cantu Ros, J.S. McEwen and P. Gaspard, Effect of ultrafast diffusion on adsorption, desorption, and reaction processes over heterogeneous surfaces, Phys. Rev. E. 83 (2011) 021604. [7] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uralc’eva, Linear and Quasi-linear Equations of Parabolic Type (Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1968). [8] C.V. Pao, Nonlinear Parabolic and Elliptic Equations (Plenum Press, New York, 1992). [9] C.V. Pao and W.H. Ruan, Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions, J. Math. Anal. Appl. 333 (2007) 472– 499. 19

[10] C.V. Pao and W.H. Ruan, Quasilinear parabolic and elliptic systems with mixed quasimonotone functions, J. Diff. Eq. 255 (2013) 1515–1553. [11] W. Pogorzelski, Properi`et`es des Int´egrales de L‘´equation Parabolique Normale, Ann. Polon. Math. 4 (1957) 61–92. [12] M. Pierre, T. Suzuki and R. Zou, Asymptotic behavior of solutions to chemical reaction diffusion systems, J. Math. Anal. Appl. 450(1) (2017) 152– 168. [13] M. Pierre, T. Suzuki and H. Umakoshi, Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria. J. Appl. Anal. Comput. 8(3) (2018) 836–858. [14] V. Skakauskas and P. Katauskis, Computational study of the dimer-trimer and trimer-trimer rections on the supported catalysts, Comput. Theor. Chem. 1070 (2015) 102–107. [15] V. Skakauskas and P. Katauskis, Numerical study of CO oxidation by N2 O reaction over supported catalysts, J. Math. Chem. 54 (2016) 1306–1320.

20