Existence and uniqueness theorem to a model of bimolecular surface reactions
We prove the existence and uniqueness of classical solutions to a coupled system of parabolic and ordinary differential equations in which the latter are determined on the boundary. This system describes the model of bimolecular surface reaction between carbon monoxide and nitrous oxide occurring on...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507594971414528 |
|---|---|
| author | Ambrazevicius, A. Амбразевичус, А. |
| author_facet | Ambrazevicius, A. Амбразевичус, А. |
| author_sort | Ambrazevicius, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:25:34Z |
| description | We prove the existence and uniqueness of classical solutions to a coupled system of parabolic and ordinary differential
equations in which the latter are determined on the boundary. This system describes the model of bimolecular surface
reaction between carbon monoxide and nitrous oxide occurring on supported rhodium in the case of slow desorption of the
products. |
| first_indexed | 2026-03-24T02:11:48Z |
| format | Article |
| fulltext |
UDC 517.9
A. Ambrazevičius (Vilnius Univ., Lithuania)
EXISTENCE AND UNIQUENESS THEOREM
TO A MODEL OF BIMOLECULAR SURFACE REACTIONS
TЕОРЕМА IСНУВАННЯ ТА ЄДИНОСТI
ДЛЯ МОДЕЛI БIМОЛЕКУЛЯРНИХ ПОВЕРХНЕВИХ РЕАКЦIЙ
We prove the existence and uniqueness of classical solutions to a coupled system of parabolic and ordinary differential
equations in which the latter are determined on the boundary. This system describes the model of bimolecular surface
reaction between carbon monoxide and nitrous oxide occurring on supported rhodium in the case of slow desorption of the
products.
Доведено iснування та єдинiсть класичних розв’язкiв зв’язаних систем параболiчних та звичайних диференцiальних
рiвнянь, останнi з яких визначенi на межi. Система описує модель бiмолекулярної поверхневої реакцiї мiж моно-
оксидом вуглецю та закисом азоту, що вiдбувається на нанесеному родiї у випадку повiльної десорбцiї продуктiв.
1. Introduction. Heterogeneous catalytic reactions are modeled by a coupled system of parabolic
and ordinary differential equations. Some of these equations are considered in the domain, while
the other equations have to be solved on a part of the boundary. The unimolecular reaction model
taking into account the reactant adsorption and desorption and both fast and slow product desorption
is considered in [1] and [2] and the existence and uniqueness of a classical solution are proved. In
[9] and [10] the same problems are solved numerically. A model of unimolecular surface reactions
involving adsorbate diffusion and rapid product desorption is studied in [3], where the existence
and uniqueness of classical solutions are proved. The model is described by a system of parabolic
differential equations, with one of them defined on a part of the boundary. In [11], the same problem
is solved numerically. A bimolecular surface reaction model, where the concentration of the reactant
on the surface is given and the product desorption is fast, is studied in [7] and [12] by using Monte
Carlo simulations.
In [4] we proved the existence and uniqueness theorem of the classical solution to the model
of bimolecular surface reactions between the carbon monoxide and nitrous oxide, CO + N2O =
= N2 + CO2, occurring on supported rhodium, Rh in the case of the rapid products desorption. In
the present paper we consider the same reaction but with a slow products desorption and prove the
existence and uniqueness theorem of the classical solution. This reaction proceeds via the following
elementary steps:
CO +K
\kappa 1
\rightleftarrows
\kappa 11
COK, N2O +K
\kappa 2
\rightleftarrows
\kappa 22
N2OK, N2OK
\kappa \ast
22\rightarrow \^N2 +OK,
COK +OK
\kappa 13\rightarrow \widehat CO2 + 2K, \^N2
\kappa 4\rightarrow N2, \widehat CO2
\kappa 5\rightarrow CO2,
(1)
where K is a free adsorption site of the catalyst surface S, CO and N2O are reactants, COK and
N2OK are adsorbates of CO and N2O, OK is the intermediate product, \^N2 and \widehat CO2 are reaction
products before the desorption, N2 and CO2 are reaction products after desorption from the catalyst
surface, \kappa i and \kappa ii, i = 1, 2, are the adsorption and desorption rate constants, \kappa 13 and \kappa \ast 22 are the
c\bigcirc A. AMBRAZEVIČIUS, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7 877
878 A. AMBRAZEVIČIUS
forward reaction rate constants, \kappa 4 and \kappa 5 are the products \^N2 and \widehat CO2 desorption rate constants.
Two first steps in this reaction are reversible, while the other four ones are irreversible.
The paper is organized as follows. In Section 2 we describe the model. In Section 3, we formulate
main results. A priori estimates are given in Section 4. Sections 5 and 6 are devoted to the uniqueness
and existence of the classical solution to problems (2) – (4).
2. The slow product desorption model. Set A1 = CO, A2 = N2O, B1 = N2, B2 = CO2,
\^B1 = \^N2, \^B2 = \widehat CO2, A1K = COK, A2K = N2OK, QK = OK. Then we have a mathematical
model of a bimolecular heterogeneous catalytic reaction of type A1+A2 = B1+B2, which proceeds
via scheme (1). In what follows we consider the case where the desorption of reaction products B1
and B2 is slow.
Suppose that the reactants A1, A2 and reactions products B1, B2 occupy a bounded domain
\Omega \subset \BbbR n, n \geq 3; a1 = a1(x, t), a2 = a2(x, t) and b1 = b1(x, t), b2 = b2(x, t) are their concentration
at point x \in \Omega at time t, respectively. Let S := \partial \Omega \subset C1+\alpha , \alpha \in (0, 1), be a surface of dimension
n - 1, and let S2 be not empty closed part of S of the same dimension, and S1 = S \setminus S2. We suppose
that \rho = \rho (\xi ) is the concentration of the adsorption sites of surface S at point \xi \in S, \rho \in C (S),
\rho (\xi ) \geq 0 for \xi \in S and \rho (\xi ) = 0 for \xi \in S1; \rho \theta i = \rho (\xi )\theta i(\xi , t) is the concentration of AiK, i = 1, 2,
at point \xi \in S2 at time t; \rho \theta 3 = \rho (\xi )\theta 3(\xi , t) is the concentration of the intermediate product QK at
point \xi \in S2 at time t; \rho \theta 4 = \rho (\xi )\theta 4(\xi , t) and \rho \theta 5 = \rho (\xi )\theta 5(\xi , t) are the concentrations of products
\^B1, \^B2 at point \xi \in S2 at time t before their desorption; \rho (1 - \theta ) is the concentration of the free
adsorption sites of S2; \theta =
\sum 5
i=1
\theta i.
Applying the mass action law and assumption that the desorption of reaction products \^B1 and
\^B2 is slow we get the Cauchy problem for \theta i = \theta i(\xi , t), i = 1, 2, . . . , 5, \xi \in S2,
\theta \prime 1 = \kappa 1a1(1 - \theta ) - \kappa 11\theta 1 - \kappa 13\rho \theta 1\theta 3, \theta 1| t=0 = \theta 10,
\theta \prime 2 = \kappa 2a2(1 - \theta ) - \kappa 22\theta 2 - \kappa \ast 22\theta 2, \theta 2| t=0 = \theta 20,
\theta \prime 3 = \kappa \ast 22\theta 2 - \kappa 13\rho \theta 1\theta 3, \theta 3| t=0 = \theta 30,
\theta \prime 4 = \kappa \ast 22\theta 2 - \kappa 4\theta 4, \theta 4| t=0 = \theta 40,
\theta \prime 5 = \kappa 13\rho \theta 1\theta 3 - \kappa 5\theta 5, \theta 5| t=0 = \theta 50.
(2)
System (2) involves the unknown values of a1 and a2 on the boundary S2. To close this system
we add equations for diffusion of reactants A1 and A2,
\partial ai
\partial t
- ki\Delta ai = 0 in \Omega \times (0, T ),
ki
\partial ai
\partial n
= 0 on S1 \times (0, T ),
ki
\partial ai
\partial n
+ \kappa i\rho ai(1 - \theta ) = \kappa ii\rho \theta i on S2 \times (0, T ),
ai| t=0 = ai0 in \Omega
(3)
for i = 1, 2.
The diffusion of products B1 and B2 can be described by the equations
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
EXISTENCE AND UNIQUENESS THEOREM TO A MODEL OF BIMOLECULAR SURFACE REACTIONS 879
\partial bi
\partial t
- \^ki\Delta bi = 0 in \Omega \times (0, T ),
\^ki
\partial bi
\partial n
= 0 on S1 \times (0, T ),
\^ki
\partial bi
\partial n
= \kappa 3+i\rho \theta 3+i on S2 \times (0, T ),
bi| t=0 = bi0 in \Omega
(4)
for i = 1, 2. Here \theta \prime i = \partial \theta i/\partial t, \theta i0 = \theta i0(\xi ), \xi \in S2, is the initial value of \theta i, i = 1, 2, . . . , 5; \Delta is
the n-dimensional Laplace operator; \partial /\partial \bfn is the outward normal derivative to S; ai0 = ai0(x) and
bi0 = bi0(x) are the initial concentrations of Ai and Bi at point x \in \Omega ; ki and \^ki are the diffusivities
of the reactant Ai and product Bi, i = 1, 2. All constants \kappa 1, \kappa 11, \kappa 13, \kappa 2, \kappa 22, \kappa \ast 22, \kappa 4, \kappa 5, ki, \^ki
are assumed to be positive.
Model (2) – (4) possesses the following three mass conservation laws:\int
\Omega
\bigl(
a1(x, s) + b2(x, s)
\bigr)
dx
\bigm| \bigm| \bigm| s=t
s=0
+
\int
S2
\rho (\xi )
\bigl(
\theta 1(\xi , s) + \theta 5(\xi , s)
\bigr)
dS\xi
\bigm| \bigm| \bigm| s=t
s=0
= 0,
\int
\Omega
\bigl(
a2(x, s) + b1(x, s)
\bigr)
dx
\bigm| \bigm| \bigm| s=t
s=0
+
\int
S2
\rho (\xi )
\bigl(
\theta 2(\xi , s) + \theta 4(\xi , s)
\bigr)
dS\xi
\bigm| \bigm| \bigm| s=t
s=0
= 0,
\int
\Omega
\bigl(
a1(x, s) + 2b2(x, s) + a2(x, s)
\bigr)
dx
\bigm| \bigm| \bigm| s=t
s=0
+
+
\int
S2
\rho (\xi )
\Bigl(
2\theta 5(\xi , s) + \theta 2(\xi , s) + \theta 3(\xi , s)
\Bigr)
dS\xi
\bigm| \bigm| \bigm| s=t
s=0
= 0.
To prove these laws, it is sufficient to integrate eqs. (3), (4) over the cylinder \Omega \times (0, t), apply the
formula of integration-by-parts, and use eqs. (2) with the boundary and initial conditions.
Thus, the bimolecular catalytic reactions can be described by system (2) – (4). Our aim is to
prove, for this system, the existence and uniqueness theorem. For every collection of continuous
functions \theta 4, \theta 5, problem (4) has a unique classic solution. Therefore, it is sufficient to prove the
solvability of problem (2), (3).
3. Main results.
Assumption 3.1. The initial functions \theta i0, i = 1, 2, . . . , 5, ai0, bi0, i = 1, 2, and given function
\rho satisfy the following conditions:
1. The functions \theta i0, i = 1, 2, . . . , 5, are continuous and nonnegative on S2, and \theta 0(\xi ) =
=
\sum 5
i=1
\theta i0(\xi ) < 1 for all \xi \in S2.
2. The functions ai0, bi0, i = 1, 2, are continuous and nonnegative in a closed domain \Omega .
3. \rho \in C (S), \rho (\xi ) \geq 0 for all \xi \in S, and \rho (\xi ) = 0 for all \xi \in S1.
Assumption 3.2. The functions ai0, bi0, i = 1, 2, are continuously differentiable on a neigh-
bourhood of the surface S.
Definition 3.1. Functions \theta i, i = 1, 2, . . . , 5, and ai, bi, i = 1, 2, form a classical solution to
problem (2) – (4) if \theta i \in C (S2 \times [0, T ]), \theta \prime i \in C (S2 \times (0, T )), the derivatives \partial ai/\partial \bfn and \partial bi/\partial \bfn
are continuous on S \times [0, T ], and ai, bi \in C2,1(\Omega \times (0, T ]) \cap C (\Omega \times [0, T ]), i = 1, 2, and they
satisfy system (2) – (4).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
880 A. AMBRAZEVIČIUS
The main result is the following theorem.
Theorem 3.1. Let \Omega be a bounded domain in \BbbR n and S = \partial \Omega be a surface of class C1+\alpha ,
\alpha \in (0, 1). Let the known functions \theta i0, ai0, bi0, and \rho satisfy Assumptions 3.1, 3.2, and \kappa 22 \geq \kappa \ast 22.
Then problems (2) – (4) has a unique classical solution.
The proof of this theorem is based on the a priori estimates formulated in the following proposi-
tions.
Lemma 3.1. Let ai, i = 1, 2, be a given continuous and nonnegative on S2 \times [0, T ] functions,
\theta i0, i = 1, . . . , 5, and \rho satisfy Assumption 3.1, and \kappa 22 \geq \kappa \ast 22. Let \theta i, i = 1, . . . , 5, be a solution1
of Cauchy problem (2). Then \theta i(\xi , t) \geq 0, i = 1, . . . , 5, and \theta (\xi , t) =
\sum 5
i=1
\theta i(\xi , t) < 1 for all
\xi \in S2, t \in [0, T ].
Lemma 3.2. Let \theta i = \theta i(\xi , t), i = 1, 2, . . . , 5, be given continuous and nonnegative on S2 \times
\times [0, T ] functions such that \theta (x, t) =
\sum 5
i=1
\theta i(\xi , t) \leq 1 for all \xi \in S2, t \in [0, T ]. Let ai0, i = 1, 2,
and \rho satisfy Assumption 3.1, and ai, i = 1, 2, be a classical solution to problem (3). Then for all
x \in \Omega , t \in [0, T ] we have the inequalities
0 \leq ai(x, t) \leq \beta i, i = 1, 2, (5)
where the constants \beta i do not depend on concrete functions \theta 1, . . . , \theta 5 satisfying the above conditions.
4. Proof of a priori estimates. Proof of Lemma 3.1. Let
\gamma =
\Bigl\{
(\theta 1, \theta 2, \theta 3, \theta 4, \theta 5) \in \BbbR 5 : \theta i = \theta i(\xi , t), i = 1, 2, . . . , 5, t \in [0, T ]
\Bigr\}
be a trajectory of system (2), which begins at the point (\theta 10(\xi ), \theta 20(\xi ), \theta 30(\xi ), \theta 40(\xi ), \theta 50(\xi )), \xi \in
\in S2. We prove that \gamma does not leave domain D, which is bounded by planes
\sum 5
j=1
\theta j = 1 and
\theta i = 0, i = 1, 2, . . . , 5.
Integrating equations (2) with respect to variable t, we get the integral equations
\theta 1e
\int t
0 (\kappa 11+\kappa 13\rho \theta 3) d\tau = \theta 10 +
t\int
0
\kappa 1a1(1 - \theta )e
\int \tau
0 (\kappa 11+\kappa 13\rho \theta 3) ds d\tau ,
\theta 2e
(\kappa 22+\kappa \ast
22)t = \theta 20 +
t\int
0
\kappa 2a2(1 - \theta )e(\kappa 22+\kappa \ast
22)\tau d\tau ,
\theta 3e
\int t
0 \kappa 13\rho \theta 1d\tau = \theta 30 +
t\int
0
\kappa \ast 22\theta 2e
\int \tau
0 \kappa 13\rho \theta 1dsd\tau ,
\theta 4e
\kappa 4t = \theta 40 +
t\int
0
\kappa \ast 22\theta 2e
\kappa 4\tau d\tau ,
\theta 5e
\kappa 5t = \theta 50 +
t\int
0
\kappa 13\rho \theta 1\theta 3e
\kappa 5\tau d\tau .
(2\ast )
1For given continuous on S2 \times [0, T ] functions ai, i = 1, 2, we say that functions \theta i, i = 1, 2, . . . , 5, form a classical
solution to Cauchy problem (2), if \theta i \in C1(S2 \times (0, T ))\cap C (S2 \times [0, T ]) and they satisfy system (2).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
EXISTENCE AND UNIQUENESS THEOREM TO A MODEL OF BIMOLECULAR SURFACE REACTIONS 881
Suppose that the trajectory \gamma leaves the domain D by crossing (or by touching) the plane \theta i = 0
for any i = 1, 2, . . . , 5 at the moment t\ast , and does not cross the other planes, that is \theta i(\xi , t\ast ) = 0,
\theta j(\xi , t
\ast ) > 0 for j \not = i, and \theta (\xi , t\ast ) < 1. Then there exists \varepsilon > 0 such that \theta i(\xi , t) < 0 for
t \in (t\ast , t\ast + \varepsilon ] and \theta j(\xi , t
\ast ) \geq 0, j \not = i, \theta (\xi , t) \leq 1 for t \in [t\ast , t\ast + \varepsilon ]. But for these t from the
ith equation of system (2\ast ) we get \theta i(\xi , t) \geq 0. The contradiction shows that \gamma does not leaves this
domain D through the plane \theta i = 0. Similarly it can be shown that the trajectory \gamma does not leave the
domain D through the intersection of several planes \theta i = 0. For example, if the trajectory \gamma leaves
the domain D through the intersection of planes \theta 1 = 0 and \theta 2 = 0 at the moment t\ast , then there
exists \varepsilon > 0 such that \theta 1(\xi , t) < 0 or \theta 2(\xi , t) < 0 and \theta j(\xi , t) > 0, j \not = 1, 2, for t \in (t\ast , t\ast + \varepsilon ],
\theta (\xi , t) \leq 1 for t \in [t\ast , t\ast + \varepsilon ]. But from the first two equations of the system (2\ast ) for these t we
get \theta 1(\xi , t) \geq 0 and \theta 2(\xi , t) \geq 0. The contradiction shows that \gamma does not leaves this domain D
through the intersection of planes \theta 1 = 0 and \theta 2 = 0. Thus \theta i(\xi , t) \geq 0, i = 1, 2, . . . , 5, for \xi \in S2,
t \in [0, T ].
Suppose that \gamma leaves the domain D by crosses or touching of the plane \theta = 1. By assumption,
\theta 0(\xi ) < 1. Then there exists the moment t\ast > 0 such that \theta i(\xi , t) \geq 0, i = 1, 2, . . . , 5, \theta (\xi , t) \leq 1
for t \in [0, t\ast ]. Summing all equations of system (2) we get
- (1 - \theta )\prime = (\kappa 1a1 + \kappa 2a2)(1 - \theta ) - \kappa 11\theta 1 - \kappa 22\theta 2 + \kappa \ast 22\theta 2 - \kappa 4\theta 4 - \kappa 5\theta 5 - \kappa 13\rho \theta 1\theta 3. (6)
Multiplying both sides of this equation by e
\int t
0 (\kappa 1a1+\kappa 2a2) ds and integrating with respect to t we get
the equation
(1 - \theta )e
\int t
0 (\kappa 1a1+\kappa 2a2)ds = 1 - \theta 0 +
t\int
0
(\kappa 11\theta 1 + (\kappa 22 - \kappa \ast 22)\theta 2 +
+ \kappa 4\theta 4 + \kappa 5\theta 5 + \kappa 13\rho \theta 1\theta 3)e
\int \tau
0 (\kappa 1a1+\kappa 2a2)ds d\tau .
By assumption of Lemma 3.1, \theta 0(\xi ) < 1 and \kappa 22 \geq \kappa \ast 22. Then for t = t\ast the right-hand side of this
equation is positive while the left one is equal to zero. The contradiction shows that \gamma does not leave
domain D through the plane \theta = 1. Hence, \theta (\xi , t) < 1 for \xi \in S2, t \in [0, T ].
Lemma 3.1 is proved.
Proof of Lemma 3.2. According to the positivity lemma (see [8, p. 19], Chapter 1, Lemma 4.1),
the functions ai, i = 1, 2, in \Omega \times [0, T ] cannot have a negative minimum. Therefore ai(x, t) \geq 0 for
all x \in \Omega and t \in [0, T ].
Let \^ai, i = 1, 2, be the solution to the problem
\partial \^ai
\partial t
- ki\Delta \^ai = 0 in \Omega \times (0, T ),
ki
\partial \^ai
\partial n
= 0 on S1 \times (0, T ),
ki
\partial \^ai
\partial n
= \kappa ii\rho on S2 \times (0, T ),
\^ai| t=0 = ai0, i = 1, 2, in \Omega .
According to the positivity lemma function \^ai - ai, i = 1, 2, in \Omega \times [0, T ] cannot have a negative
minimum. Therefore ai(x, t) \leq \^ai(x, t) for all x \in \Omega and t \in [0, T ] and ai(x, t) \leq \beta i for all x \in \Omega
and t \in [0, T ], where \beta i = \mathrm{m}\mathrm{a}\mathrm{x}x\in \Omega , t\in [0,T ] \^ai(x, t).
Lemma 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
882 A. AMBRAZEVIČIUS
5. Uniqueness of classical solution.
Theorem 5.1. Problem (2), (3) cannot have two different classical solutions.
Proof. Let \^\theta i, i = 1, . . . , 5, \^ai, i = 1, 2, and \~\theta i, i = 1, . . . , 5, \~ai, i = 1, 2, form two classical
solutions to problem (2), (3). Set \theta i = \^\theta i - \~\theta i, i = 1, . . . , 5, ai = \^ai - \~ai, i = 1, 2, and \theta =
\sum 5
i=1
\theta i.
Then for \theta i, i = 1, . . . , 5, we get Cauchy problem
\theta \prime 1 = \kappa 1a1(1 - \^\theta ) - \kappa 1\~a1\theta - \kappa 11\theta 1 - \kappa 13\rho (\theta 1\^\theta 3 + \~\theta 1\theta 3), t \in (0, T ),
\theta \prime 2 = \kappa 2a2(1 - \^\theta ) - \kappa 2\~a2\theta - (\kappa 22 + \kappa \ast 22)\theta 2, t \in (0, T ),
\theta \prime 3 = \kappa \ast 22\theta 2 - \kappa 13\rho (\theta 1\^\theta 3 + \~\theta 1\theta 3), t \in (0, T ),
\theta \prime 4 = \kappa \ast 22\theta 2 - \kappa 4\theta 4, t \in (0, T ),
\theta \prime 5 = \kappa 13\rho (\theta 1\^\theta 3 + \~\theta 1\theta 3) - \kappa 5\theta 5, t \in (0, T ),
\theta i| t=0 = 0, i = 1, 2, . . . , 5, for \xi \in S2. Integrating these equations with respect to variable t, we get
the integral equations
\theta 1 =
t\int
0
\kappa 1a1(1 - \^\theta ) - \kappa 1\~a1\theta - \kappa 11\theta 1 - \kappa 13\rho (\theta 1\^\theta 3 + \~\theta 1\theta 3) ds, t \in (0, T ),
\theta 2 =
t\int
0
\kappa 2a2(1 - \^\theta ) - \kappa 2\~a2\theta - (\kappa 22 + \kappa \ast 22)\theta 2 ds, t \in (0, T ),
\theta 3 =
t\int
0
\kappa \ast 22\theta 2 - \kappa 13\rho (\theta 1\^\theta 3 + \~\theta 1\theta 3) ds, t \in (0, T ),
\theta 4 =
t\int
0
\kappa \ast 22\theta 2 - \kappa 4\theta 4 ds, t \in (0, T ),
\theta 5 =
t\int
0
\kappa 13\rho (\theta 1\^\theta 3 + \~\theta 1\theta 3) - \kappa 5\theta 5 ds, t \in (0, T ).
Let | | | \theta | | | =
\sum 5
i=1
| \theta i| . Then
| | | \theta | | | \leq
t\int
0
(\kappa 1| a1| + \kappa 2| a2| ) ds+ C
t\int
0
| | | \theta | | | ds,
where
C =
2\sum
i=1
\kappa i \~mi +\mathrm{m}\mathrm{a}\mathrm{x}\{ \kappa 11 + 3\kappa 13\rho , \kappa 22 + 3\kappa \ast 22, \kappa 4, \kappa 5\} ,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
EXISTENCE AND UNIQUENESS THEOREM TO A MODEL OF BIMOLECULAR SURFACE REACTIONS 883
\rho = \mathrm{m}\mathrm{a}\mathrm{x}
\xi \in S2
\rho (\xi ), \~mi = \mathrm{m}\mathrm{a}\mathrm{x}
\xi \in S2, t\in [0,T ]
| \~ai(\xi , t)| , i = 1, 2.
Using the Gronwall lemma we get
| | | \theta (\xi , t) | | | \leq eCt
t\int
0
\bigl(
\kappa 1| a1(\xi , s)| + \kappa 2| a2(\xi , s)|
\bigr)
ds. (7)
Using that \^\theta \leq 1 from (3) for each i = 1, 2, we have
1
2
\int
\Omega
a2i dx+
\tau \int
0
\int
\Omega
ki| \nabla ai| 2 dx dt =
\tau \int
0
\int
S2
\bigl(
\kappa i\rho (\~ai\theta + (\^\theta - 1)ai) + \kappa ii\rho \theta i
\bigr)
ai dS dt \leq
\leq C1
\tau \int
0
\int
S2
\bigl(
| a1(\xi , t)| + | a2(\xi , t)|
\bigr)
| | | \theta (\xi , t) | | | dS dt,
C1 = \rho \mathrm{m}\mathrm{a}\mathrm{x}
i=1,2
\bigl\{
\kappa i \~mi + \kappa ii
\bigr\}
.
Adding these equalities and using the inequalities
\tau \int
0
\int
S2
| a1(\xi , t)| | | | \theta (\xi , t) | | | dSdt \leq eC\tau
\int
S2
\tau \int
0
| a1(\xi , t)| dt
\tau \int
0
(\kappa 1| a1(\xi , t)| + \kappa 2| a2(\xi , t)| )dtdS \leq
\leq \tau eC\tau
\left( \kappa 1 \tau \int
0
\int
S2
a21(\xi , t) dS dt+
\kappa 2
2
\tau \int
0
\int
S2
\bigl(
a21(\xi , t) + a22(\xi , t)
\bigr)
dS dt
\right)
and
\tau \int
0
\int
S2
| a2(\xi , t)| | | | \theta (\xi , t) | | | dSdt \leq eC\tau
\int
S2
\tau \int
0
| a2(\xi , t)| dt
\tau \int
0
(\kappa 1| a1(\xi , t)| + \kappa 2| a2(\xi , t)| )dt dS \leq
\leq \tau eC\tau
\left( \kappa 2 \tau \int
0
\int
S2
a22(\xi , t) dS dt+
\kappa 1
2
\tau \int
0
\int
S2
\bigl(
a21(\xi , t) + a22(\xi , t)
\bigr)
dS dt
\right)
we obtain
1
2
\int
\Omega
2\sum
i=1
a2i dx+
\tau \int
0
\int
\Omega
2\sum
i=1
ki| \nabla ai| 2 dx dt \leq C2
\tau \int
0
\int
S2
2\sum
i=1
a2i dS dt.
For every \varepsilon > 0, we have the estimate\int
S
a2 dx \leq \varepsilon
\int
\Omega
| \nabla a| 2 dx+ C\varepsilon
\int
\Omega
a2 dx,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
884 A. AMBRAZEVIČIUS
where the constant C\varepsilon is independent of the function a, and C\varepsilon \rightarrow \infty as \varepsilon \rightarrow 0. Therefore,
\int
\Omega
2\sum
i=1
a2i dx+
\tau \int
0
\int
\Omega
2\sum
i=1
ki| \nabla ai| 2 dx dt \leq C2
\tau \int
0
\int
\Omega
2\sum
i=1
a2i dx dt.
From here by the Gronwall lemma we get
\tau \int
0
\int
\Omega
2\sum
i=1
a2i dx dt \leq 0.
Hence, ai = 0 for i = 1, 2. Now estimate (7) shows that \theta i = 0 for i = 1, 2, 3, 4, 5.
Theorem 5.1 is proved.
6. Existence of classical solution. In this section, we prove that problem (2), (3) has a classical
solution. Let \Omega 0 = \Omega if a10 = 0 and a20 = 0 in some neighborhood of the surface S, and \Omega 0 \supset \Omega
if a10 or a20 is continuously differentiable on some neighbourhood of the surface S. In the last case,
we extend the functions a10 and a20 to \Omega 0 \setminus \Omega preserving the same smoothness. Let
\Gamma k(x, t) =
1\bigl(
4\pi kt
\bigr) n/2 e - | x| 2
4kt , x \in \BbbR n, t > 0,
be the fundamental solution to the equation at - k\Delta a = 0, k > 0. Then, for any continuous on
S2\times [0, T ] functions \theta 1, \theta 2, . . . , \theta 5 and continuous on S function \rho , problem (3) has a unique solution
ai \in C2,1(\Omega \times (0, T ])\cap C (\Omega \times [0, T ]), i = 1, 2, which can be presented by the formula (see [5])
ai(x, t) =
t\int
0
\int
S
\Gamma ki(x - \xi , t - \tau )\varphi i(\xi , \tau ) dS\xi d\tau +
\int
\Omega 0
\Gamma ki(x - y, t)ai0(y) dy, (8)
where \varphi i, i = 1, 2, is a continuous and bounded solution on S \times [0, T ] to the Volterra integral
equation
1
2
\varphi i(\eta , t) +
t\int
0
\int
S
\biggl(
\partial \Gamma ki(\eta - \xi , t - \tau )
\partial \bfn \eta
+
1
ki
\sigma i(\eta , t, \theta )\Gamma ki(\eta - \xi , t - \tau )
\biggr)
\varphi i(\xi , \tau ) dS\xi d\tau =
=
1
ki
\psi i(\eta , t, \theta ) -
\int
\Omega 0
\biggl(
\partial \Gamma ki(\eta - x, t)
\partial \bfn \eta
+
1
ki
\sigma i(\eta , t, \theta )\Gamma ki(\eta - x, t)
\biggr)
ai0(x) dx, (9)
\sigma i(\xi , t, \theta ) =
\left\{ 0 if (\xi , t) \in S1 \times [0, T ],
\kappa i\rho (\xi )
\bigl(
1 - \theta (\xi , t)
\bigr)
if (\xi , t) \in S2 \times [0, T ],
\psi i(\xi , t, \theta ) =
\left\{ 0 if (\xi , t) \in S1 \times [0, T ],
\kappa ii\rho (\xi )\theta i(\xi , t) if (\xi , t) \in S2 \times [0, T ],\bigm| \bigm| \varphi i(\xi , t)
\bigm| \bigm| \leq Mi, \xi \in S, t \in [0, T ].
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
EXISTENCE AND UNIQUENESS THEOREM TO A MODEL OF BIMOLECULAR SURFACE REACTIONS 885
Here constant Mi is independent of functions \theta 1, \theta 2, . . . , \theta 5 such that \theta i(\xi , t) \geq 0, i = 1, 2, . . . , 5,
and \theta (\xi , t) :=
\sum 5
i=1
\theta i(\xi , t) \leq 1 for all (\xi , t) \in S2 \times [0, T ].
Let ai1, i = 1, 2, defined by formulas (8), and \varphi i1, i = 1, 2, be solutions to problem (3) and
the integral equation (9) with functions \theta i = \theta i0, i = 1, 2, . . . , 5. Then by Lemma 3.2 functions ai1,
i = 1, 2, are nonnegative, ai1(x, t) \leq \beta i for x \in \Omega and t \in [0, T ], and\bigm| \bigm| \varphi i1(\xi , t)
\bigm| \bigm| \leq Mi for \xi \in S, t \in [0, T ].
Assume that \theta i1, i = 1, 2, . . . , 5, form a solution to Cauchy problem (2) with ai = ai1, i = 1, 2.
Then by Lemma 3.1 functions \theta i1, i = 1, 2, . . . , 5, are nonnegative and
\sum 5
i=1
\theta i1(\xi , t) < 1, for all
\xi \in S2, t \in [0, T ].
Let ai2, i = 1, 2, defined by formulas (8), and \varphi i2, i = 1, 2, be solutions to problem (3) and
the integral equation (9) with functions \theta i = \theta i1, i = 1, 2, . . . , 5. Then by Lemma 3.2 functions ai2,
i = 1, 2, are nonnegative, ai2(x, t) \leq \beta i for x \in \Omega and t \in [0, T ], and\bigm| \bigm| \varphi i2(\xi , t)
\bigm| \bigm| \leq Mi for \xi \in S, t \in [0, T ], i = 1, 2.
Assume that \theta i2, i = 1, 2, . . . , 5, form a solution to Cauchy problem (2) with ai = ai2, i = 1, 2.
Then by Lemma 3.1 functions \theta i2, i = 1, 2, . . . , 5, are nonnegative and
\sum 5
i=1
\theta i2(\xi , t) < 1 for all
\xi \in S, t \in [0, T ].
Proceeding with this argument, we get the sequences
\{ aij\} \infty j=1, i = 1, 2, \{ \varphi ij\} \infty j=1, i = 1, 2, \{ \theta ij\} \infty j=1, i = 1, 2, . . . , 5,
which are uniformly bounded:
aij(x, t) \geq 0 for x \in \Omega , t \in [0, T ], i = 1, 2, j = 1, 2, . . . ,
aij(x, t) \leq \beta i for x \in \Omega , t \in [0, T ], i = 1, 2, j = 1, 2, . . . ,
| \varphi ij(\xi , t)| \leq Mi for \xi \in S, t \in [0, T ], i = 1, 2, j = 1, 2, . . . ,
\theta ij(\xi , t) \geq 0 for \xi \in S2, t \in [0, T ], i = 1, 2, . . . , 5, j = 1, 2, . . . ,
5\sum
i=1
\theta ij(\xi , t) < 1 for \xi \in S2, t \in [0, T ], j = 1, 2, . . . .
Now we prove that they are equicontinuous. Functions aij are defined by the formula
aij(x, t) =
t\int
0
\int
S
\Gamma ki(x - \xi , t - \tau )\varphi ij(\xi , \tau ) dS\xi d\tau +
\int
\Omega 0
\Gamma ki(x - y, t)ai0(y) dy.
The potential of a simple layer (see [5] or [6])
t\int
0
\int
S
\Gamma ki(x - \xi , t - \tau )\varphi ij(\xi , \tau ) dS\xi d\tau
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
886 A. AMBRAZEVIČIUS
belongs to the Hölder space C\lambda
\bigl(
\Omega \times [0, T ]
\bigr)
with \lambda \in (0, 1). Hence, the sequences \{ aij\} \infty j=1,
i = 1, 2, are equicontinuous.
Functions \theta ij , i = 1, 2, . . . , 5, are solutions to the system of integral equations
\theta 1j(\xi , t) = \theta 10(\xi ) +
t\int
0
\Biggl[
\kappa 1a1j(\xi , s)
\Biggl(
1 -
5\sum
i=1
\theta ij(\xi , s)
\Biggr)
- \kappa 11\theta 1j(\xi , s) -
- \kappa 13\rho (\xi )\theta 1j(\xi , s)\theta 3j(\xi , s)
\Biggr]
ds,
\theta 2j(\xi , t) = \theta 20(\xi ) +
t\int
0
\Biggl[
\kappa 2a2j(\xi , s)
\Biggl(
1 -
5\sum
i=1
\theta ij(\xi , s)
\Biggr)
- (\kappa 22 + \kappa \ast 22)\theta 2j(\xi , s)
\Biggr]
ds,
\theta 3j(\xi , t) = \theta 30(\xi ) +
t\int
0
\Bigl[
\kappa \ast 22\theta 2j(\xi , s) - \kappa 13\rho (\xi )\theta 1j(\xi , s)\theta 3j(\xi , s)
\Bigr]
ds,
\theta 4j(\xi , t) = \theta 40(\xi ) +
t\int
0
\Bigl[
\kappa \ast 22\theta 2j(\xi , s) - \kappa 4\theta 4j(\xi , s)
\Bigr]
ds,
\theta 5j(\xi , t) = \theta 50(\xi ) +
t\int
0
\Bigl[
\kappa 13\rho (\xi )\theta 1j(\xi , s)\theta 3j(\xi , s) - \kappa 5\theta 5j(\xi , s)
\Bigr]
ds.
Therefore,
| \theta 1j(\xi , t) - \theta 1j(\xi , \tau )| \leq | t - \tau |
\bigl(
\kappa 1\beta 1 + \kappa 11 + \kappa 13\rho
\bigr)
, \rho = \mathrm{m}\mathrm{a}\mathrm{x}
\xi \in S2
\rho (\xi ),
| \theta 2j(\xi , t) - \theta 2j(\xi , \tau )| \leq | t - \tau | (\kappa 2\beta 2 + \kappa 22 + \kappa \ast 22),
| \theta 3j(\xi , t) - \theta 3j(\xi , \tau )| \leq | t - \tau | (\kappa \ast 22 + \kappa 13\rho ),
| \theta 4j(\xi , t) - \theta 4j(\xi , \tau )| \leq | t - \tau | (\kappa \ast 22 + \kappa 4),
| \theta 5j(\xi , t) - \theta 5j(\xi , \tau )| \leq | t - \tau | (\kappa 13\rho + \kappa 5).
Moreover,
5\sum
i=1
| \theta ij(\xi , t) - \theta ij(\eta , t)| \leq
5\sum
i=1
| \theta i0(\xi ) - \theta i0(\eta )| + 3\kappa 13T | \rho (\xi ) - \rho (\eta )| +
+ C
t\int
0
5\sum
i=1
| \theta ij(\xi , s) - \theta ij(\eta , s)| ds+
t\int
0
2\sum
i=1
\kappa i| aij(\xi , s) - aij(\eta , s)| ds
and
5\sum
i=1
| \theta ij(\xi , t) - \theta ij(\eta , t)| \leq eCT
\Biggl(
5\sum
i=1
| \theta i0(\xi ) - \theta i0(\eta )| + 3\kappa 13T | \rho (\xi ) - \rho (\eta )|
\Biggr)
+
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
EXISTENCE AND UNIQUENESS THEOREM TO A MODEL OF BIMOLECULAR SURFACE REACTIONS 887
+
eCT - 1
C
\mathrm{m}\mathrm{a}\mathrm{x}
s\in [0,T ]
2\sum
i=1
\kappa i
\bigm| \bigm| aij(\xi , s) - aij(\eta , s)
\bigm| \bigm|
for all \xi , \eta \in S2, t, \tau \in [0, T ]. Here C = m + \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\kappa 11 + 3\rho \kappa 13, \kappa 22 + 3\kappa \ast 22, \kappa 4, \kappa 5
\bigr\}
. These
estimates show that the sequence \{ \theta ij\} \infty j=1, i = 1, 2, . . . , 5, is equicontinuous.
Functions \varphi ij , i = 1, 2, are solutions to integral equation (9) with \theta i = \theta ij - 1. The potential of a
double-layer (see [5] or [6]),
t\int
0
\int
S
\partial \Gamma ki(\eta - \xi , t - \tau )
\partial \bfn \eta
\varphi ij(\xi , \tau ) dS\xi d\tau
belongs to the Hölder space C\lambda (S \times [0, T ]) with \lambda < 2\alpha /3. Therefore, the sequences \{ \varphi ij\} \infty j=1,
i = 1, 2, are equicontinuous. According to the Arzelà – Ascoli theorem we can select uniformly
converging subsequences from sequences \{ aij\} \infty j=1, i = 1, 2, \{ \varphi ij\} \infty j=1, i = 1, 2, and \{ \theta ij\} \infty j=1,
i = 1, 2, . . . , 5. Since problem (2), (3) cannot possess two classical solutions, we claim that the
sequences \{ aij\} \infty j=1, i = 1, 2, \{ \varphi ij\} \infty j=1, i = 1, 2, and \{ \theta ij\} \infty j=1, i = 1, 2, . . . , 5, converge uniformly.
Set
ai(x, t) = \mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
aij(x, t), x \in \Omega , t \in [0, T ], i = 1, 2,
\varphi i(\xi , t) = \mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\varphi ij(\xi , t), \xi \in S, t \in [0, T ], i = 1, 2,
\theta i(\xi , t) = \mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\theta ij(\xi , t), \xi \in S2, t \in [0, T ], i = 1, 2, . . . , 5.
Formula (8) holds for the limit functions ai, i = 1, 2. Therefore, the limit functions ai \in C2,1( \Omega \times
\times (0, T ]) \cap C (\Omega \times [0, T ]) are solutions to problem (3). The limit functions \theta i, i = 1, 2, . . . , 5, are
solutions to the system of integral equations
\theta 1(\xi , t) = \theta 10(\xi ) +
t\int
0
\Biggl[
\kappa 1a1(\xi , s)
\Biggl(
1 -
5\sum
i=1
\theta i(\xi , s)
\Biggr)
- \kappa 11\theta 1(\xi , s) - \kappa 13\rho (\xi )\theta 1(\xi , s)\theta 3(\xi , s)
\Biggr]
ds,
\theta 2(\xi , t) = \theta 20(\xi ) +
t\int
0
\Biggl[
\kappa 2a2(\xi , s)
\Biggl(
1 -
5\sum
i=1
\theta i(\xi , s)
\Biggr)
- (\kappa 22 + \kappa \ast 22)\theta 2j(\xi , s)
\Biggr]
ds,
\theta 3(\xi , t) = \theta 30(\xi ) +
t\int
0
\Bigl[
\kappa \ast 22\theta 2(\xi , s) - \kappa 13\rho (\xi )\theta 1(\xi , s)\theta 3(\xi , s)
\Bigr]
ds,
\theta 4(\xi , t) = \theta 40(\xi ) +
t\int
0
\Bigl[
\kappa \ast 22\theta 2(\xi , s) - \kappa 4\theta 4(\xi , s)
\Bigr]
ds,
\theta 5(\xi , t) = \theta 50(\xi ) +
t\int
0
\Bigl[
\kappa 13\rho (\xi )\theta 1(\xi , s)\theta 3(\xi , s) - \kappa 5\theta 5(\xi , s)
\Bigr]
ds.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
888 A. AMBRAZEVIČIUS
Therefore, \theta i, i = 1, 2, . . . , 5, are uniformly differentiable with respect to variable t and form a
solution to Cauchy problem (2). Hence, problem (2), (3) has a classical solution. According to
Theorem 5.1, this solution is unique.
Acknowledgment. The author thanks Prof. V. Skakauskas for the formulation of the problem
(system (2) – (4)) and fruitful discussions.
References
1. Ambrazevičius A. Solvability of a coupled system of parabolic and ordinary differential equations // Centr. Eur. J.
Math. – 2010. – 8, № 3. – P. 537 – 547.
2. Ambrazevičius A. Existence and uniqueness theorem to a unimolecular heterogeneous catalytic reaction model //
Nonlinear Anal. Model. Control. – 2010. – 15, № 4. – P. 405 – 421.
3. Ambrazevičius A. Solvability theorem for a model of a unimolecular heterogeheous reaction with adsorbate diffusion //
J. Math. Sci. – 2012. – 184, № 4. – P. 383 – 398 (transl. from Probl. Math. Anal. – 2012. – 65. – P. 13 – 26).
4. Ambrazevičius A. Solvability theorem for a mathematical bimolecular reaction model // Acta Appl. Math. – 2015. –
140. – P. 95 – 109.
5. Friedman A. Partial differential equations of parabolic type. – Englewood Clifs, NJ: Prentice Hall, 1964.
6. Ladyzhenskaya O. A., Solonnikov V. A., Uralceva N. N. Linear and quasilinear equation of parabolic type // Amer.
Math. Soc. Transl. – 1968 (English transl.).
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1999. – 83, № 18. – P. 3673 – 3676.
8. Pao C. V. Nonlinear parabolic and elliptic equations. – New York: Plenum Press, 1992.
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Nonlinear Anal. Model. Control. – 2010. – 15, № 3. – P. 351 – 360.
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Received 20.04.15,
after revision — 06.06.17
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 7
|
| id | umjimathkievua-article-1743 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:48Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/13/189a1c00e5075f85900ba2b845037013.pdf |
| spelling | umjimathkievua-article-17432019-12-05T09:25:34Z Existence and uniqueness theorem to a model of bimolecular surface reactions Теорема iснування та єдиностi для моделi бiмолекулярних поверхневих реакцiй Ambrazevicius, A. Амбразевичус, А. We prove the existence and uniqueness of classical solutions to a coupled system of parabolic and ordinary differential equations in which the latter are determined on the boundary. This system describes the model of bimolecular surface reaction between carbon monoxide and nitrous oxide occurring on supported rhodium in the case of slow desorption of the products. Доведено iснування та єдинiсть класичних розв’язкiв зв’язаних систем параболiчних та звичайних диференцiальних рiвнянь, останнi з яких визначенi на межi. Система описує модель бiмолекулярної поверхневої реакцiї мiж монооксидом вуглецю та закисом азоту, що вiдбувається на нанесеному родiї у випадку повiльної десорбцiї продуктiв. Institute of Mathematics, NAS of Ukraine 2017-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1743 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 7 (2017); 877-888 Український математичний журнал; Том 69 № 7 (2017); 877-888 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1743/725 Copyright (c) 2017 Ambrazevicius A. |
| spellingShingle | Ambrazevicius, A. Амбразевичус, А. Existence and uniqueness theorem to a model of bimolecular surface reactions |
| title | Existence and uniqueness theorem to a model of bimolecular surface reactions |
| title_alt | Теорема iснування та єдиностi для моделi бiмолекулярних поверхневих реакцiй |
| title_full | Existence and uniqueness theorem to a model of bimolecular surface reactions |
| title_fullStr | Existence and uniqueness theorem to a model of bimolecular surface reactions |
| title_full_unstemmed | Existence and uniqueness theorem to a model of bimolecular surface reactions |
| title_short | Existence and uniqueness theorem to a model of bimolecular surface reactions |
| title_sort | existence and uniqueness theorem to a model of bimolecular surface reactions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1743 |
| work_keys_str_mv | AT ambrazeviciusa existenceanduniquenesstheoremtoamodelofbimolecularsurfacereactions AT ambrazevičusa existenceanduniquenesstheoremtoamodelofbimolecularsurfacereactions AT ambrazeviciusa teoremaisnuvannâtaêdinostidlâmodelibimolekulârnihpoverhnevihreakcij AT ambrazevičusa teoremaisnuvannâtaêdinostidlâmodelibimolekulârnihpoverhnevihreakcij |