Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
-
Yuan-Hao Liu
and
Suobing Zhang
Abstract
This article is concerned with the stability of time-periodic traveling fronts for reaction–diffusion equations with time-periodic degenerate monostable and ignition nonlinearities. Based on the existence of pyramidal traveling fronts established by Bu et al., we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces and establish some important estimates. Then, we show the asymptotic stability of pyramidal traveling fronts by using the sub-super solution method combined with the comparison principle.
1 Introduction
In this article, we investigate the following reaction–diffusion equation:
where
For the nonlinear parabolic equation (reaction–diffusion equation) and the variations, the well-posedness of solutions, including existence, uniqueness, and dynamical behavior, has been investigated by many authors. For example, Xu and Su [40] studied the semilinear pseudo-parabolic equation. Xu and Su [40] proved the global existence, nonexistence, and asymptotic behavior of solutions with low initial energy and the critical initial energy by introducing a family of potential wells. Moreover, they also obtained finite-time blowup with high initial energy by the comparison principle. Based on the model described in [40], and considering a porous medium, Lian et al. [17] considered a model with a singular potential term
In this article, we are more concerned with the global solution of parabolic partial differential equations. That is, a solution with a definition for all times
Motivated by [5,41,42], this article studies the traveling fronts of reaction–diffusion equations with time-periodic degenerate monostable and ignition nonlinearities. We begin by giving some important estimates. In particular, we use the periodic planar traveling fronts and the periodic V-shaped traveling fronts to characterize some qualitative properties of the periodic pyramidal traveling fronts. Furthermore, using these estimates, along with the super-sub solution method and the comparison principle, we show the asymptotic stability of the periodic pyramidal traveling fronts. Different from the reaction–diffusion equations without periodicity, the presence of the time variable
According to [2,26–28], equation (1.1) admits a periodic planar traveling front
and
For
The aim of this article is to investigate the stability of pyramidal traveling fronts of (1.1). Assuming that the traveling wave solution propagates along the
We write the solution of (1.5) and (1.6) as
Now, we introduce the notion of a pyramid. Let
Define
then
The boundary of
The edges of the pyramid are represented as
And
and
It is easy to verify that
The existence of time-periodic pyramidal traveling fronts for reaction–diffusion equations with time-periodic degenerate monostable nonlinearity has been established by Bu et al. [4]. For the case of time-periodic ignition nonlinearity, one can easily check with a similar proof. More precisely, we have the following theorem.
Theorem 1.1
[4, Theorem 1.1] Assume that (F1) and (F2) hold. For each
and
Now, we give the main result of this article.
Theorem 1.2
Let
and
then the solution
2 Preliminaries
In this section, we establish some key results. First, we present the existence and stability of V-shaped traveling fronts for time-periodic reaction–diffusion equations in
for
It is clear that
is a subsolution of (2.1). In addition, we have
Theorem 2.1
[41, Theorem 1.2], [42, Theorem 1.1] Assume that
Theorem 2.2
[41, Theorem 1.4], [42, Theorem 1.3] Assume that
for any
Second, we introduce the mollified pyramid and present some of its properties. Set
Besides, we have
where
The set
where
The following two lemmas were shown by Taniguchi [31], which give some important properties of the functions
Lemma 2.3
Let
and
Lemma 2.4
There exist two positive constants
In addition, for integers
and
Third, we consider the following equations:
where
We define
Fourth, we present some properties of the reaction term
Besides, by assumption (F1), we know that there exists a positive constant
Finally, we construct an auxiliary function
which will be used in constructing supersolutions.
3 Key estimates
In this section, we provide some important estimates that will be used in the proof of asymptotic stability. First, we show that for any
Lemma 3.1
Let
Proof
By the periodicity of
By (1.2), we have
From (1.3) and the fact that
we know that there exists a constant
Define
This completes the proof.□
From the definition of
Corollary 3.2
There exist constants
Set
Let
We consider the following Cauchy problem:
where
Let
and
Lemma 3.3
Let
where
for any
Proof
Let
Let
By (3.1) and the maximum principle, we have
Now, we estimate
for
Define
and
Clearly,
Since
For any
where
Same as the proof in [5, Lemma 3.1], we have
Then, we obtain
Substituting
In the following, we will show that the time-periodic pyramidal traveling front converges to the two-dimensional time-periodic V-shaped traveling front along the edges at infinity. More precisely, for a time-periodic pyramidal traveling front, as we move along any edge toward infinity, the function values on the cross-sections perpendicular to this edge converge in a certain manner to some two-dimensional time-periodic V-shaped traveling front. For this purpose, we first investigate the behavior of the two-dimensional time-periodic V-shaped traveling front on each edge. Consider the edge
where
Thus, the direction of traveling fronts
Let
The angle between
and thus
The speed of
which coincides with the speed of
and
where
Now, let
By [41, Lemma 4.2] and [42, Lemma 4.1], we obtain
in
Then, we have
Since each
The following lemma establishes some properties of
where
Lemma 3.4
The function
and
Proof
Theorem 2.1 implies that for each
Thus,
Take
Consequently,
Then, it follows from Theorem 1.1, we obtain that (3.9) holds. This completes the proof.□
The following lemma demonstrates that
Theorem 3.5
Assume that the
for some
Proof
Define
Then,
Step 1. For any given
Without loss of generality, we can assume
with
Recall that
The change of variables is given by
The coefficient matrix in (3.10) is an orthogonal matrix of determinant +1. Then, we have
Set
It is clear that
is a solution to (1.7) and (1.8). Let
Then, by setting
we have
According to Lemma 3.4, we have
Following from Lemma 3.4, Theorem 1.1, and the assumption on
By the definition of
Choose a function
Then,
We consider (3.11) for
which is equivalent to consider (3.12) for
respectively. It follows from the definition of
By Theorem 2.1, we have
Applying Theorem 2.2, we obtain
which implies
Since
Thus, for any given
Set
Then,
Since
Let
Clearly,
Note that
if
for any fixed
for any fixed integer
We have now established the estimate for
and fix integer
Step 2. For any
By the definition of
Applying Lemma 3.3 to
for any
Combining the estimates of Steps 1 and 2, we obtain
This completes the proof.□
The following lemma follows directly from Theorem 3.5.
Lemma 3.6
Let
Furthermore, suppose the initial value
for some
Proof
Since
In the following, we present some properties of
Lemma 3.7
Proof
It is obvious that
Since
we have
Additionally, by
we have proved the lemma.□
Lemma 3.8
Proof
The proof is similar to that of [42, Lemma 4.6] and is omitted.□
Lemma 3.9
Furthermore, for any
Proof
First, we prove (3.19). Since
for some
Now, (3.16) gives
where
for
Apply the interior
Then, we obtain
and thus
We have
Thus, it follows from [41, Lemma 4.2] and [42, Theorem 4.1], we obtain (3.19) hold.
Now, we proceed to prove (3.18).
Assume
Thus, we obtain
Apply the interior
for any
Since
then we have
Thus, (3.18) holds. This completes the proof.□
The following lemma shows that the solution to (1.5) and (1.6) with initial value
Lemma 3.10
Let
Proof
Note that
Thus, for any
Since
for
Following from Lemma 3.6, we have for any
Combining (3.20) and (3.21), we have
Then, by the definition of
Combining with
Since
The following lemma establishes a positive lower bound for the partial derivatives of
Lemma 3.11
Assume that
For any
where
Proof
The proof is similar to that of [5, Lemma 3.6] and is omitted.□
Lemma 3.12
Let
for some
Proof
Recall that
Then, we have
for
Similarly, we have
We now proceed to estimate
where
Let
Then, we have
Here, recall that
4 Stability of pyramidal traveling fronts
In this section, we study the asymptotic stability of pyramidal traveling fronts when
where
Since both
Lemma 4.1
[4, Lemma 3.1] For each
is a supersolution of (1.7) and (1.8) on
Lemma 4.2
For any
Proof
By the definition of
For any
as
In the following, we construct some supersolutions for (1.5).
Lemma 4.3
Let
there exists a positive constant
is a supersolution of (1.5), where
Proof
From (1.2) and some computations, we have
To prove the lemma, we consider three cases.
Case 1.
Since
for
By Lemma 2.3, we have
Then, for the reaction team
where
Set
provided that
Case 2.
Since
Then, by (2.7), we have
Let
provided that
Case 3.
Furthermore, we denote
From (1.2) and the definition of
Then, by the boundedness of
Then, for
provided that
This completes the proof.□
Lemma 4.4
Let
there exists a positive constant
is a supersolution of (1.5), where
Proof
The proof is very similar to the proof of Lemma 4.3 and we omit it.□
Fix
is well-defined, where
Lemma 4.5
Let V and
Proof
By the periodicity of
Assume that
For any
where
For
with
where
That is, there exists a constant
By (1.9), there exists a constant
Set
Note that
which implies
Combining the above equation with equation (1.9), we obtain
Applying Theorem 3.5 to
Define
then we have
or
For the former case, since
for
Thanks to
and
there exists
Define
Set
For
Thus, we have
Using Lemma 3.6, (3.19), and (4.7), if
Since
in
for
which infers
For
Finally, we obtain
for
Let
for
which contradicts the definition of
We now give the proof of Theorem 1.2.
Proof of Theorem 1.2
By Lemma 3.9, we can define
Let
which implies
By Lemma 3.6, there exists a sufficiently large integer
Thus, there exists
which implies
Let
Consequently, we obtain
By
Taking into account (4.15) and (4.16), we obtain
for
By Lemma 4.4 and the comparison principle, we have
where
where
for
for
for
for
On the other hand, since
Thus, it follows from Lemma 3.10 that there exists a constant
Combining with (4.20) and (4.21), we obtain
This completes the proof.□
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and many useful suggestions. They also thank Mingyou Zhang for helpful discussions and suggestions.
-
Funding information: Yuan-Hao Liu and Zhen-Hui Bu were supported by the National Natural Science Foundation of China (Nos. 12101499 and 12471163).
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. All authors have contributed more or less equally to the contribution of this manuscript.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
