1. Introduction
In the current paper, we focus on stochastic reaction–diffusion equations driven by Gaussian noise
where
and
as initial data are both Heaviside functions. In this paper, it always holds that
- (H1)
, ;
- (H2)
is decreasing for u, , and for ; is decreasing for v, , and for ;
- (H3)
There exists for any , and ; it holds that , ;
- (H4)
, .
In general, (H1) and (H2) imply that System (
1) is cooperative, so its corresponding dynamical system is monotonic and a comparison method can be used in this system. (H4) ensures the noise is moderate, otherwise the solution of Equation (
1) tends to zero as
.
Under conditions (H1)–(H4), System (
1) poses an only positive stable equilibrium denoted by
. If
,
, Zhao and Øksendal [
1] investigated the pathwise property and ergodicity of a stochastic reaction–diffusion equation in a scalar scale under conditions (H2)–(H4),
Inspired by Benth and Gjessing [
2], they successfully obtained the stochastic Feynman–Kac formula by constructing exponential martingale and revealed that the existence of a travelling wave solution and its asymptotic wave speed depend on the strength of noise. In detail, if
and the noise is strong, the solution to Equation (
2) almost surely tends to zero. If
and the noise is weak, the travelling wave solution of Equation (
2) converges to the travelling wave solution of deterministic reaction–diffusion equation. Moreover, if
exist and the noise is moderate, the wavefront marker is
.
If
, Freidlin [
3] studied the asymptotic behavior of the Cauchy problem under conditions (H1)–(H3),
Until now, many papers have been concerned with stochastic travelling wave solutions in the scalar equation. Zhao [
4] studied the wave speed of a stochastic KPP equation driven by white noise. Huang and Wang [
5,
6,
7,
8,
9] investigated the asymptotic behavior of a stochastic reaction–diffusion equation driven by various noises. Indeed, a way to deal with the coupling terms is the crux in the research of travelling wave solution of stochastic reaction–diffusion equations. In this paper, with monotonic random dynamical system theory and comparison principle, the boundedness of solution to Equation (
1) is obtained and used to construct a sup-solution and a sub-solution under conditions (H1)–(H4). Hence, via the SCP (Support Compactness Propagation) property proposed by Shiga [
10] and two sufficient conditions proposed by Tribe [
11], the existence of a travelling wave solution can be obtained. Again, with the boundedness of solution, we can estimate the upper bound and the lower bound of wave speed by a sup-solution and a sub-solution, respectively.
Throughout this paper, we set as the space of temper distributions, as the -algebra on , and as the white noise probability space. We denote by the expectation with respect to . We denote by . Here are some notations:
;
;
;
and f is continuous;
;
is continuous, and as ;
;
is the space of nonnegative functions with compact support;
for some is the space of functions with exponential decay;
is the wavefront marker.
Lemma 1 ([
11])
. A set is called relatively compact if and only if- (a)
K is equicontinuous on a compact set;
- (b)
.
Lemma 2 ([
11])
. is (relatively) compact if and only if it is (relatively) compact in for all . Lemma 3 ([
11]
(Kolmogorov tightness criterion))
. For , we define then, with the above conditions, we know that is compact in , where a is a constant.- (1)
If are -valued processes, with tight and there are such that for all , ,then are tight. - (2)
Similarly, if are -valued processes, with tight, and there are such that for all ,then are tight.
2. Asymptotic Behavior of a Travelling Wave Solution
At the beginning of our work, we offer the definition of stochastic travelling wave solution in law.
values are equipped with the
metric,
and
are measurable subsets of
, and the three spaces are all Polish spaces and are compact. We consider a stochastic reaction–diffusion equation with a Heaviside function as follows:
Definition 1. A stochastic travelling wave solution is a solution to to Equation (4) with values in and for which the centered process is a stationary process with respect to time, and the law of a stochastic travelling wave solution is the law of on . We denote by
and
; then, Equation (
1) can be rewritten as
Lemma 4. For any Heaviside functions and as initial data, for, , there exists a unique solution to Equation (5) in law with the formwhere is the Green function. Proof. We denote by
, since
and
are Lipschitz continuous, perform some truncations and let
; then,
is Lipschitz continuous. Hereto, the truncated Equation (
7) can be constructed:
where
, and
as
. We refer to [
12]. There exists a unique solution
to Equation (
7) in law and
. According to the Kolmogorov tightness criterion, we can show that for, a.e.,
there is a unique solution
to Equation (
5) such that
converges to
as
. □
Lemma 5 ([
12])
. All solutions to (5) with initial date have the same law which is denoted by , and map is continuous. For any Heaviside function , law forms a strong Markov family. Next, we perform several estimations about which play an important role in constructing travelling wave solution and estimating its asymptotic wave speed.
Theorem 1. For any Heaviside functions and as initial data, if (H1)–(H4) hold, for any fixed and, , it is true thatwhere is a constant. Proof. We consider Equation (
1):
We denote by
; we have
We let
; with (H1) and (H3), we know that there exists
such that
combination with (H2) and (
11) gives
Frequently, it can be determined that
We denote
;
is decreasing for
, and
for
. We let
be the solution to Equation (
13):
According to monotone random dynamical systems theory [
13] and its corresponding comparison principle [
14], it can be determined that
and
We let
be the solution to the following equation:
We refer to [
12] and use a stochastic Feynman–Kac formula. We have
For any fixed
, for any
, multiplying
in (
14) and integrating over
R, it can be determined that
We let
; then, we have
It can be deduced that
Furthermore, we have
Then, we let
, and we have
Therefore, we perform combination with (
15) and obtain
Moreover, since the initial data
and
are both Heaviside functions, we take the expectation and obtain
where
. □
Theorem 2. For any Heaviside functions and as initial data, if (H1)–(H4) hold, for, , any fixed, it it true thatwhere is constant. Proof. We let
using the Itô formula, and we obtain
Then, integrating both sides in
and taking the expectation implies
Hence, with Young inequality and Gronwall inequality, we have
where
is a constant. □
With the boundedness of , the sup-solution can be constructed to describe how fast the support of can spread and the SCP property can be obtained.
Lemma 6. We let be the solution to (1) with initial data as a Heaviside function. We suppose for some such that is supported outside ; then, for any , Proof. Since
is bounded and we construct a sup-solution solving the following equation:
where
is a constant such that
and
, then we refer to [
11,
12]. The proof can be completed. □
Then, we show that satisfy the Kolmogorov tightness criterion and , which is dedicated to constructing a tight probability measure sequence and furthermore obtain the existence of a travelling wave solution.
Lemma 7. For any Heaviside functions and as initial data, if (H1)–(H4) hold, for, , any fixed, , if , there exits such that Proof. Direct calculation shows that
Since
and
is bounded, for
and
, we have
and
With Hölder inequality, for
I and
, we have
and
Combining all the inequalities above completes the proof:
□
In order to construct a travelling weave solution, according to the two sufficient conditions proposed by Tribe [
11], we are required to show that the wavefront marker is bounded for all
and the translation of solution with respect to a wavefront marker is stationary. However, it is quite difficult to deal with
directly, so we turn to a new suitable wavefront marker for help. We define
as the law of the unique solution to Equation (
5) with initial data
. For a probability measure
on
, we define
define a new wavefront marker
,
and
then,
is an approximation to
. We let
,
, and define
Now, we summarize the steps of constructing a travelling wave solution. First, we show that the new wavefront marker is bounded (see Lemma 8) to ensure the shifting does not destroy the tightness (see Lemma 7). Based on this, we construct a tight probability measure sequence (see Lemma 9) and show that any limit point is nontrivial (see Theorem 3), where is the law of a travelling wave solution.
Lemma 8. For any Heaviside functions and as initial data, for, , any , , , there exists such that Proof. With the comparison principle, we construct a sup-solution solving Equation (
28):
where
is a constant such that
and
; thus, we determine that
and
hold on
uniformly, and for, a.e.,
the solution to Equation (
28) can be expressed by
Without generality, we assume that
and take
for an example. We have
according to the definition of
, we know
meanwhile, we have
Combining (
30) with (
31) implies that
On the other hand, Jensen’s inequality offers
additionally, we can obtain such estimation:
Rearranging the inequalities above implies
We combine (
32) with (
33), and the proof can be completed. □
Via the boundedness of wavefront marker , we can construct the tight sequence with .
Lemma 9. For any Heaviside functions and as initial data, for, , the sequence is tight.
Proof. Similarly, we start discussion with
. Since
, it can be determined that
, and furthermore, we have
According to Lemma 8, it can be easily determined that
as
. In addition, for any given
, we choose
to make
I as close to
as desired. In addition, we know
thus, for a given
, we can choose
such that
as close to one as desired for
T and
d that are sufficiently large, which means that sequence
is tight. We refer to Lemma 3.9 in [
12]. The proof can be completed. □
Theorem 3. For any Heaviside functions and as initial data, if (H1)–(H4) hold, for, , any fixed, there is a travelling wave solution to Equation (1), and is the law of travelling wave solution. Proof. Refer to Theorem 3.10 in [
12]. The proof can be completed. □
Based on the existence of a stochastic travelling wave solution, we can estimate the wave speed by the comparison principle in the following. First, we construct the sup-solution
where
is Lipschitz continuous and decreasing for
, and there exists
, such that
for
. Since
is the solution to (
1) and bounded, it holds that
and
. Then, we construct the sub-solution
where
,
. According to the comparison theorem of wave speed (see Lemma 4.2 in [
12]), the wave speed of travelling wave solution to Equation (
1) can be estimated.
Theorem 4. For any Heaviside functions and as initial data, if (H1)–(H4) hold, we denote by c the wave speed of the travelling wave solution to Equation (1); then,where , . Proof. According the definition of wave speed,
the wave speed of stochastic reaction–diffusion equations is the maximum value between the two sub-systems; thus, with the comparison theorem of wave speed and referring to Theorems 4.1 and 4.2 in [
12], the proof can be completed. □