Nonlinear Stability Analysis of Stationary Solutions for a Special Class
of Reaction-Diffusion Systems with Respect to Small Perturbations
QINGXIA LI1, XINYAO YANG2, ZIYAN ZHANG1
1Department of Mathematics and Computer Science, Fisk University
Nashville, Tennessee 37208,
USA
2Department of Applied Mathematics, Xi’an Jiaotong Liverpool University
Suzhou, Jiangsu 215123,
CHINA
Abstract: We prove that the stationary solution of a class of reaction-diffusion systems is stable in the
intersection of the Sobolev space and an exponentially weighted space
. Particular attention is
given to a special case, the combustion model. The stationary solution considered here is the end state of the
traveling front associated with the system, and thus the present result complements recent work by A.
Ghazaryan, Y. Latushkin and S. Schecter, where the stability of the traveling fronts was investigated.
Key-Words: Dynamical Systems, Systems Theory, Reaction-Diffusion systems; Traveling waves;
Stationary solutions; Essential spectrum; Exponential weight; Nonlinear stability.
Received: April 14, 2022. Revised: October 29, 2022. Accepted: November 21, 2022. Published: December 31, 2022.
1 Introduction
The theory of reaction-diffusion equations emerged
in the first half of the last century and has been
influenced by various applications such as thermal
explosions and the propagation of chemical waves.
It brings together the theory of heat conduction and
mass diffusion on the one hand and has a wide range
of applications to chemical and biological kinetics
on the other. Specifically, in the late 1930s,
Kolmogorov-Petrovskii-Piskunov [6] and Fischer
[3] proposed reaction-diffusion waves;
subsequently, Zeldowitsch and Frank-Kamenetzki
[10] studied them in conjunction with combustion
theory. In this type of study, the reaction-diffusion
equation
(1)
is considered over the infinite domain ,
and can be either the temperature or the
concentration of the reactant. A traveling wave is a
solution of the system (1) of the form
, where is a constant, i.e., the speed of
the wave. This type of solution propagates with a
constant speed and a certain shape and describe the
asymptotic behavior of the system. The existence,
stability and bifurcation problems of traveling
waves are associated with many applications and
mathematical models and have therefore been
studied intensively in recent decades. In general,
when analyzing the reaction-diffusion equation in
unbounded domains, the invertibility of the limiting
operator is required. This condition implies that the
essential spectrum does not contain the origin, so we
often need to study the essential spectrum of the
linear operators of the system. In the practical
application of this technique, especially in the study
of some models in combustion and chemical
kinetics, one often finds that the essential spectrum
may contain the origin. In such cases, many
conventional methods and theories are no longer
applicable.
In this paper we will illustrate how to deal with
this case in terms of the stability of the stationary
solution of reaction-diffusion waves in one-
dimensional space of a special class, and lead to
some questions worth investigating. We will
introduce the settings and some definitions in
Section 2 and study the spectrum of the operator
obtained by linearizing the equation with respect to
the stationary solution in Section 3.1. Section 3.2
focuses on the nonlinear terms in the system and
some nonlinear estimates needed to prove the main
theorem. The proof of the stability of the stationary
solution is given in Section 3.3, see Theorem 3.16.
Finally, in Section 4 we give a generalization of the
type of reaction-diffusion systems considered in
[4,5].
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2 Problem Formulation
We first briefly introduce a combustion model in
that includes two equations:
(2)
where the parameters and satisfy and
, and in the nonlinear terms is taken in
the form of Arrihenius exponential:
（3）
Let
and
, then the system (2) can be rewritten
in the vector form as
(4)
Given a fixed vector , the corresponding
system written in the moving coordinate frame
can be reduced to the equation
.
(5)
We are concerned with the traveling wave
solution of (5). In general, considering the
practical model, such reaction-diffusion waves
will approach the stationary states and
exponentially as tends to infinity, that is,
there exist constants and
such that for and
for . The stability of
the stationary solution with respect to small
perturbations is usually determined by the
spectrum of the linear operator. The situation is
more complex in the case of traveling waves,
because these are families of solutions, and the
corresponding linear operator will have a zero
eigenvalue. In this case, the discussion may
involve a transfer of stability, i.e., the
convergence of the solution of the non-stationary
problem to a stationary solution in the family of
solutions.
It is obvious that the system (5) has two types
of stationary solutions: one when is equal
to a real constant and , and the other
when and is equal to a real
constant. In particular, we can choose ,
, which is the state corresponding to
completely burned reactants, and , ,
which corresponds to unburned substances. In
other words, we choose and
, see [4] to see why and are chosen
this way. The one dimensional gasless
combustion model of a solid fuel described by the
system
(6)
has been studied in detail in [4], in which ,
is the temperature, represents the concentration
of unburned fuel. The authors of [4] investigated a
traveling wave solution for
where is the speed of the front. Furthermore,
approaches the end states exponentially.
However, the traveling wave is not spectrally stable
in . The authors introduced a weight function
, where is positive and small, such that the
perturbation of the traveling wave belonging to this
weighted space approaches exponentially near the
right end state, that is, as . In the weighted
space, the nonlinear terms in (6) do not yield a
locally Lipschitz mapping. To prove the stability of
the traveling wave, the authors proved that
perturbations of the traveling wave that are small in
both the weighted norm and the unweighted norm
will decay exponentially to the traveling wave in the
weighted norm. As we will see in what follows, the
study of the stability of the left end state of the front
encounters similar difficulties.
We will now consider a time-independent
solution of (5) of the form , where we assume
that the function depends only on the scalar
variable . We can perturb the function by
either adding a function that depends only on , that
is, by considering the solution of (5) with
initial condition
(7)
with some from an appropriate
function space; or by adding a function that depends
on , that is, consider the solution of (5)
with the initial condition
with some from an appropriate
function space. The two types of perturbations lead
to the spectral analysis of two different operators
acting on or respectively. We
have already studied the second type of perturbation
in [7]. In this paper we will study the first type as
described in (7), noting that a similar approach as in
[5] is mainly used here.
In particular, the study is related to the essential
spectrum of the linear operator of the system. If the
essential spectrum is in the right half-plane, it can be
shifted to the left half-plane by introducing some
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exponentially weighted space. Given a real
parameter , we shall say that is a
weight function of class if for .
The weighted space with the weight function is
defined as
(8)
We denote the norm of on the unweighted space
by and the norm on the weighted space
by .
Since we will use only perturbations of the first
type, we need a solution of the form
, where belongs to an appropriate space
of functions on . With this notation we will have
the following equation for the perturbation :
(9)
Introduction of the nonlinear term
,
the equation (9) can be rewritten as follows:
. (10)
Since
we therefore have
.
We now define the linear differential expression
with constant coefficients by
. (11)
A major difficulty is that the nonlinear term in
(10) does not give a locally Lipschitz mapping on
the weighted space
. To fix this problem, we
introduce a new space:
. (12)
with .
3 Stability of the Left End State
In this section, we will consider perturbing the right
end state of (5) with the perturbation as
described in (7) and investigate the stability of the
end state.
3.1 Spectral of the Linear Operators
To determine the stability of the perturbation as in
(10), we need spectral information about the linear
operator associated with (11). Consider the system
of differential expressions given by (11). We will
now define several differential operators associated
with .
We define the linear operator on by the
formula and the domain of as the set of
where . For the space
, the domain of in is the set of
where .
We will show below that the spectrum of
touches the imaginary axis so that the equilibrium
solution is not spectrally stable in . A way
out of this problem is then to use a weighted space
.
We define the operator on
as the
linear operator given by the formula , and
the domain of is the set
,
see formula (8). Similarly, we define the weighted
space
, the domain
of on
is the set:
.
We denote by the linear operator on given
by where the domain of is the set of
on satisfying , where
and are the respective domains
defined above.
In the remaining part of this subsection, we will
collect several elementary facts about the spectrum
of the differential operators and on the
respective spaces. We recall here that for a general
closed densely defined operator , the resolvent set
is the set of such that has a
bounded inverse. The complement of is the
spectrum . It is the union of the discrete
spectrum , which is the set of isolated points
in that are eigenvalues of of finite algebraic
multiplicity, and the essential spectrum ,
which is the rest. We will use the Fourier transform
to find on . First, we notice that the
operator on is similar to the operator of
multiplication on by the matrix-valued
function , where
(13)
see e.g. [2. Section 6.5]. The spectrum of on
is the closure of the union over of the
spectra of the matrices . Hence the spectrum of
is equal to the closure of the set of for
which there exists such that
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It is a collection of curves of the form ,
where are the eigenvalues of the matrices
.
The spectrum of on is equal to its
spectrum on , which is proved in the
following lemma.
Lemma 3.1. The linear operator with constant
coefficients associated with the differential
expression in (11) has the same spectrum on
and .
Proof. We will denote the operator associated with
by on and by on . Recall
that has the domain and spectrum .
Therefore, the operator
is an isomorphism. Using the identity
for all , we get
. Thus, we can conclude that
as claimed. □
By Lemma 3.1 and the preceding discussion, for
the operator associated with the differential
expression
(14)
the spectrum of on and is
(15)
Hence, the spectrum of on is the union of
the two curves and
where , therefore
. Thus the spectrum of on and
touches the imaginary axis.
Next, we will tackle on
, which
can be described as follows. Let be or
and . The linear
operator defined by is an
isomorphism from
to . Define the linear
operator on , with domain
if , or domain if
. It is therefore similar to on
and
hence has the same spectrum.
Assume that belongs to the weighted
space with the weight function . It follows that
with
By substituting into the formula for
and
multiplying by and noticing that
, we
can rewrite the linear differential expression
.
Via the Fourier transfom, the operator on
is similar to the operator of multiplication on
by the matrix-valued function
Hence the spectrum of on equal to that of
multiplication by on .
Thus, we find that the spectrum of the operator
is the union of the two curves
and
for all . Then
Re
Re
The linear operator is an operator with
constant coefficients, so . We
also have the following analogue of Lemma 3.1.
Lemma 3.2. The linear operator associated with
the differential expression, that is defined in (14),
has the same spectrum on
and
For , we will have Re
so that the spectrum has been
moved to the left of the imaginary axis. We
summarize this result as the following proposition.
Proposition 3.3. On the unweighted space ,
one has Re
the spectrum of
will touch the imaginary axis. On the weighted
space
, if , then the spectrum of
will be bounded away from the imaginary axis
and Re
for some .
In the system (10) we have the following
triangular structure,
Let
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(16)
(17)
and for , let be the operators on
defined by , the domain of on
is the set of where .
Lemma 3.4. Consider the operators , and
associated with the differential expressions (14),
(16), and (17), respectively,
1) The operator generates a bounded
semigroup on ;
2) the operator on satisfies
Re
;
3) the following is true on :
(a) Re
;
(b) Re;
(c) there exist and such that
for .
Proof. We claim that the semigroups generated by
the operators , , on and are
similar (this gives yet another way to prove that
has the same spectrum on and for
). We denote on as and on
as .
Recall that the Fourier transform is an
isomorphism of onto
, where the
weight function is for .
The operator of multiplication by the function
is an isomorphism of
onto . Under the
Fourier transform followed by this isomorphism of
onto , the operator of differentiation
on is similar to the operator of multiplication
by on . Using this, we have
, where is defined in (13). The
operator of multiplication by on is similar
to the operator of differentiation on via the
Fourier transform, and thus we have
. It follows that
and thus the operators on and
associated with the same constant-coefficient
differential expression are similar. Therefore the
semigroups they generate are similar, proving the
claim.
The operator generates a bounded semigroup
on by Proposition A.1(1) of [6]. Thus, 1) is
proved because on is similar to on
.
Using the Fourier transform, we can find that the
spectrum of on is the curve
and the spectrum of on is the curve
. Thus Re
and Re
on
. It is also true on , proving statements
2) and 3) (a), (b).
Statement 3)(c) is a direct consequence of 2), see
[5, Lemma 3.13]. □
3.2 Lipschitz Property of the Nonlinear Term
In this subsection, we will mainly focus on the
nonlinear term in (10) and will show that the
nonlinear term yields a locally Lipschitz mapping
on the intersection space. This exposition is quite
elementary by nature, and we present it here
because it is a necessary condition for proving the
existence of solutions of (10). In particular, we
substitute into the nonlinear term
to obtain
We introduce the notation
, then can
be written as
(18)
where is defined as in the equation (3).
In order to prove that is a locally Lipschitz
mapping on an appropriate space, we will use below
the inclusion when . To prove
this inclusion, we first show that
for . Indeed, by L'Hospital's rule for ,
(19)
On the other hand, if then
and it
follows that
are all continuous functions for . Using (19),
we can conclude that approaches as
for all and thus is continuous for
all . The required inclusion follows.
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We recall notation (12), that is,
with the norm .
In order to prove the Lipschitz property of on
, we will also need the following elementary
proposition.
Proposition 3.5.
(1) If , then ;
furthermore, there exists a constant such
that .
(2) If , then
; furthermore,
there exists a constant such that
.
(3) If , then ; also there exists a
constant such that
.
Proof. Assertion (1) is a well-known result of
Sobolev spaces, see [1, Theorem 5.23]. Assertion
(2) can be proved by
To show assertion (3), let . Then by
assertion (1),
and by assertion (2),
Therefore and
. □
Let and let denote the space of
bounded functions with the sup
norm, which we now denote . More
generally, let denote the space of
functions such that , , , are
all bounded functions, with the -norm:
Proposition 3.6. Let . Then the
formula defines mappings from
to and from
to . The first is
Lipschitz on any set of the form ; the
second is Lipschitz on any set of the form
.
Proof. We have
(20)
Therefore,
and
Also, differentiating the equation (20) we have
Thus
and since by the Sobolev
embedding theorem, we have
similarly,
If and are both bounded by the
constant , then , and due to the
equation (20) there exists a constant
depending on , such that
Similarly, if are bounded by the
constant , then ,
and
thus
.
□
Proposition 3.7. Let . Consider the
formula (21)
(1) Formula (21) defines a mapping from
to that is locally Lipschitz on any set of
the form .
(2) Formula (21) defines a mapping from to
that is locally Lipschitz on any set of the form
.
Proof. See Proposition 5.6 in [4]. □
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Proposition 3.8. Let
with from (3).
(1) The formula defines a
mapping from to that is
Lipschitz on any set of the form
and there is a constant depending
on such that .
(2) If , and
is given by the
equation (18), then defines a mapping
from to that is Lipschitz on
any set of the form and
.
Proof. (1) The Lipschitz property follows from
Proposition 3.6. Since the mapping is Lipschitz, and
, we then have
that .
(2) Since , assertion (2)
follows from assertion (1) and Proposition 3.7. (2).
□
Proposition 3.9. Consider the formula
(22)
where is given by the equation (18).
(1) If , then
, also is
Lipschitz on any set of the form
and there exists a constant depending
on such that .
(2) Formula (18) for defines a mapping
from to that is Lipschitz on any set of the
form and
.
Proof. To prove assertion (1), we use the following
inequality:
for some depending on . Assertion (2) can
be proved similarly to Proposition 3.8. (2) by using
Proposition 3.7. (2). □
3.3 Nonlinear Stability Analysis
In this subsection, we prove the stability of the right
end state of the system (5) on . The operator
generates a strongly continuous semigroup on
. The nonlinear term yields a locally Lipschitz
mapping on by Proposition 3.9. Therefore we
can apply the following standard result.
Lemma 3.10. Let be a Banach space. Consider
the system
where is locally Lipschitz continuous in and
the operator generates a
semigroup on .
For any the system has a unique mild
solution with the initial value . The solution is
defined for the time in the maximal interval
where .
Proof. See [8, Theorem 6.1.4]. □
Next, we recall yet another standard fact.
Consider a system of the form
where the operator is defined by the
formula with the domain and
generates a -semigroup on , and is a locally
Lipschitz mapping from into .
Let be given by
so that the set is open in , and the map
from to is continuous. We have the
following lemma.
Lemma 3.11. For each , if , then
there exists depending on and , with
, such that the following holds: if satisfies
(23)
and , then the solution of the
system (10) is defined and satisfies
(24)
Proof. The proof is the same as the proof in [9,
Theorem 46.4]. □
If are fixed, let denote the
supremum of all such that (24) holds for all
whenever (23) is satisfied. In addition, we
obtain the following results by Proposition 3.3:
Lemma 3.12. Let
be the operator defined in subsection 3.1.
There exists which satisfies
Re (25)
Moreover, there exists a constant such that
for .
Proof. Recall equation (11):
.
The operator generates an analytic semigroup
provided and a strongly continuous
semigroup provided . As shown in [4], in both
cases the differential operator associated with the
differential expression in (11) enjoys the spectral
mapping property, that is, the boundary of the
spectrum of the semigroup operator is
controlled by the boundary of the spectrum of the
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semigroup generator for any . By
Proposition 3.3 we can choose such that
Re. Furthermore, by the
above mentioned semigroup property, see, e.g.
Proposition 4.3 in [4], there exists such that
. □
We are ready to start the stability analysis. We
will first show that the solution of (10) is
exponentially decaying in the norm.
Proposition 3.13. Let satisfies (25). Then
there exist and such that for every
and every with , the
following holds for the mild solution of
(10) with the initial value : If satisfies
(23) such that satisfies (24) for
, then
(26)
Proof. Since is a mild solution of (10) on , it
satisfies the integral equation
(27)
Since by assumption, it is clear that
is in
by Proposition 3.9, so we have
Next, we replace by in (27) and choose
such that Re
and define .
By Lemma 3.12, there exists such that
for all . Pick any ,
for , and notice that if , then
for all by Lemma
3.11.
With the aid of Propostion 3.9. (1), there exists a
constant depending on such that for all
, using that when
, it follows that
For each , and , if ,
then for all by
Lemma 3.1. Then, for all , we have
Applying Gronwall's inequality for ,
we conclude that the inequality
implies, by Gronwall's inequality, that
so that
By choosing
, we can
conclude that (26) holds for all . □
We now show that the solution of (10) is bounded
in the norm, and the component is
exponentially decaying in the norm.
Proposition 3.14. Let be chosen as in Lemma
3.4. (3), and be given by Proposition 3.13.
Assume that , where satisfies (25). Then
there exist constants and such
that for every and every with
, the following holds: If , and
satisfies (23) such that satisfies
(24), then the following estimates hold:
(28)
(29)
Proof. We write (10) as a non-autonomous linear
system on :
(30)
(31)
where , are defined in (16) and (17),
is a fixed solution of (10), and
Note that is the solution of (30)-(31) with
the value at , that is
.
With the help of Proposition 3.8. (2), we can find
a constant such that
(32)
and
(33)
if .
The solution of (31) in can be written as
We then choose some and
such that
Re
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By Lemma 3.4. (3), there exists such that
.For each
and , if then
By Lemma 3.11, we can use (33) to obtain the
following estimate for :
We then compute
Applying Gronwall's inequality to , we
infer that
Let
Then for it
follows that
for
(34)
which proves (29).
We now give the proof of equation (28). The
solution of (30) in satisfies
First, since generates a bounded semigroup by
Lemma 3.4. (1), there exists a constant , such
that . Using equation (32)
and the fact that for
, we infer that
Also, using the fact that
, we have, for a constant
independent of , that
Then we use (34) to obtain
for some . In conclusion, there exists a
constant such that for and
, the inequalities (28) and (29) hold if
. □
We now complete the proof of the nonlinear
stability of the end state .
Remark 3.15. We claim that the end state of (5)
is stable in . The proof of the stability of
is, in fact, contained in the next theorem and relies
on on the following bootstrap argument based on
Proposition 3.13 and Proposition 3.14. Indeed,
these propositions yield the existence of constants
and such that for every
and every , there exists such that
for every the inequalities
and (35)
hold for the solution of (10) with initial value
as long as . Let us show that for
each , there exists an such that if
then for all , that is,
the end state of (5) is stable in . Indeed,
assuming with no loss of generality, set
and assume . Then
by using (35), and thus the
solution with the initial value satisfies
(35) again for , again by
Propositions 3.13 and 3.14. So, these propositions
can be applied for all , proving the stability.
In addition, as long as these propositions are
applicable, we obtain a more refined information
about the behavior of the solution, such as its
boundedness in norm and the exponential
decay in norm, see items (3)-(4) of the next
theorem. We now proceed with a more formal
exposition of the stability statement.
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Given an initial value , let
be the solution of (10) in with
, which we have shown to exist on
by Lemma 3.10. We shall show that
is defined and bounded in norm, and
exponentially decaying in norm for all time
. We note that the small constant in the next
theorem can be chosen as where is
chosen as in Proposition 3.14.
Theorem 3.16. There exist constants and
such that for each , we can find
such that if , then for all
the following holds:
(1) is defined;
(2) ;
(3) ;
(4) ;
(5)
Proof. Choose as in Lemma 3.12. Choose
as indicated in Proposition 3.14. Let be a constant
satisfying with and given
as in Propositions 3.13 and 3.14. Let
and set . Assume satisfies
. Since , the solution
exists and satisfies statements (2)-(5) in the
theorem for by Propositions 3.13
and 3.14. We claim that , so that the
proof is finished as soon as the claim is justified. To
do this, for any we consider the
solution with the initial data . Note that
staments (3)-(5) for yield
and thus Lemma 3.11 applies
and gives for .
Therefore, we proved that if then
for all . This
shows that and therefore
implies and thus
as claimed.
□
4 Conclusion
We now generalize the above results to a more
general system with the general nonlinearity
and the coefficient matrix given by
(36)
where
with all , and the function is
smooth, see Hypothesis 4.1 below. We will present
here the stability result of an -independent
stationary solution to the system (36) and
its perturbation depending only on
from summarizing the stability analysis of the
model system (5).
Hypothesis 4.1.
(a) In appropriate variables
, , we assume that
for some constant matrix ,
Moreover,
where each is a nonnegative diagonal
matrix of size , and
for .
(b) The function is from to .
(c) For the linear operator associated with the
differential expression
there exists such that Re
on
.
(d) The operator associated with the differential
expression
generates a bounded semigroup on
and .
(e) The operator associated with the differntial
expressin
satisfies Re
on
and .
By the discussion in Section 2, replacing the
spatial variable by the moving
variable in (36), we obtain
(37)
We will now rewrite the equation for the
perturbation of the stationary solution in
the form amenable for the subsequence analysis. We
seek a solution to (37) of the form
, with this notation, satisfies
(38)
We can show that for the system (36) satisfying
Hypothesis 4.1, if the initial values of the
perturbation of the stationary solution are
sufficiently small in both the weighted and
unweighted norms, the perturbation will converge
exponentially in the global time domain in the
weighted norm and remain bounded in the
unweighted norm. Note that the hypotheses we have
given are sufficient to cover the conditions we need
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.32
Qingxia Li, Xinyao Yang, Ziyan Zhang
E-ISSN: 2224-266X
304
Volume 21, 2022
in the proof, so this proof will be very similar to the
one in Section 3, and we will not repeat it here.
In summary, in this paper we have discussed the
stability of the stationary solution of a class of
reaction-diffusion equations in multidimensional
space and have summarized the characteristics of
this class of equations. Although the approach of
this discussion is essentially similar to the one in [5]
for equations in one-dimensional space, it can be
constituted together with the discussion in [7] for
the stability of the stationary solution of this class of
reaction-diffusion equations in multidimensional
space with respect to two types of perturbations. It is
worth noting that there are still many unsolved
problems in this type of nonlinear stability analysis,
such as the stability analysis of traveling wave
solutions for this class of equations in
multidimensional space, which we mentioned in
Section 2, is still a difficult problem, and this is a
direction that may need to be covered in future
studies.
Acknowledgement:
The authors thank Y. Latushkin at the University of
Missouri-Columbia, for providing research
questions and ideas, and for all the help he gave us.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Qingxia Li organized the writing of this article,
provided input, and helped with funding acquisition.
Xinyao Yang completed the first draft of the paper
and the main mathematical analysis part.
Ziyan Zhang completed the proofreading of the
paper and verified the main calculations in the
paper.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
Research funding was provided by the National
Natural Science Foundation of China [Young
Scholar 11901468]; Xi'an Jiaotong-Liverpool
University [KSF-E-35]; and the National Science
Foundation [NSF-HRD 2112556].
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.32
Qingxia Li, Xinyao Yang, Ziyan Zhang
E-ISSN: 2224-266X
305
Volume 21, 2022
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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Creative Commons Attribution License 4.0
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