Abstract
In the current series of two papers, we study the long time behavior of nonnegative solutions to the following random Fisher–KPP equation,
where \(\omega \in \Omega \), \((\Omega , {\mathcal {F}},{\mathbb {P}})\) is a given probability space, \(\theta _t\) is an ergodic metric dynamical system on \(\Omega \), and \(a(\omega )>0\) for every \(\omega \in \Omega \). We also study the long time behavior of nonnegative solutions to the following nonautonomous Fisher–KPP equation,
where \(a_0(t)\) is a positive locally Hölder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on \(a(\omega )\), we show that the constant solution \(u=1\) of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval \([2\sqrt{{\underline{a}}}, 2\sqrt{{\bar{a}}}]\), where \({\underline{a}}\) and \({\bar{a}}\) are the least and the greatest means of \(a(\cdot )\), respectively, and hence the spreading speed interval is linearly determinate. It is shown that the solution of (1) with a nonnegative initial function which is bounded away from 0 for \(x\ll -1\) and is 0 for \(x\gg 1\) propagates at the speed \(2\sqrt{{\hat{a}}}\), where \({\hat{a}}\) is the mean of \(a(\cdot )\). Under some assumption on \(a_0(\cdot )\), we also show that the constant solution \(u=1\) of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that \(a(\omega )\) and \(a_0(t)\) are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher–KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).
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1 Introduction and Statements of the Main Results
The current series of two papers is concerned with the long time behavior of nonnegative solutions to the following random Fisher–KPP equation,
where \(\omega \in \Omega \), \((\Omega , {\mathcal {F}},{\mathbb {P}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\) is an ergodic metric dynamical system on \(\Omega \), \(a:\Omega \rightarrow (0,\infty )\) is measurable, and \(a^\omega (t):=a(\theta _t\omega )\) is locally Hölder continuous for every \(\omega \in \Omega \). It also considers the long time behavior of nonnegative solutions to the following nonautonomous Fisher–KPP equation,
where \(a_0:{{\mathbb {R}}}\rightarrow (0,\infty )\) is locally Hölder continuous. Among others, (1.1) and (1.2) are used to model the population growth of a species in biology. In such case, u(t, x) denotes the population density of the species. Thanks to the biological reason, we are only interested in nonnegative solutions of (1.1) and (1.2).
Observe that (1.1) [resp. (1.2)] with \(a(\omega )\equiv 1\) (resp. with \(a_0(t)\equiv 1\)) becomes
Equation (1.3) is called in literature Fisher–KPP equation due to the pioneering works of Fisher [13] and Kolmogorov et al. [25] on traveling wave solutions and take-over properties of (1.3). It is clear that the constant solution \(u=1\) of (1.3) is asymptotically stable with respect to strictly positive perturbations. Fisher in [13] found traveling wave solutions \(u(t,x)=\phi (x-ct)\) of (1.3) \((\phi (-\infty )=1,\phi (\infty )=0, \phi (s)>0)\) of all speeds \(c\ge 2\) and showed that there are no such traveling wave solutions of slower speed. He conjectured that the take-over occurs at the asymptotic speed 2. This conjecture was proved in [25] for some special initial distribution and was proved in [3] for general initial distributions. More precisely, it is proved in [25] that for the nonnegative solution u(t, x) of (1.3) with \(u(0,x)=1\) for \(x<0\) and \(u(0,x)=0\) for \(x>0\), \(\lim _{t\rightarrow \infty }u(t,ct)\) is 0 if \(c>2\) and 1 if \(c<2\). It is proved in [3] that for any nonnegative solution u(t, x) of (1.3), if at time \(t=0\), u is 1 near \(-\infty \) and 0 near \(\infty \), then \(\lim _{t\rightarrow \infty }u(t,ct)\) is 0 if \(c>2\) and 1 if \(c<2\). In literature, \(c^*=2\) is called the spreading speed for (1.3).
A huge amount of research has been carried out toward various extensions of traveling wave solutions and take-over properties of (1.3) to general time and space independent as well as time and/or space dependent Fisher–KPP type equations. See, for example, [2, 3, 11, 15, 24, 41, 48], etc., for the extension to general time and space independent Fisher–KPP type equations; see [4, 5, 7, 14, 22, 26,27,28,29, 31, 37, 38, 49, 50], and references therein for the extension to time and/or space periodic Fisher–KPP type equations; and see [5, 8,9,10, 16, 21, 30, 32,33,34,35,36, 43,44,47, 51, 52], and references therein for the extension to quite general time and/or space dependent Fisher–KPP type equations. The reader is referred to [12, 17, 53], etc. for the study of Fisher–KPP reaction diffusion equations with time delay.
All the existing works on (1.1) [resp. (1.2)] assumed \(\inf _{t\in {{\mathbb {R}}}} a^\omega (t)>0\) and \(a^\omega (\cdot )\in L^\infty ({{\mathbb {R}}})\) (resp. \(\inf _{t\in {{\mathbb {R}}}} a_0(t)>0\) and \(\sup _{t\in {{\mathbb {R}}}} a_0(t)<\infty \)). The objective of the current series of two papers is to study the long time behavior, in particular, the stability of positive constant solutions, the spreading speeds, and the transition fronts of (1.1) [resp. (1.2)] without the assumption \(\inf _{t\in {{\mathbb {R}}}} a^\omega (t)>0\) and \(a^\omega (\cdot )\in L^\infty ({{\mathbb {R}}})\) (resp. without the assumption \(\inf _{t\in {{\mathbb {R}}}} a_0(t)>0\) and \(\sup _{t\in {{\mathbb {R}}}} a_0(t)<\infty \)). It will also discuss the applications of the results established for (1.1) to Fisher–KPP equations whose growth rate and/or carrying capacity are perturbed by real noises.
In this first part of the series, we investigate the stability of positive constant solutions and the spreading speeds of (1.1) and (1.2). We first consider the stability of positive constant solutions and spreading speeds of (1.1) and then consider the stability of positive constant solutions and spreading speeds of (1.2). In the second part of the series, we will study the existence and stability of transition fronts of (1.1) and (1.2).
In the following, we state the main results of the current paper. Let
with norm \(\Vert u\Vert _\infty =\sup _{x\in {{\mathbb {R}}}}|u(x)|\) for \(u\in C_{\mathrm{unif}}^b({{\mathbb {R}}})\). For given \(u_0\in X:=C_{\mathrm{unif}}^b({{\mathbb {R}}})\) and \(\omega \in \Omega \), let \(u(t,x;u_0,\omega )\) be the solution of (1.1) with \(u(0,x;u_0,\omega )=u_0(x)\). Note that, for \(u_0\in X\) with \(u_0\ge 0\), \(u(t,x;u_0,\omega )\) exists for \(t\in [0,\infty )\) and \(u(t,x;u_0,\omega )\ge 0\) for all \(t\ge 0\). Note also that \(u\equiv 0\) and \(u\equiv 1\) are two constant solutions of (1.1). Let
and
Observe that
and that
Then by the countability of the set \({{\mathbb {Q}}}\) of rational numbers, both \({\hat{a}}_{\inf }(\omega )\) and \({\hat{a}}_{\sup }(\omega )\) are measurable in \(\omega \).
Throughout this paper, we assume that the following standing assumption holds.
(H1) \(0< {\hat{a}}_{\inf }(\omega )\le {\hat{a}}_{\sup }(\omega )<\infty \) for a.e. \(\omega \in \Omega \).
Note that (H1) implies that \({\hat{a}}_{\inf }(\cdot ),a(\cdot ),{\hat{a}}_{\sup }(\cdot )\in L^1 (\Omega , {\mathcal {F}},{\mathbb {P}})\), and that there are \({{\hat{a}}}, {\underline{a}}, {\bar{a}}\in {{\mathbb {R}}}^+\) and a measurable subset \(\Omega _0\subset \Omega \) with \({{\mathbb {P}}}(\Omega _0)=1\) such that
(see Lemma 2.1). Throughout this paper, \({{\hat{a}}}\) is referred to as the mean or average of \(a(\cdot )\), and \({\underline{a}}\) and \({\overline{a}}\) are referred to as the least mean and the greatest mean of \(a(\cdot )\), respectively.
Our main result on the stability of the constant solution \(u\equiv 1\) of (1.1) reads as follows.
Theorem 1.1
For every \(u_0\in C^{b}_{\mathrm{uinf}}({{\mathbb {R}}})\) with \(\inf _{x\in {{\mathbb {R}}}}u_0(x)>0\) and for every \(\omega \in \Omega \), we have that
where \(M(u_0):=\max \{1,\Vert u_0\Vert _{\infty }\}\cdot \max \Big \{\Big |1-\frac{1}{\min \{1,\inf _{x\in {{\mathbb {R}}}} u_0(x)\}} \Big |, \Big |1-\frac{1}{\max \{1,\sup _{x\in {{\mathbb {R}}}} u_0(x)\}} \Big |\Big \}\). Hence if \(\int _0^{\infty }a(\theta _s\omega )ds=\infty \), then
In particular, if (H1) holds, then for every \(0<{\tilde{a}}<{\underline{a}}\), every \(u_0\in C^{b}_{\mathrm{uinf}}({{\mathbb {R}}})\) with \(\inf _{x}u_0(x)>0\), and almost all \(\omega \in \Omega \), there is positive constant \(M>0\) such that
If \(a(\theta _{\cdot }\omega )\in L^{1}(0,\infty )\), then the constant equilibrium solution, \(u\equiv 1\), of (1.1) is not asymptotically stable.
To state our main results on the spreading speeds of (1.1), let
Let
Definition 1.1
For given \(\omega \in \Omega \), let
and
Let
\([c_{\inf }^*(\omega ),c_{\sup }^*(\omega )]\) is called the spreading speed interval of (1.1) with respect to compactly supported initial functions.
The following theorem shows that the spreading speed interval of (1.1) with respect to compactly supported initial functions is deterministic and is linearly determinate, that is, \( [c_{\inf }^*(\omega ),c_{\sup }^*(\omega )]=[{\underline{c}}^*,\bar{c}^*]\) for all \(\omega \in \Omega _0\).
Theorem 1.2
Assume that (H1) holds. Then the following hold.
-
(i)
For any \(\omega \in \Omega _0\), \(c_{\sup }^*(\omega )=\bar{c}^*\).
-
(ii)
For any \(\omega \in \Omega _0\), \(c_{\inf }^{*}(\omega )={\underline{c}}^*\).
The above theorem concerns the spreading speeds of solutions of (1.1) with compactly supported nonnegative initial functions. To consider the spreading speeds of solutions of (1.1) with front-like initial functions, let
Definition 1.2
For given \(\omega \in \Omega \), let
and
Let
\([{{\tilde{c}}}_{\inf }^*(\omega ),{{\tilde{c}}}_{\sup }^*(\omega )]\) is called the spreading speed interval of (1.1) with respect to front-like initial functions.
We have the following theorem on the spreading speeds of the solutions with front-like initial functions.
Theorem 1.3
Assume that (H1) holds. Then the following hold.
-
(i)
For any \(\omega \in \Omega _0\), \({{\tilde{c}}}_{\sup }^*(\omega )=\bar{c}^*\).
-
(ii)
For any \(\omega \in \Omega _0\), \({{\tilde{c}}}_{\inf }^{*}(\omega )={\underline{c}}^*\).
We also have the following theorem on the take-over property of the solutions of (1.1) with front-like initial functions and with the initial function \(u_0^*(x)=1\) for \(x< 0\) and \(u_0^*(x)=0\) for \(x>0\). Note that \(u(t,x;u_0^*,\omega )\) exists for all \(t>0\) (see [25, Theorem 1]).
Theorem 1.4
-
(i)
For a.e. \(\omega \in \Omega \),
$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{x(t,\omega )}{t}= {\hat{c}}^*, \end{aligned}$$(1.10)where \(x(t,\omega )\) is such that \(u(t,x(t,\omega );u_0^*,\omega )=\frac{1}{2}\). Moreover,
$$\begin{aligned} \lim _{t\rightarrow \infty }\sup _{x\ge ({\hat{c}}^*+h)t}u(t,x;u_0^*,\omega )=0, \forall \ h>0, \ \text {a.e }\ \omega \end{aligned}$$(1.11)and
$$\begin{aligned} \lim _{t\rightarrow \infty }\inf _{x\le ({\hat{c}}^*-h)t}u(t,x;u_0^*,\omega )=1, \forall \ h>0, \ \text {a.e }\ \omega . \end{aligned}$$(1.12) -
(ii)
For any \(u_0\in {{\tilde{X}}}_c^+\), it holds that
$$\begin{aligned} \lim _{t\rightarrow \infty }\sup _{x\ge ({\hat{c}}^*+h)t}u(t,x;u_0,\omega )=0, \forall \ h>0, \ \text {a.e }\ \omega \end{aligned}$$(1.13)and
$$\begin{aligned} \lim _{t\rightarrow \infty }\inf _{x\le ({\hat{c}}^*-h)t}u(t,x;u_0,\omega )=1, \forall \ h>0, \ \text {a.e }\ \omega . \end{aligned}$$(1.14)
Consider now (1.2). Define \({\underline{a}}_0\) and \({\overline{a}}_0\) by
Let (H2) be the following standing assumption.
(H2) \(0< {\underline{a}}_0\le {\overline{a}}_0<\infty \).
The assumption (H2) is the analogue of (H1). We will give some example for \(a_0(\cdot )\) satisfying (H2) in Sect. 5. Assume (H2). Let
For given \(u_0\in C_{\mathrm{unif}}^b({{\mathbb {R}}})\) with \(u_0\ge 0\) and \(s\in {{\mathbb {R}}}\), let \(u(t,x;u_0,\sigma _s a_0)\) be the solution of
with \(u(0,x;u_0,\sigma _s a_0)=u_0(x)\), where \(\sigma _s a_0(t)=a_0(s+t)\).
We have the following theorem on the spreading speeds of (1.2).
Theorem 1.5
Assume (H2). Then for every \(u_0\in X_c^+\),
and
We conclude the introduction with the following four remarks.
First, the results in Theorems 1.2–1.5 are new. If \(a_0(t)\) is periodic with period T, then \({\underline{a}}_0={\bar{a}}_0={{\hat{a}}}_0:=\frac{1}{T}\int _0^T a_0(\tau )d\tau \) and hence \({\underline{c}}_0^*={\bar{c}}_0^*=2\sqrt{{{\hat{a}}}_0}\). More generally, if \(a_0(t)\) in globally Hölder continuous and is uniquely ergodic in the sense that the space \(H(a_0)\) is compact and the flow \((H(a_0),\sigma _t)\) is uniquely ergodic, where \(H(a_0)=\mathrm{cl}\{\sigma _s a_0\,|\, s\in {{\mathbb {R}}}\}\) with open compact topology and \(\sigma _s a_0(\cdot )=a_0(\cdot +s)\), then \({\underline{a}}_0={\bar{a}}_0={{\hat{a}}}_0:=\lim _{T\rightarrow \infty }\frac{1}{T}\int _0^T a_0(\tau )d\tau \) and hence \({\underline{c}}_0^*={\bar{c}}_0^*=2\sqrt{{{\hat{a}}}_0}\). Therefore the existing results on spreading speeds of (1.2) in the time periodic and time almost periodic cases are recovered. The current paper provides a new and simpler proof in these special cases.
Second, by Theorems 1.2 and 1.3 ,
for any \(\omega \in \Omega _0\). Hence \([{\underline{c}}^*,{\bar{c}}^*]\) is called the spreading speed interval of (1.1), which is deterministic and is determined by the linearized equation of (1.1) at \(u\equiv 0\). Theorem 1.4 is an extension of the take-over property proved in [3] and [25] for (1.3). In order to prove Theorem 1.4 we are first led to prove that \(x(t,\omega ) \) is a subadditive process (see Lemma 5.4 for more detail). The fact that \(x(t,\omega )\) is a subadditive process is interesting. Its proof relies on comparison between various translation of the solution and on a zero-number argument enabling to bound the width of the interface. It is our belief that this result will open the way to other applications in the future.
Third, the results established for (1.1) and (1.2) can be applied to the following general random Fisher–KPP equation,
where \(r:\omega \rightarrow (-\infty ,\infty )\) and \(\beta :\Omega \rightarrow (0,\infty )\) are measurable with locally Hölder continuous sample paths \(r^\omega (t):=r(\theta _t\omega )\) and \(\beta ^\omega (t):=\beta (\theta _t\omega )\), and to the following nonautonomous Fisher–KPP equation,
where \(r_0:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) and \(\beta _0:{{\mathbb {R}}}\rightarrow (0,\infty )\) are locally Hölder continuous. Note that (1.19) models the population growth of a species with random perturbations on its growth rate and carrying capacity, and (1.20) models the population growth of a species with deterministic time dependent perturbations on its growth rate and carrying capacity.
In fact, under some assumptions on \(r(\omega )\) and \(\beta (\omega )\), it can be proved that
is an random equilibrium of (1.19). Let \({{\tilde{u}}}=\frac{u}{Y(\theta _t\omega )}\) and drop the tilde, (1.19) becomes (1.1) with \(a(\theta _t\omega )=\beta (\theta _t\omega )\cdot Y(\theta _t\omega )\), and then the results established for (1.1) can be applied to (1.19). For example, consider the following random Fisher–KPP equation,
where \(\omega \in \Omega \), \((\Omega , {\mathcal {F}},{\mathbb {P}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\) is an ergodic metric dynamical system, \(\xi :\Omega \rightarrow {{\mathbb {R}}}\) is measurable, and \(\xi _t(\omega ):=\xi (\theta _t\omega )\) is locally Hölder continuous (\(\xi _t\) denotes a real noise or a colored noise). Let \({\hat{\xi }}_{\inf }(\omega )\) and \({\hat{\xi }}_{\sup }(\omega )\) be defined as in (1.4) and (1.5) with \(a(\cdot )\) being replaced by \(\xi (\cdot )\), respectively. Assume that \(\xi _t(\cdot )\) satisfies the following (H3).
(H3) \(\xi :\Omega \rightarrow {{\mathbb {R}}}\) is measurable; \(\int _\Omega |\xi (\omega )|d{\mathbb {P}}(\omega )<\infty \) and \(\int _\Omega \xi (\omega )d{\mathbb {P}}(\omega )=0\); \(-1<{ {\hat{\xi }}_{\inf }(\omega )}\le {{\hat{\xi }}_{\sup }(\omega )}<\infty \) and\({ \inf _{t\in {{\mathbb {R}}}}\xi (\theta _{t}\omega )}>-\infty \) for a.e. \(\omega \in \Omega \); and \(\xi ^\omega (t):=\xi (\theta _t\omega )\) is locally Hölder continuous.
Assume (H3). By the arguments of Lemma 2.1, there are \({\underline{\xi }},{\overline{\xi }}\in {{\mathbb {R}}}\) such that \({\hat{\xi }}_{\inf }(\omega )={{\underline{\xi }}}\) and \({\hat{\xi }}_{\sup }(\omega )={{\overline{\xi }}}\) for a.e. \(\omega \in \Omega \). It can be proved that
is a spatially homogeneous asymptotically stable random equilibrium of (1.21) (see Theorem 3.2 and Corollary 3.1). It can also be proved that for any \(u_0\in X_c^+\),
and
for a.e. \(\omega \in \Omega \). where \(u(t,x;u_0,\theta _s\omega )\) is the solution of (1.21) with \(\omega \) being replaced by \(\theta _s\omega \) and \(u(0,x;u_0,\theta _s\omega )=u_0(x)\) (see Corollary 4.1).
Fourth, it is interesting to study the spreading properties of (1.1) with (H1) being replaced by the following weaker assumption,
(H1)\('\) \(0<{{\hat{a}}}:=\int _\Omega a(\omega )d{\mathbb {P}}(\omega )<\infty \).
We plan to study this general case somewhere else, which would have applications to the study of the spreading properties of the following stochastic Fisher–KPP equation,
where \(W_t\) denotes the standard two-sided Brownian motion (\(dW_t\) is then the white noise). In fact, let \( \Omega :=\{\omega \in C({{\mathbb {R}}},{{\mathbb {R}}})\ |\ \omega (0)=0\ \}\) equipped with the open compact topology, \({\mathcal {F}}\) be the Borel \(\sigma -\)field and \({\mathbb {P}}\) be the Wiener measure on \((\Omega , {\mathcal {F}})\). Let \(W_t\) be the one dimensional Brownian motion on the Wiener space \((\Omega ,{\mathcal {F}},{\mathbb {P}})\) defined by \(W_t(\omega )=\omega (t)\). Let \(\theta _t\omega \) be the canonical Brownian shift: \((\theta _t\omega )(\cdot )=\omega (t+\cdot )-\omega (t)\) on \(\Omega \). It is easy to see that \(W_t(\theta _s\omega )=W_{t+s}(\omega )-W_s(\omega )\). If \(\frac{\sigma ^2}{2}<1\), then it can be proved that
is a spatially homogeneous stationary solution process of (1.23). Let \({{\tilde{u}}}=\frac{u}{Y(\theta _t\omega )}\) and drop the tilde, (1.23) becomes (1.1) with \(a(\theta _t\omega )= Y(\theta _t\omega )\). The reader is referred to [18,19,20, 23, 39, 40] for some study on the front propagation dynamics of (1.24). Note that Theorem 1.4 (i) is an analogue of [20, Theorem 1].
It is important to note that the authors of the work [10] studied the asymptotic spreading speeds for space-time heterogeneous equations of the form
where \(f(t,x,0)=f(t,x,1)=0\). We note that Theorem 1.5 improves [10, Proposition 3.9], since \(\inf _{t\in {{\mathbb {R}}}}a_0(t)=0\) and \(\sup _{t\in \mathbb {R}}a_0(t)=+\infty \) are allowed here. Moreover, the techniques developed in the current work are different from the ones in [10]. Certainly, it should be mentioned that (1.25) is more general than (1.2).
The rest of the paper is organized as follows. In Sect. 2, we present some preliminary lemmas, which will be used in the proofs of main results of the current paper in later sections. In Sect. 3, we establish some results about the stability of the positive constant equilibrium solution \(u\equiv 1\) of (1.1) (resp. (1.2)) and prove Theorem 1.1. In Sect. 4, we study the spreading properties of solutions of (1.1) with nonnegative and compactly supported initial functions or front like initial functions and prove Theorems 1.2 and 1.3 . We investigate in Sect. 4 the take-over property of (1.1) and prove Theorem 1.4. We consider spreading properties of (1.2) in Sect. 5.
2 Preliminary Lemmas
In this section, we present some preliminary lemmas to be used in later sections of this paper as well as in the second part of the series.
Lemma 2.1
(H1) implies that \({\hat{a}}_{\inf }(\cdot ),a(\cdot ), { {{\hat{a}}}_{\sup }(\cdot )}\in L^1 (\Omega , {\mathcal {F}},{\mathbb {P}})\) and that there are \({\underline{a}}, {\bar{a}}, {{\hat{a}}}\in {{\mathbb {R}}}^+\) and a measurable subset \(\Omega _0\subset \Omega \) with \({{\mathbb {P}}}(\Omega _0)=1\) such that \(\theta _t\Omega _0=\Omega _0\) for all \(t\in {{\mathbb {R}}}\), \({{\hat{a}}}_{\inf }(\omega ) ={\underline{a}}\) and \({{\hat{a}}}_{\sup }(\omega )={\bar{a}}\) for all \(\omega \in \Omega _0\), and \(\lim _{t\rightarrow \pm \infty }\frac{1}{t}\int _0^t a(\theta _\tau \omega )d\tau ={{\hat{a}}}\) for all \(\omega \in \Omega _0\).
Proof
First, let
and
Then \({\Omega _\infty \cup } \cup _{n=1}^\infty \Omega _n=\Omega \). By (H1), there is \({\bar{n}}\in {{\mathbb {N}}}\) such that \({{\mathbb {P}}}(\Omega _{{\bar{n}}})>0\). By (1.6), \(\theta _t\Omega _n=\Omega _n\) for all \(t\in {{\mathbb {R}}}\) and \(n\in {{\mathbb {N}}}\). Then by the ergodicity of the metric dynamical system \((\Omega , {\mathcal {F}},{\mathbb {P}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\), we have \({\mathbb {P}}(\Omega _{{\bar{n}}})=1\). This implies that \({\hat{a}}_{\sup }(\cdot )\in L^1 (\Omega , {\mathcal {F}},{\mathbb {P}})\), and then \({\hat{a}}_{\inf }(\cdot )\in L^1 (\Omega , {\mathcal {F}},{\mathbb {P}})\). Moreover, by (1.6),
and
It then follows that there are \({\underline{a}},{\bar{a}}\in {{\mathbb {R}}}\) and a measurable subset \(\Omega _1\subset \Omega \) with \({{\mathbb {P}}}(\Omega _1)=1\) such that \(\theta _t\Omega _1=\Omega _1\) for all \(t\in {{\mathbb {R}}}\), and \({{\hat{a}}}_{\inf }(\omega )={\underline{a}}\) and \({{\hat{a}}}_{\sup }(\omega )={\bar{a}}\) for all \(\omega \in \Omega _1\).
Next, for given \(n\in {{\mathbb {N}}}\), let
Then \(a_n(\cdot )\in L^1 (\Omega , {\mathcal {F}},{\mathbb {P}})\), \(0<a_1(\omega )\le a_2(\omega )\le \cdots \), and \(\lim _{n\rightarrow \infty } a_n(\omega )=a(\omega )\). By the ergodicity of the metric dynamical system \((\Omega , {\mathcal {F}},{\mathbb {P}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\), we have that for a.e. \(\omega \in \Omega \),
This together with the Monotone Convergence Theorem implies that
Therefore, \(a(\cdot )\in L^1 (\Omega , {\mathcal {F}},{\mathbb {P}})\), and moreover, by the ergodicity of the metric dynamical system \((\Omega , {\mathcal {F}},{\mathbb {P}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\), there are \({{\hat{a}}}\in {{\mathbb {R}}}\) and a measurable subset \(\Omega _2\subset \Omega \) with \({{\mathbb {P}}}(\Omega _2)=1\) such that \(\theta _t \Omega _2=\Omega _2\) for all \(t\in {{\mathbb {R}}}\), and
The lemma thus follows with \(\Omega _0=\Omega _1\cap \Omega _2\). \(\square \)
Lemma 2.2
Suppose that \(b\in C({{\mathbb {R}}}, (0,\infty ))\) and that \(0<\underline{ b}\le \overline{ b}<\infty \), where
Then
Proof
The proof of this lemma follows from a proper modification of the proof of [33, Lemma 3.2]. For the sake of completeness we give a proof here. Let \(0<\gamma <{\underline{b}}\). By \({\overline{b}}<\infty \), there is \(T>0\) such that
Define
It is clear that \(B\in W^{1,\infty }_{\mathrm{loc}}({{\mathbb {R}}}) \cap L^\infty ({{\mathbb {R}}})\) with
Furthermore, it follows from (2.2) that \(\Vert B\Vert _{\infty }\le 2T{\overline{b}}\) and that \(\gamma <\varepsilon _k\) for every \(k\in {\mathbb {Z}}\). Hence (2.3) implies that
Since \(\gamma \) is arbitrarily chosen less than \({\underline{b}}\) we deduce that
On the other hand for each given \(B\in W^{1,\infty }_{\mathrm{loc}}({{\mathbb {R}}})\cap L^{\infty }({{\mathbb {R}}})\) and \(t>s\) we have
Hence
This completes the proof of the lemma. \(\square \)
In the following, let \(b\in C({{\mathbb {R}}}, (0,\infty ))\) be given and satisfy that \(0<{\underline{b}}\le {\overline{b}}<\infty \). Consider
For given \(u_0\in C_{\mathrm{unif}}^b({{\mathbb {R}}})\) with \(u_0\ge 0\), let \(u(t,x;u_0,b)\) be the solution of (2.4) with \(u(0,x;u_0,b)=u_0(x)\).
For every \(0<\mu <{\underline{\mu }}^*:=\sqrt{{\underline{b}}}\), \(x\in {{\mathbb {R}}}\), \(t\in {{\mathbb {R}}}\) and \(\omega \in \Omega \), let
and
Then the function \( \phi ^{\mu }\) satisfies
Lemma 2.3
Let
Then
Proof
It follows directly from the comparison principle for parabolic equations. \(\square \)
Lemma 2.4
For every \(\mu \) with \(0<\mu<{\tilde{\mu }}<\min \{2\mu , {\underline{\mu }}^*\}\), there exist \(\{t_k\}_{k\in {{\mathbb {Z}}}}\) with \(t_k<t_{k+1}\) and \(\lim _{k\rightarrow \pm \infty }t_k=\pm \infty \), \(B_b\in W^{1,\infty }_{\mathrm{loc}}({{\mathbb {R}}})\cap L^{\infty }({{\mathbb {R}}})\) with \(B_b(\cdot )\in C^1((t_k,t_{k+1}))\) for \(k\in {{\mathbb {Z}}}\), and a positive real number \(d_b\) such that for every \(d\ge d_{b}\) the function
satisfies
for \(t\in (t_k,t_{k+1})\), \(x\ge C(t,b,\mu )+ \frac{\ln d}{{{\tilde{\mu }}}-\mu }+\frac{ B_b(t)}{\mu },\,\, k\in {{\mathbb {Z}}}\).
Proof
First of all, for given \(0<\mu<{\tilde{\mu }}<\min \{2\mu , {\underline{\mu }}^*\}\), let \(0<\delta \ll 1\) such that \((1-\delta ){\underline{b}}>{\tilde{\mu }}\mu \). It then follows from the arguments of Lemma 2.2 that there exist \(T>0\) and \(B_b\in W^{1,\infty }_{\mathrm{loc}}({{\mathbb {R}}})\cap L^{\infty }({{\mathbb {R}}})\) such that \(B_b\in C^1((t_k,t_{k+1}))\), where \(t_k=kT\) for \(k\in {{\mathbb {Z}}}\), and
Next, fix the above \(\delta >0\) and \(B_b(t)\). Let \(d>1\) to be determined later. Let \(\xi (t,x)=x-C(t;b,\mu )\). We have
for \(t\in (t_k,t_{k+1})\).
Observe now that
For this choice of d, if \( \phi ^{\mu ,d,B_b}(t,x)\ge 0\), which is equivalent to \(\xi (t,x)=x-C(t;b,\mu )\ge \frac{\ln d}{{{\tilde{\mu }}}-\mu }+\frac{ B_b(t)}{\mu }\), then \( \xi (t,x)\ge 0 \) and each term in the expression at the right hand side of (2.8) is less or equal to zero. The lemma thus follows. \(\square \)
Recall that \(u_0^*(x)=1\) for \(x< 0\) and \(u_0^*(x)=0\) for \(x>0\). By [25, Theorem 1], the solution of (2.4) with initial function \(u_0^*\), denoted by \(u(t,x;u_0^*,b)\), exists for \(t>0\).
Lemma 2.5
Suppose that \(u_\epsilon \in C_{\mathrm{unif}}^b({{\mathbb {R}}})\) with \(u_\epsilon \ge 0\) and \(\lim _{\epsilon \rightarrow 0}\int _{-\infty }^\infty |u_\epsilon (x)-u_0^*(x)|dx =0.\) Then for each \(t>0\),
Proof
See [25, Theorem 8]. \(\square \)
Lemma 2.6
For given \(u_i\in C_{\mathrm{unif}}^b({{\mathbb {R}}})\) with \(u_i\ge 0\) \((i=1,2)\), if \(u_1(x)-u_2(x)\) has exactly one simple zero \(x_0\) and \(u_1(x)>u_2(x)\) for \(x<x_0\) and \(u_1(x)<u_2(x)\) for \(x>x_0\), then for any \(t>0\), there is \(\xi (t)\in [-\infty ,\infty ]\) such that
Proof
Let \(v(t,x)=u(t,x;u_1,b)-u(t,x;u_2,b)\). Then v(t, x) satisfies
where \(q(t,x)=b(t)-b(t)(u(t,x;u_1,b)+u(t,x;u_2,b))\). Note that v(0, x) has exactly one simple zero \(x_0\) and \(v(0,x)>0\) for \(x<x_0\), \(v(x)<0\) for \(x>x_0\). The lemma then follows from [1, Theorems A,B]. \(\square \)
Let x(t, b) and \(x_+(t,b)\) be such that
Lemma 2.7
For any \(t>0\), there holds
Proof
First, let \(\phi _n(x)=\min \{1-\frac{1}{n},\phi ^\mu (0,x;b)\}\). Then \(\lim _{n\rightarrow \infty }\phi _n(x)=\phi _+^\mu (0,x;b)\) uniformly in \(x\in {{\mathbb {R}}}\). Then for any given \(t>0\),
uniformly in \(x\in {{\mathbb {R}}}\). Let \(x_+^n(t,b)\) be such that \(u(t,x_+^n(t,b);\phi _n,b)=\frac{1}{2}.\) We have
Next, for given \(n\ge 1\), let \(u_\epsilon ^*(x)\) be a nonincreasing function such that \(u_\epsilon ^*\in C_{\mathrm{unif}}^b({{\mathbb {R}}})\); \(u_\epsilon ^*(x)=1\) for \(x\ll -1\) and \(u_\epsilon ^*(x)=0\) for \(x\gg 0\); \(u_\epsilon ^*(x)-\phi _n(x+h)\) has exactly one simple zero for any \(h\in {{\mathbb {R}}}\); and
Let \(x_\epsilon (t,b)\) be such that
By Lemma 2.6, for any \(t>0\),
By Lemma 2.5, for any \(t>0\),
Letting \(\epsilon \rightarrow 0\), we get
Letting \(n\rightarrow \infty \), the lemma follows. \(\square \)
Lemma 2.8
Let \(F: {{\mathbb {R}}}\times \Omega \rightarrow {{\mathbb {R}}}\) be measurable in \(\omega \in \Omega \) and continuous hemicompact in \(x\in {{\mathbb {R}}}\) (i.e for every \(\omega \in \Omega \), \(F(\cdot ,\omega )\) is continuous in x and any sequence \(\{x_n\}_{n\ge 1}\subset {{\mathbb {R}}}\) with \(|x_n-F(x_n,\omega )|\rightarrow 0\) as \(n\rightarrow \infty \) has a convergent subsequence). Then F has a deterministic fixed point (i.e there is \(X: \Omega \rightarrow {{\mathbb {R}}}\) such that \(F(X(\omega ),\omega )=X(\omega )\)) if and only if F has random fixed point (i.e there is a measurable function \(X: \Omega \rightarrow {{\mathbb {R}}}\) such that \(F(X(\omega ),\omega )=X(\omega )\)).
Proof
See [42, Lemma 4.7] \(\square \)
Lemma 2.9
Let \(f : {{\mathbb {R}}}\times \Omega \rightarrow (0,1) \) be a measurable function such that for every \(\omega \in \Omega \) the function \(f^{\omega }:=f(\cdot ,\omega ) : {{\mathbb {R}}}\rightarrow (0,1)\) is continuously differentiable and strictly decreasing. Assume that \(\lim _{x\rightarrow -\infty }f^{\omega }(x)=1\) and \(\lim _{x\rightarrow \infty }f^{\omega }(x)=0\) for every \(\omega \in \Omega \). Then for every \(a\in (0,1)\) the function \(\Omega \ni \omega \mapsto f^{\omega ,-1}(a)\in {{\mathbb {R}}}\) is measurable, where \(f^{\omega ,-1}\) denotes the inverse function of \(f^{\omega }\).
Proof
Let \(a\in (0,1)\) be given. Note that for every \(\omega \in \Omega \), we have that \(f^{\omega ,-1}(a)\) is the unique fixed point of the function
Note that
Hence the function \(F(x,\omega )\) is hemicompact in x. By Lemma 2.8, the function \(\Omega \ni \omega \mapsto f^{\omega ,-1}(a)\) is measurable. The lemma is thus proved. \(\square \)
3 Stability of Positive Random Equilibrium Solutions
In this section, we establish some results about the stability of the positive constant equilibrium solution \(u\equiv 1\) of (1.1) (resp. (1.2)). We also study the existence and stability of positive random equilibria of (1.21). The results obtained in this section will play a role in later sections for the investigation of spreading speeds and take-over property of solutions of (1.1) [resp. (1.2)].
3.1 Stability of the Positive Constant Equilibrium Solution \(u\equiv 1\) of (1.1)
In this subsection, we establish some results about the stability of the positive constant equilibrium solution \(u\equiv 1\) of (1.1) [resp. (1.2)]. Observe that \(u(t,x)=v(t,x-C(t;\omega ))\) with \(C(t;\omega )\) being differential in t solves (1.1) if and only if v(t, x) satisfies
where \(c(t;\omega )=C'(t;\omega )\). In this subsection, we also study the stability of the positive constant equilibrium solution \(u\equiv 1\) of (3.1).
We first prove Theorem 1.1.
Proof of Theorem 1.1
First, for given \(u_0\in C^{b}_{\mathrm{uinf}}({{\mathbb {R}}})\) with \(\inf _{x\in {\mathbb {R}}}u_0(x)>0\) and \(\omega \in {\Omega }\), let \({\underline{u}}_0:=\min \{1, \inf _{x\in {\mathbb {R}}}u_0(x)\}\) and \({\overline{u}}_0:=\max \{1,\sup _{x\in {\mathbb {R}}}u_0(x)\}\). By the comparison principle for parabolic equations, we have that
and
Since \({\underline{u}}_0\) and \({\overline{u}}_0\) are positive numbers, by the uniqueness of solutions of (1.1) and its corresponding ODE with a given initial function, we have that
Next, let \({\underline{u}}(t)=\Big (\frac{1}{u(t,0;{\underline{u}}_0,\omega )} -1\Big )e^{\int _0^ta(\theta _s\omega )ds}\) and \({\overline{u}}(t)=\Big (1-\frac{1}{u(t,0;{\overline{u}}_0,\omega )}\Big ) e^{\int _0^ta(\theta _s\omega )ds}\). It can be verified directly that
Hence,
which is equivalent to
and
Now, by (3.2)–(3.5), we have that
which implies that inequality (1.8) holds. Taking \(u_0\) to be a positive constant with \( 0<u_0<1\), it follows from (3.4) that
If \(\Vert a(\theta _{\cdot }\omega )\Vert _{L^{1}(0,\infty )}<\infty \), then \(\lim _{t\rightarrow \infty }u(t,x;{\underline{u}}_0,\omega )=\frac{1}{1+(\frac{1}{{\underline{u}}_0}-1)e^{-\Vert a(\theta _{\cdot }\omega )\Vert _{L^{1}(0,\infty )}}}<1\), which completes the proof of the theorem. \(\square \)
Remark 3.1
-
(1)
Theorem 1.1 guarantees the exponential stability of the trivial constant equilibrium solution \(u\equiv 1\) of (1.1) with respect to the solutions \(u(t,x;u_0,\omega )\) with \(\inf _{x\in {\mathbb {R}}^n}u_0(x)>0\) provided that (H1) holds. This result will be useful in the later sections.
-
(2)
Let \(v(t,x;u_0,\omega )\) be the solution of (3.1) with \(v(0,x;u_0,\omega )=u_0(x)\). The result in Theorem 1.1 also holds for \(v(t,x;u_0,\omega )\).
Let
Next, we prove the following theorem about the stability of \(u\equiv 1\).
Theorem 3.1
Assume (H1). Suppose that \(v(t,x;\omega )\) with \(0<v(t,x;\omega )<1\), is an entire solution of (3.1) which is nonincreaing in x. For given \(\omega \in \Omega \) with \(0<{{\underline{c}}(\omega )\le {\overline{c}}(\omega )}<\infty \), if there is \(x^*\in {{\mathbb {R}}}\) such that \(\inf _{t\in {{\mathbb {R}}}} v(t,x^*;\omega )>0\), then \(\lim _{x\rightarrow -\infty } v(t,x;\omega )=1\) uniformly in \(t\in {{\mathbb {R}}}\).
To prove the above theorem, we first prove a lemma.
Lemma 3.1
Let \(u_0,u_n\in C_{\mathrm{unif}}^b({{\mathbb {R}}})\) be such that \(0\le u_n(x)\le u_0(x)\le 1\). Let \(v(t,x;u_0,\theta _{t_0}\omega )\) (respectively \(v(t,x;u_n,\theta _{t_0}\omega )\)) denote the solution of (3.1) with \(\omega \) being replaced by \(\theta _{t_0}\omega \) and with initial function \( u_0\) (respectively \(u_n \)). If \(\lim _{n\rightarrow \infty } u_n(x)=u_0(x)\) locally uniformly in \(x\in {{\mathbb {R}}}\), then for any fixed \(t>0\) with \(-\infty<\inf _{t_0\in {{\mathbb {R}}}}\int _{0}^{t} c(\tau +t_0;\omega )\le \sup _{t_0\in {{\mathbb {R}}}} \int _0^t c(\tau +t_0;\omega )<\infty \), we have
uniformly in \(t_0\in {{\mathbb {R}}}\) and locally uniformly in \(x\in {{\mathbb {R}}}\).
Proof
Fix \(\omega \in \Omega \). For every \(n\ge 1\), the function \(v^n(t,x;t_0):=v(t,x;u_0,\theta _{t_0}\omega )-v(t,x;u_n,\theta _{t_0}\omega )\) is non-negative and satisfies
It follows that, for every \(n\ge 1\), \({{\tilde{v}}}^n(t,x;t_0):=v^n(t,x-\int _{t_0}^{t_0+t}c(\tau ;\omega )d\tau );t_0)\) satisfies
By the comparison principle for parabolic equations,
Note that \(\lim _{n\rightarrow \infty } \big ( e^{\int _{t_0}^{t_0+t}a(\theta _{\tau }\omega )d\tau }e^{t\Delta } v^n(0,\cdot +\int _0^tc(\tau +t_0)d\tau )\big )(x)=0\) locally uniformly in \(x\in {{\mathbb {R}}}\) and uniformly in \(t_0\in {{\mathbb {R}}}\). Hence \(\lim _{n\rightarrow \infty } v^n(t,x;t_0)=0\) uniformly in \(t_0\in {{\mathbb {R}}}\) and locally uniformly in \(x\in {{\mathbb {R}}}\). \(\square \)
We now prove Theorem 3.1.
Proof of Theorem 3.1
Fix \(\omega \in \Omega \) with \(-\infty<{\underline{c}}(\omega )\le {\overline{c}}(\omega )<\infty \) and assume that there is \(x^*\in {{\mathbb {R}}}\) such that \(\inf _{t\in {{\mathbb {R}}}} v(t,x^*;\omega )>0\).
Consider the constant function \(u_0\equiv \inf _{t\in {{\mathbb {R}}}} v(t,x^*;\omega )\). We first note from the hypotheses of Theorem 3.1 that \(u_0>0\). Next, let \({{\tilde{u}}}_0(\cdot )\) be uniformly continuous, \(0\le {{\tilde{u}}}_0(x)\le u_0\), \({{\tilde{u}}}_0(x)=u_0\) for \(x\le x^*-1\), and \({{\tilde{u}}}_0(x)=0\) for \(x\ge x^*\). For any \(R>0\), it holds that \({\tilde{u}}_0(x-n)=u_0\) for every \(|x|\le R\) and \(n\ge R+1+|x^*|\). This shows that \(\lim _{n\rightarrow \infty } {{\tilde{u}}}_0(x-n)=u_0\) locally uniformly in \(x\in {{\mathbb {R}}}\). By (H1) and the arguments of Theorem 1.1,
uniformly in \(t_0\in {{\mathbb {R}}}\) and \(x\in {{\mathbb {R}}}\). Hence, for any \(\epsilon >0\), there is \(T>0\) such that
and
By Lemma 3.1, there is \(N>1\) such that
This implies that
Note that
and
Hence
The theorem thus follows. \(\square \)
3.2 Existence and Stability of Positive Random Equilibria of (1.21)
In this subsection, we first study the existence and stability of positive random equilibria of (1.21), and then show that (1.21) can be transferred to (1.1).
To this end, we consider the following corresponding ODE,
Throughout this subsection, we assume that (H3) holds. For given \(u_0\in {{\mathbb {R}}}\), let \(u(t;u_0,\omega )\) be the solution of (3.6) with \(u(0;u_0,\omega )=u_0\). It is known that
Theorem 3.2
\(Y(\omega )=\frac{1}{\int _{-\infty }^0 e^{ s+\int _0^s \xi (\theta _\tau \omega )d\tau }ds}\) is a random equilibrium of (3.6), that is, \(u(t;Y(\omega ),\omega )=Y(\theta _t\omega )\) for \(t\in {{\mathbb {R}}}\) and \(\omega \in \Omega \).
Proof
First, we note that
Second, note that
Hence \(u(t;Y(\omega ),\omega )=Y(\theta _t\omega )\) and then \(Y(\omega )\) is a random equilibrium of (3.6). \(\square \)
Observe that \(0<Y(\omega )<\infty \). Let \({{\tilde{u}}}=\frac{u}{Y(\theta _t\omega )}\) and drop the tilde. We have
Clearly, (3.7) is of the form (1.1) with \(a(\omega )=Y(\omega )\). Let \(\hat{Y}_{\inf }(\omega )\) and \(\hat{Y}_{\sup }(\omega )\) be defined as in (1.4) and (1.5) with \(a(\cdot )\) being replaced by \(Y(\cdot )\), respectively.
Lemma 3.2
\(Y(\omega )\) satisfies the following properties.
-
(1)
For a.e. \(\omega \in \Omega \), \(0<{ \inf _{ t\in {{\mathbb {R}}}}Y(\theta _t\omega )\le \sup _{t\in {{\mathbb {R}}}}Y(\theta _t\omega )}<\infty \).
-
(2)
For a.e. \(\omega \in \Omega \), \(\lim _{t\rightarrow \infty } \frac{\ln Y(\theta _t\omega )}{t}=0\).
-
(3)
For a.e. \(\omega \in \Omega \), \(\lim _{t\rightarrow \infty } \frac{\int _0^t Y(\theta _s\omega )ds}{t}=1\).
-
(4)
\({\hat{Y}_{\inf }}(\omega )=1+{\underline{\xi }}>0\), and \({\hat{Y}_{\sup }}(\omega )=1+{\overline{\xi }}<\infty \) for a.e. \(\omega \in \Omega \).
Proof
(1) First, note that
By (H3), for every \(\lambda \in (0,1)\) and a.e. \(\omega \in \Omega \) there is \(T_{\lambda }\gg 1\),
It then follows that
That is,
The first inequality of (3.9) combined with (3.8) yields that
Hence
Next, let \(\xi _{\inf }(\omega )=\inf _{t\in {{\mathbb {R}}}}\xi (\theta _t\omega )\). Observe that
This combined with the second inequality in (3.9) yield that
It easily follows from (3.10) and (3.11) that
The result (1) then follows.
(2) It follows from (1).
(3) Note that
Integrating both sides with respect to t, we obtain that
The result (3) follows from (2) and the fact that \(\lim _{t\rightarrow \infty }\frac{1}{t}\int _0^t\xi (\theta _s\omega )ds=0\) for a.e. \(\omega \in \Omega \).
(4) Observe that (3.12) implies that
and
for a.e. \(\omega \in \Omega \). It follows from (1) that
Hence we have that \( {\hat{Y}_{\inf }(\omega )}=1+{\underline{\xi }}>0\) for a.e. \(\omega \in \Omega \). Similar arguments yield that \( {\hat{Y}_{\sup }(\omega )}=1+{\overline{\xi }}\) for a.e. \(\omega \in \Omega \). \(\square \)
Corollary 3.1
For given \(u_0\in C^{b}_{\mathrm{uinf}}({{\mathbb {R}}})\) with \(\inf _{x}u_0(x)>0\), for a.e. \(\omega \in \Omega \),
uniformly in \(t_0\in {{\mathbb {R}}}\), where \(u(t,x;u_0,\theta _{t_0}\omega )\) is the solution of (1.21) with \(u(0,x;u_0,\theta _{t_0}\omega )=u_0(x)\).
Proof
It follows from Theorems 1.1, 3.2, and Lemma 3.2. \(\square \)
4 Deterministic and Linearly Determinate Spreading Speed Interval
In this section, we discuss the spreading properties of solutions of (1.1) with nonempty compactly supported initials or front like initials and prove Theorems 1.2 and 1.3 .
We first prove some lemmas.
Lemma 4.1
Let \(\omega \in \Omega _0\). If there is a positive constant \(c(\omega )>0\) such that
then \(c^*_{\inf }(\omega )\ge c(\omega )\). Therefore it holds that
Proof
Let \(\omega \in \Omega _0\) and \(c(\omega )\) satisfy (4.1). Let \(0<c<c(\omega )\) and \(u_0\in X_c^+\) be given. Choose \({\tilde{c}}\in (c, c(\omega ))\). It follows from (4.1) that
There is \(T\gg 1\) such that
Suppose by contradiction that there is \((s_n,t_n,x_n)\in {{\mathbb {R}}}\times {{\mathbb {R}}}^+\times {{\mathbb {R}}}\) with \(|x_n|\le ct_n\) for every \(n\ge 1\) and \(t_n\rightarrow \infty \) such that
Let \(0<\varepsilon <1\) be fixed. By (H1), Theorem 1.1 implies that there is \({\tilde{T}}_\varepsilon >T\) such that
Observe that \( ({\tilde{c}}-c)(t_n-{\tilde{T}}_\varepsilon )-2c{\tilde{T}}_\varepsilon \rightarrow \infty \) as \(n\rightarrow \infty \). Then there is \(n_\varepsilon \) such that
For every \(n\ge n_\varepsilon \), let \(u_{0n}\in C^{b}_{\mathrm{unif}}({{\mathbb {R}}})\) with \(\Vert u_{0n}\Vert _{\infty }\le \frac{m_{{\tilde{c}}}}{2}\) and
Since \(|x|\le ({\tilde{c}}-c)(t_n-{\tilde{T}}_\varepsilon )-c{\tilde{T}}_\varepsilon \) implies that \(|x+x_n|\le {\tilde{c}}(t_n-{\tilde{T}}_\varepsilon )\) for every \(n\ge n_\varepsilon \), it follows from (4.3) to (4.6) that
By the comparison principle for parabolic equations, we have
where \({\tilde{s}}_n=s_n+t_n-{\tilde{T}}_\varepsilon \).
Observe from the definition of \(u_{0n}\) that \(u_{0n}(x)\rightarrow \frac{m_{{\tilde{c}}}}{2}\) as \(n\rightarrow \infty \) locally uniformly in \(x\in {{\mathbb {R}}}\). It then follows from Lemma 3.1 that for every \(t>0\),
By (4.5), we have that
This combined with (4.7) and (4.8) yields that
On the other hand, since \(\Vert u_0(\cdot +x_n)\Vert _{\infty }=\Vert u_0\Vert _{\infty }\) for every \(n\ge 1\), it follows from the comparison principle for parabolic equations that
This together with (4.5) implies that
which combined with (4.9) yields that
Letting \(\varepsilon \rightarrow 0\), we obtain that
which contradicts to (4.4). Thus we have that
This implies that \(c^*_{\inf }(\omega )\ge c(\omega )\).
Therefore, we have that
On the other hand, it is clear from the definition of \(C^{*}_{\sup }(\omega )\) that
The lemma is thus proved. \(\square \)
Lemma 4.2
Let \(b>0\) be a positive number and \(v_0\in X_c^+\). Let \(v(t,x;v_0,b)\) be the solution of
Then
Proof
It follows from [3, Page 66, Corollary 1]. \(\square \)
Lemma 4.3
Assume (H1). Then for every \(\omega \in \Omega _0\),
Therefore, \( c^{*}_{\inf }(\omega )\ge 2\sqrt{{\underline{a}}}, \quad \forall \ \omega \in \Omega _0.\)
Proof
First, fix \(\omega \in \Omega _0\) and \(u_0\in X_c^+\). Let \(0<c<2\sqrt{{\underline{a}}}\) be given. Choose \(b>c\) and \(0<\delta <1\) such that \( c<2\sqrt{b}<2\sqrt{\delta {\underline{a}}}.\) By the proof of Lemma 2.2, there are \(\{t_{k}\}_{k\in {{\mathbb {Z}}}}\) with \(t_k<t_{k+1}\), \(t_{k}\rightarrow \pm \infty \) as \(k\rightarrow \pm \infty \) and \(A\in W^{1,\infty }_{loc}({{\mathbb {R}}})\cap L^\infty ({{\mathbb {R}}})\) such that \(A\in C^1(t_k,t_{k+1})\) for every k and
Let \( \sigma =\frac{(1-\delta )e^{-\Vert A\Vert _{\infty }}}{\Vert u_0\Vert _{\infty }+1}\) and \(v(t,x;b)=v(t,x;u_0,b)\). By Lemma 4.2, we have that
Next, for given \(s\in {{\mathbb {R}}}\), let \({\tilde{v}}(t,x;s)=\sigma e^{A(t+s)}v(t,x;b)\). By the comparison principle for parabolic equations, we have that
Hence, it follows from the definition of \(\sigma \) that
Thus for any \(s\in {{\mathbb {R}}}\),
Note that
By the comparison principle for parabolic equations again, we have that
This combined with (4.11) yields that
Hence (4.10) holds. By (4.10) and Lemma 4.1, we have \( c^{*}_{\inf }(\omega )\ge 2\sqrt{{\underline{a}}}, \quad \forall \ \omega \in \Omega _0.\) \(\square \)
Now, we prove Theorem 1.2.
Proof of Theorem 1.2
(i) We first prove \( c^*_{\sup }(\omega )\le 2\sqrt{{\overline{a}}}\) for all \(\omega \in \Omega _0.\)
Suppose that \(\mathrm{supp}(u_0)\subset (-R,R)\). For every \(\mu >0\), let \(C_\mu (t,s)=\int _s^{s+t}\frac{\mu ^2+a(\theta _\tau \omega )}{\mu }d\tau \) and \(\phi ^\mu (x)=\Vert u_0\Vert _{\infty }e^{-\mu ( x-R)} \) and \({\tilde{\phi }}^{\mu }_{\pm }(t,x;s)=\phi ^\mu (\pm x-C_\mu (t,s))\) for every \(x\in {{\mathbb {R}}}\) and \(t\ge 0\). Then
and
By the comparison principle for parabolic equations, we have
This implies that
For any \(c>{\bar{c}}^*=2\sqrt{{\bar{a}}}=\inf _{\mu >0}\frac{\mu ^2+\sqrt{{\bar{a}}}}{\mu }\), choose \(\mu >0\) such that \(c>\frac{\mu ^2+\sqrt{{\bar{a}}}}{\mu }>{\bar{c}}^*\). By the above arguments, we have
Hence for any \(\omega \in \Omega _0\),\(c^*_{\sup }(\omega )\le 2\sqrt{{\overline{a}}}\).
Next, we prove that \(c_{\sup }^*(\omega )\ge 2\sqrt{{\overline{a}}}\) for all \(\omega \in \Omega _0.\) We prove this by contradiction.
Assume that there is \(\omega \in \Omega _0\) such that \(c_{\sup }^*(\omega )< 2\sqrt{{\overline{a}}}\). Then there is \(0<\delta <1\) such that
Note that
Then there is \(0<\delta ^{'}<1\) and \(\{t_n\}\), \(\{s_n\}\) such that \(\lim _{n\rightarrow \infty } t_n-s_n=\infty \) and
Choose \(c\in (c^*_{\sup }(\omega ), 2\sqrt{\delta {\overline{a}}})\). Set \(L=\frac{2\pi }{\sqrt{4{\bar{a}}\delta -c^2}}\) and
Then \(w^+(x)\) satisfies
and \(0<w^+(x)<1\) for \(0<x<L\).
For any given \(u_0\in X_c^+\), by the assumption that \(c>c_{\mathrm{sup}}^*(\omega )\),
Hence there is \(T>0\) such hat
and then
Observe that \(u(t,x;u_0,\theta _s\omega )\ge u(t,x;\frac{u_0}{1+\Vert u_0\Vert _{\infty }},\theta _s\omega ) \) and
This implies that
Let \(v(t,x;s)=u(t,x+ct;u_0,\theta _{s-T}\omega )\). By (4.15),
Let \(w(t,x;s)=e^{-\int _s ^{s-T+t} \big (\delta ^{'} a(\theta _\tau \omega )-\delta {\bar{a}}\big )d\tau } v(t,x;s)\). Then
By (4.16) and the comparison principle for parabolic equations, we have
This implies that for \(0\le x\le L\),
By (4.14),
which contradicts (4.17). Therefore, \(c_{\mathrm{sup}}^*(\omega )\ge {\bar{c}}^*\) and then \(c_{\mathrm{sup}}^*(\omega )={\bar{c}}^*\) for any \(\omega \in \Omega _0\). (i) thus follows.
(ii) By Lemma 4.3, \(c_{\inf }^*(\omega )\ge {\underline{c}}^*\) for every \(\omega \in \Omega _0\). It then suffices to prove that \(c_{\mathrm{inf}}^*(\omega )\le {\underline{c}}^*\) for every \(\omega \in \Omega _0\). We prove this by contradiction.
Assume that there is \(\omega \in \Omega _0\) such that \(c_{\mathrm{inf}}^*(\omega )>{\underline{c}}^*\). Choose \(c\in ({\underline{c}}^*,c_{\mathrm{inf}}^*(\omega ))\) and \(\delta >1\) such that \(c>2\sqrt{\delta {\underline{a}}}\). Then
Hence there are \(\{t_n\}\) and \(\{s_n\}\) such that \(\lim _{n\rightarrow \infty }t_n-s_n=\infty \) and
Let \({\underline{\mu }}=\sqrt{\delta {\underline{a}}}\). Then
Choose \(u_0\in X_c^+\) such that
By the assumption that \(c<c_{\mathrm{inf}}^*(\omega )\), there is \(T>0\) such that for any \(t\ge T\) and \(s\in {{\mathbb {R}}}\),
This implies that for any \(n\ge 1\) with \(t_n-s_n\ge T\),
Observe that \(u(t,x;u_0,\theta _{s_n}\omega )\) satisfies
It then follows from the comparison principle for parabolic equations that
and then for \(x=c(t_n-s_n)\), we have
which contradicts to (4.19). Therefore \(c_{\inf }^*(\omega )\le {\underline{c}}^*\) for any \(\omega \in \Omega _0\) and (ii) follows. \(\square \)
The following corollary follows directly from Lemma 3.2 and Theorem 1.2.
Corollary 4.1
Assume (H3). Let \(Y(\omega )\) be the random equilibrium solution of (1.21) given in (1.22). Then for any \(u_0\in X_c^+\),
and
for a.e. \(\omega \in \Omega \). where \(u(t,x;u_0,\theta _s\omega )\) is the solution of (1.21) with \(\omega \) being replaced by \(\theta _s\omega \) and \(u(0,x;u_0,\theta _s\omega )=u_0(x)\).
Finally, we prove Theorem 1.3.
Proof
(i) It is clear that \({{\tilde{c}}}_{\mathrm{sup}}^*(\omega )\ge c_{\mathrm{sup}}^*(\omega )={\bar{c}}^*\) for any \(\omega \in \Omega _0\). It then suffices to prove that \({{\tilde{c}}}_{\mathrm{sup}}^*(\omega )\le {\bar{c}}^*\) for any \(\omega \in \Omega _0\).
To this end, fix \(\omega \in \Omega _0\). For every \(\mu >0\), let \(C_\mu (t,s)=\int _s^{s+t}\frac{\mu ^2+a(\theta _\tau \omega )}{\mu }d\tau \) and \({\tilde{\phi }}^{\mu }_{+}(t,x;s)=e^{-\mu (x-C_\mu (t,s))}\) for every \(x\in {{\mathbb {R}}}\) and \(t\ge 0\). Note that for any \(u_0\in {{\tilde{X}}}_c^+\), there is \(M_0>0\) such that
Note also that
Hence, by the comparison principle for parabolic equations, we have that
This implies that
For any \(c>{\bar{c}}^*=2\sqrt{{\bar{a}}}=\inf _{\mu >0}\frac{\mu ^2+\sqrt{{\bar{a}}}}{\mu }\), choose \(\mu >0\) such that \(c>\frac{\mu ^2+\sqrt{{\bar{a}}}}{\mu }>{\bar{c}}^*\). By the above arguments, we have
Hence for any \(\omega \in \Omega _0\), we have \({{\tilde{c}}}^*_{\sup }(\omega )\le 2\sqrt{{\overline{a}}}\). (i) thus follows.
(ii) First, it is clear that \({{\tilde{c}}}_{\inf }^*(\omega )\ge c_{\mathrm{inf}}^*(\omega )={\underline{c}}^*\). It then suffices to prove that \({{\tilde{c}}}_{\mathrm{inf}}^*(\omega )\le {\underline{c}}^*\) for any \(\omega \in \Omega _0\). This can be proved by the similar arguments as those in Theorem 1.2 (ii). \(\square \)
The following corollary follows directly from Lemma 3.2 and Theorem 1.3.
Corollary 4.2
Assume (H3). Let \(Y(\omega )\) be the random equilibrium solution of (1.21) given in (1.22). Then for any \(u_0\in {\tilde{X}}_c^+\),
and
for a.e. \(\omega \in \Omega \). where \(u(t,x;u_0,\theta _s\omega )\) is the solution of (1.21) with \(\omega \) being replaced by \(\theta _s\omega \) and \(u(0,x;u_0,\theta _s\omega )=u_0(x)\).
5 Take-Over Property
In this section, we investigate the take-over property of (1.1) and prove Theorem 1.4. We first prove some lemmas.
Recall that
and that, for \(t>0\), \(x(t,\omega )\in {{\mathbb {R}}}\) is such that
Note that, by Lemma 2.9, for each \(t>0\), \(x(t,\omega )\) is measurable in \(\omega \). Note also that for \(\omega \in \Omega \), the mapping \((t,t_0)\ni (0,\infty )\times {{\mathbb {R}}}\rightarrow u(t,\cdot ;u_0^*,\theta _{t_0}\omega )\in C_{\mathrm{unif}}^b({{\mathbb {R}}})\) is continuous and hence \(x(t,\theta _{t_0}\omega )\) is continuous in \((t,t_0)\in (0,\infty )\times {{\mathbb {R}}}\).
Suppose that (H1) holds. Let \(\omega \in \Omega _0\), and \(0<\mu<{\tilde{\mu }}<\min \{2\mu ,{\underline{\mu }}^*\}\) be given, where \({\underline{\mu }}^*=\sqrt{{\underline{a}}}\). Let \(b(t)=a(\theta _t\omega )\). Put
and
where \(c(t;b,\mu )\) and \(C(t;b,\mu )\) are as in (2.5), and \(B_b\) and \(d_b\) are as in Lemma 2.4. Note that we can choose \(d_{\theta _{t_0}\omega }=d_\omega \) and \(A_{\theta _{t_0}\omega }(t)=A_\omega (t+t_0)\) for any \(t_0\in {{\mathbb {R}}}\). Let
Note that for any given \(t\in {{\mathbb {R}}}\),
We introduce the following function
It is clear from Lemma 2.4, and the comparison principle for parabolic equations, that
Lemma 5.1
For every \(\omega \in \Omega _0\), \(\lim _{x\rightarrow -\infty } u(t,x+C(t,\theta _{t_0}\omega ,\mu ); \phi _+^\mu (0,\cdot ;\theta _{t_0}\omega ),\theta _{t_0}\omega )=1\) uniformly in \(t>0\) and \(t_0\in {{\mathbb {R}}}\), and \(\lim _{x\rightarrow \infty } u(t,x+C(t,\theta _{t_0}\omega ,\mu );\phi _+^\mu (0,\cdot ;\theta _{t_0}\omega ),\theta _{t_0}\omega )=0\) uniformly in \(t>0\) and \(t_0\in \Omega \).
Proof
First, it follows from Lemma 2.3 that
Second, define \(v(t,x;\theta _{t_0}\omega )=u(t,x+C(t,\theta _{t_0}\omega ,\mu ); \phi _+^\mu (0,\cdot ;\theta _{t_0}\omega ),\theta _{t_0}\omega )\) and
It follows from (5.1) and (5.3) that
Moreover, \(x\mapsto v(t,x;\theta _{t_0}\omega )\) is decreasing and
where \(c(t;\omega ,\mu )=C'(t;\omega ,\mu )\). By the arguments of Theorem 3.1, we have that
uniformly in \(t>0,t_0\in {{\mathbb {R}}}\). \(\square \)
Lemma 5.2
For each \(t>0\), there is \(m(t)\le n(t)\in {{\mathbb {R}}}\) such that
and hence \(x(t,\omega )\) is integrable in \(\omega \).
Proof
First, let
We have that \(0\le u_{0n}^*(x)\le 1\) and \(u_{0n}^*(x)\rightarrow 1\) as \(n\rightarrow \infty \). By Lemma 3.1, for every \(\omega \in \Omega _0\) and \(t>0\)
uniformly in \(t_0\in {{\mathbb {R}}}\) and locally uniformly in \(x\in {{\mathbb {R}}}\). Observe that
and the mapping \({{\mathbb {R}}}\ni x\mapsto u(t,x;u_{0}^*,\theta _{t_0}\omega ) \) is decreasing. Thus, there is \(N(t,\omega )\in {\mathbb {N}}\) such that
This implies that
Let
We have that \(\Omega _0\ni \omega \mapsto m(t,\omega )\in {{\mathbb {R}}}^+\) is measurable and \(m(t,\theta _\tau \omega )=m(t,\omega )\) for any \(\tau \in {{\mathbb {R}}}\). By the ergodicity of the metric dynamical system \((\Omega _0, {\mathcal {F}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\), we have that \(m(t,\omega )=m(t) \quad \mathrm{for}\,\, a.e \ \text {in} \ \omega .\)
Next, let \({\tilde{u}}_{0n}^*(x)=u_0^*(x+n)\). We have that \(0\le u_{0n}^*(x)\le 1\) and \({\tilde{u}}_{0n}^*(x)\rightarrow 0\) as \(n\rightarrow \infty \). By Lemma 3.1 again, for every \(\omega \in \Omega _0\) and \(t>0\),
uniformly in \(t_0\in {{\mathbb {R}}}\) and locally uniformly in \(x\in {{\mathbb {R}}}\). Observe that
and the mapping \({{\mathbb {R}}}\ni x\mapsto u(t,x;u_{0}^*,\theta _{t_0}\omega ) \) is decreasing. Thus, there is \({\tilde{N}}(t,\omega )\in {\mathbb {N}}\) such that
This implies that
Let
By (5.4), we have that \(-\infty< x(t,\omega )\le n(t,\omega )\le N(t,\omega )<\infty \). Hence \(\Omega _0\ni \omega \mapsto m(t,\omega )\in {{\mathbb {R}}}^+\) is measurable and \(m(t,\theta _\tau \omega )=m(t,\omega )\) for any \(\tau \in {{\mathbb {R}}}\). By the ergodicity of the metric dynamical system \((\Omega _0, {\mathcal {F}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\), we have that \(m(t,\omega )=m(t) \quad \mathrm{for}\,\, a.e \ \text {in} \ \omega .\) \(\square \)
Let \(x_+(t,\omega ,\mu )\) be such that
Lemma 5.3
For any \(t>0\), there holds
Proof
It follows from Lemma 2.7. \(\square \)
Lemma 5.4
There is \({\hat{M}}>0\) such that
for all \(t,s\ge 0\) and a.e. \(\omega \in \Omega \).
Proof
First, let \({{\tilde{x}}}(t,\omega )\) and \({{\tilde{x}}}_+(t,\omega )\) be such that
respectively. Since the function \(x\mapsto u(t, x;u_0,\omega ) \) is decreasing, we have
Moreover, for each \(t>0\), \({{\tilde{x}}}(t,\omega )\) is measurable in \(\omega \), and for each \(\omega \in \Omega \), \({{\tilde{x}}}(t,\theta _{t_0}\omega )\) is continuous in \((t,t_0)\in (0,\infty )\times {{\mathbb {R}}}\). By Lemma 5.3,
Let
Note that
and
By Lemma 5.1, there is a positive constant \(K(\omega )\) such that
This combined with (5.7) implies that \(M(\omega )<\infty \).
Note that the function \(\Omega _0\ni \omega \mapsto M(\omega )\in {{\mathbb {R}}}^+\) is measurable and invariant. By the ergodicity of the metric dynamical system \((\Omega _0, {\mathcal {F}},\{\theta _t\}_{t\in {{\mathbb {R}}}})\), we have that there are an invariant measurable set \({\tilde{\Omega }}\) with \({{\mathbb {P}}}({{\tilde{\Omega }}})=1\) and a positive constant \({\hat{M}}\) such that
Second, note that
Hence,
This implies that
It then follows from (5.9) that
The lemma follows. \(\square \)
We now prove Theorem 1.4.
Proof of Theorem 1.4
(i) We first prove that there is \(c^*\) such that (1.10) holds with \({\hat{c}}^*\) being replaced by \(c^*\). To this end, let \(y(t,\omega )=-x(t,\omega )+{ {\hat{M}}}\) where \({\hat{M}}\) is given by Lemma 5.4. Then, by Lemma 5.4
a.e in \(\omega \). By Lemma 5.2, \(y(t,\cdot )\in L^{1}(\Omega ) \). It then follows from the subadditive ergodic theorem that there is \(c^*\in {{\mathbb {R}}}\) such that
Next, we claim that (1.11) and (1.12) hold with \({\hat{c}}^*\) being replaced by \(c^*\). In fact, by (5.5), (5.8), and Lemma 5.1,
and
Therefore, (1.11) and (1.12) hold with \({\hat{c}}^*\) being replaced by \(c^*\).
Now, we prove that \(c^*={\hat{c}}^*\). By the comparison principle for parabolic equations,
Hence
This implies that
Letting \(t\rightarrow \infty \), we obtain that
Taking \(\mu =\sqrt{{\hat{a}}}\), we obtain that
It then remains to prove that
We prove this by contradiction.
Assume that \(c^*< {\hat{c}}^*=2\sqrt{{\hat{a}}}\). Then there are \(h>0\) and \(0<\delta <1\) such that
By (1.11), for a.e. \(\omega \in \Omega \),
Fix such \(\omega \). There are \(0<\delta ^{'}< 1\) and \(T>0\) such that
and
As in the proof of Theorem 1.2(i), let \(L=\frac{2\pi }{\sqrt{4{ {\hat{a}}}\delta -c^2}}\) and
By the similar arguments as those in Theorem 1.2(i), we have
for \(0\le x\le L\) and \(t\ge T\), where \(\alpha =\sup _{0\le x\le L}u(T,x+cT;u_0^*,\omega )\). This implies that
which is a contradiction. Hence \(c^*= {\hat{c}}^*=2\sqrt{{\hat{a}}}\).
(ii) For any given \(u_0\in {{\tilde{X}}}_c^+\), there are \(0<\alpha \le 1\le \beta \) and \(x_-<x_+\) such that
By the comparison principle for parabolic equations, we have
This together with (1.11) implies that there is a measurable set \(\Omega _1\subset \Omega \) with \({{\mathbb {P}}}(\Omega _1)=1\) such that
and
We claim that
Indeed, let \(\omega \in \Omega _1\) and \(h>0\) be fixed. Let \(\{x_n\}\) and \(\{t_n\}\) with \(t_n\rightarrow \infty \) and \(x_n\le ({\hat{c}}^*-h)t_n\) be such that
For every \(0<\varepsilon \ll \frac{1}{2}\), Theorem 1.1 implies that there is \(T_\varepsilon >0\) such that
Consider a sequence of \(u_{0n}\in C^b_{\mathrm{unif}}({{\mathbb {R}}})\) satisfying that
Note that
By (5.10), there is \(N_1\gg 1\) such that
By the comparison principle for parabolic equations, we then have that
In particular, taking \(t=T_\varepsilon \) and \(x=0\), we obtain
Note that \(u_{0n}(x)\rightarrow \frac{\alpha }{2}\) as \(n\rightarrow \infty \). Letting \(t\rightarrow \infty \) in (5.14), it follows from (5.13) and Lemma 3.1 that
Letting \(\varepsilon \rightarrow 0\) in the last inequality, it follows from (5.12) that
It is clear that
The Claim thus follows and (ii) is proved. \(\square \)
The following corollary follows directly from Lemma 3.2 and Theorem 1.4.
Corollary 5.1
Assume (H3). Let \(Y(\omega )\) be the random equilibrium solution of (1.21) given in (1.22) and let \(U^*_0(x;\omega )=Y(\omega )\) for \(x<0\) and \(U^*_0(x;\omega )=0\) for \(x>0\). Then,
where \(X(t,\omega )\) is such that \(u(t,X(t,\omega );U^*_0(\cdot ;\omega ),\omega )=\frac{1}{2}Y(\omega )\), and
and
where \(u(t,x;U^*_0(\cdot ;\omega ),\omega )\) is the solution of (1.21) with \(u(0,x;U^*_0(\cdot ;\omega ),\omega )=U^*_0(x;\omega )\).
6 Spreading Speeds of Nonautonomous Fisher–KPP Equations
In this section we consider the nonautonomous Fisher–KPP equation (1.2) and prove Theorem 1.5.
Proof of Theorem 1.5
First, we prove (1.17). To this end, for given \(0<c<2\sqrt{{\underline{a}}_0}\), choose \(b>c\) and \(0<\delta <1\) such that \( c<2\sqrt{b}<2\sqrt{\delta {\underline{a}}_0}.\) By the proof of Lemma 2.2, there are \(\{t_{k}\}_{k\in {{\mathbb {Z}}}}\) with \(t_k<t_{k+1}\), \(t_{k}\rightarrow \pm \infty \) as \(k\rightarrow \pm \infty \) and \(A\in W^{1,\infty }_{loc}({{\mathbb {R}}})\cap L^\infty ({{\mathbb {R}}})\) such that \(A\in C^1(t_k,t_{k+1})\) for every k and
Let \( \sigma =\frac{(1-\delta )e^{-\Vert A\Vert _{\infty }}}{\Vert u_0\Vert _{\infty }+1}\) and v(t, x; b) be the solution of the PDE
By Lemma 4.2, we have that
For given \(s\in {{\mathbb {R}}}\), let \({\tilde{v}}(t,x;s)=\sigma e^{A(t+s)}v(t,x;b)\). By the similar arguments to those in Lemma 4.3, it can be proved that
This combined with (6.1) yields that
By the arguments in Lemma 4.1, it can be proved that
(1.17) then follows.
Next, we prove (1.18). To this end, for any given \(u_0\in X_c^+\), suppose that \(\mathrm{supp}(u_0)\subset (-R,R)\). For every \(\mu >0\), let \(C_\mu (t,s)=\int _s^{s+t}\frac{\mu ^2+a_0(\tau \omega )}{\mu }d\tau \) and \(\phi ^\mu (x)=\Vert u_0\Vert _{\infty }e^{-\mu ( x-R)} \) and \({\tilde{\phi }}^{\mu }_{\pm }(t,x;s)=\phi _{\pm }^\mu (\pm x-C_\mu (t,s))\) for every \(x\in {{\mathbb {R}}}\) and \(t\ge 0\). It is not difficult to see that
and
By the comparison principle for parabolic equations, we then have that
This implies that
For any \(c>2\sqrt{{\bar{a}}_0}=\inf _{\mu >0}\frac{\mu ^2+\sqrt{{\bar{a}}_0}}{\mu }\), choose \(\mu >0\) such that \(c>\frac{\mu ^2+\sqrt{{\bar{a}}_0}}{\mu }>{\bar{c}}^*\), we have
(1.18) then follows. \(\square \)
We conclude this section with some example of explicit function \(a_0(t)\) satisfying (H2).
Define the sequences \(\{l_{n}\}_{n\ge 0}\) and \(\{L_n\}_{n\ge 0}\) inductively by
Define \(a_0(t)\) such that \(a_0(-t)=a_0(t)\) for \(t\in {{\mathbb {R}}}\) and
for \(n\ge 0\), where \(g_{2n}(t)=1\) and \(g_{2n+1}(t)=2\) for \(n\ge 0\), and \(f_0(t)=1\), for \(n\ge 1\), \(f_{n}\) is Hölder’s continuous on \([l_{n},L_{n}]\), \(f_n(l_n)=g_n(l_n)\), \(f_{n}(L_{n})=g_n(L_{n})\), and satisfies
and
It is clear that \(a_0(t)\) is locally Hölder’s continuous, \( \inf _{t\in {{\mathbb {R}}}} a_{0}(t)=0\), and \(\sup _{t\in {{\mathbb {R}}}} a_{0}(t)=\infty \). Moreover, it can be verified that
Hence \(a_0(t)\) satisfies (H2).
References
Angenent, S.B.: The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390, 79–96 (1988)
Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J. (ed.) Partail Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 466, pp. 5–49. Springer, New York (1975)
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33–76 (1978)
Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model.I. Species persistence. J. Math. Biol. 51(1), 75–113 (2005)
Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems, I—Periodic framework. J. Eur. Math. Soc. 7, 172–213 (2005)
Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems, II—General domains. J. Am. Math. Soc. 23, 1–34 (2010)
Berestycki, H., Hamel, F., Roques, L.: Analysis of periodically fragmented environment model: II—biological invasions and pulsating traveling fronts. J. Math. Pures Appl. 84, 1101–1146 (2005)
Berestycki, H., Hamel, F.: Generalized Travelling Waves for Reaction–Diffusion Equations. Perspectives in Nonlinear Partial Differential Equations. Contemporary Mathematics, vol. 446, pp. 101–123. American Mathematical Society, Providence (2007)
Berestycki, H., Hamel, F.: Generalized transition waves and their properties. Commun. Pure Appl. Math. 65(5), 592–648 (2012)
Berestycki, H., Nadin, G.: Asymptotic spreading for general heterogeneous Fisher-KPP type equations (2015)hal-01171334v3 (preprint)
Bramson, M.: Convergence of Solutions of the Kolmogorov Equations to Traveling Waves. Memoirs of the American Mathematical Society, vol. 44 (1983)
Faria, T., Huang, W., Wu, J.: Travelling waves for delayed reaction-diffusion equations with global response. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462(2065), 229–261 (2006)
Fisher, R.: The wave of advance of advantageous genes. Ann. Eugen. 7, 335–369 (1937)
Freidlin, M., Gärtner, J.: On the propagation of concentration waves in periodic and ramdom media. Soviet Math. Dokl. 20, 1282–1286 (1979)
Hamel, F.: Qualitative properties of monostable pulsating fronts : exponential decay and monotonicity. J. Math. Pures Appl. (9) 89, 355–399 (2008)
Heinze, S., Papanicolaou, G., Stevens, A.: A variational principle for propagation speeds in inhomogeneous media. SIAM J. Appl. Math. 62, 129–148 (2001)
Hernández, E., Wu, J.: Traveling wave front for partial neutral differential equations. Proc. Am. Math. Soc. 146(4), 1603–1617 (2018)
Huang, Z., Liu, Z.: Random traveling wave and bifurcations of asymptotic behaviors in the stochastic KPP equation driven by dual noises. J. Differ. Equ. 261(2), 1317–1356 (2016)
Huang, Z., Liu, Z.: Stochastic traveling wave solution to stochastic generalized KPP equation. NoDEA Nonlinear Differ. Equ. Appl. 22(1), 143–173 (2015)
Huang, Z., Liu, Z., Wang, Z.: Stochastic traveling wave solution to a stochastic KPP equation. J. Dyn. Differ. Equ. 28(2), 389–417 (2016)
Huang, J.H., Shen, W.: Speeds of spread and propagation for KPP models in time almost and space periodic media. SIAM J. Appl. Dyn. Syst. 8, 790–821 (2009)
Hudson, W., Zinner, B.: Existence of Traveling Waves for Reaction Diffusion Equations of Fisher Type in Periodic Media, Boundary Value Problems for Functional-Differential Equations, pp. 187–199. World Scientifi Publishing, River Edge (1995)
Ji, C., Jiang, D., Shi, N.: Analysis of a predator–prey model with modified Lesile–Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 359, 482–498 (2009)
Kametaka, Y.: On the nonlinear diffusion equation of Kolmogorov–Petrovskii–Piskunov type. Osaka J. Math. 13, 11–66 (1976)
Kolmogorov, A., Petrowsky, I., Piskunov, N.: A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Bjul. Moskovskogo Gos. Univ. 1, 1–26 (1937)
Kong, L., Shen, W.: Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity. J. Dyn. Differ. Equ. 26(1), 181–215 (2014)
Liang, X., Yi, Y., Zhao, X.-Q.: Spreading speeds and traveling waves for periodic evolution systems. J. Diff. Equ. 231(1), 57–77 (2006)
Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60(1), 1–40 (2007)
Liang, X., Zhao, X.-Q.: Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259(4), 857–903 (2010)
Matano, H.: Traveling waves in spatially random media. RIMS Kokyuroku 1337, 1–9 (2003)
Nadin, G.: Traveling fronts in space–time periodic media. J. Math. Pures Appl. 92(9), 232–262 (2009)
Nadin, G.: Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(4), 841–873 (2015)
Nadin, G., Rossi, L.: Propagation phenomena for time heterogeneous KPP reaction–diffusion equations. J. Math. Pures Appl. (9) 98(6), 633–653 (2012)
Nadin, G., Rossi, L.: Transition waves for Fisher–KPP equations with general time-heterogeneous and space-periodic coeffcients. Anal. PDE 8(6), 1351–1377 (2015)
Nadin, G., Rossi, L.: Generalized transition fronts for one-dimensional almost periodic Fisher–KPP equations. Arch. Ration. Mech. Anal. 223, 1239–1267 (2017)
Nolen, J., Roquejoffre, J.-M., Ryzhik, L., Zlatoš, A.: Existence and non-existence of Fisher–KPP transition fronts. Arch. Ration. Mech. Anal. 203(1), 217–246 (2012)
Nolen, J., Rudd, M., Xin, J.: Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds. Dyn. PDE 2, 1–24 (2005)
Nolen, J., Xin, J.: Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete Contin. Dyn. Syst. 13(5), 1217–1234 (2005)
Øksendala, B., Vage, G., Zhao, H.: Asymptotic properties of the solutions to stochastic KPP equations. Proc. R. Soc. Edinb. A 130, 1363–1381 (2000)
Øksendala, B., Vage, G., Zhao, H.: Two properties of stochastic KPP equations: ergodicity and pathwise property. Nonlinearity 14, 639–662 (2001)
Sattinger, D.H.: On the stability of waves of nonlinear parabolic systems. Adv. Math. 22, 312–355 (1976)
Shen, W.: Traveling waves in diffusive random media. J. Dyn. Differ. Equ. 16, 1011–1060 (2004)
Shen, W.: Variational principle for spatial spreading speeds and generalized wave solutions in time almost and space periodic KPP models. Trans. Am. Math. Soc. 362(10), 5125–5168 (2010)
Shen, W.: Existence, uniqueness, and stability of generalized traveling solutions in time dependent monostable equations. J. Dyn. Differ. Equ. 23(1), 1–44 (2011)
Shen, W.: Existence of generalized traveling waves in time recurrent and space periodic monostable equations. J. Appl. Anal. Comput. 1(1), 69–93 (2011)
Shen, W.: Stability of transition waves and positive entire solutions of Fisher–KPP equations with time and space dependence. Nonlinearity 30(9), 3466–3491 (2017)
Tao, T., Zhu, B., Zlatoš, A.: Transition fronts for inhomogeneous monostable reaction–diffusion equations via linearization at zero. Nonlinearity 27(9), 2409–2416 (2014)
Uchiyama, K.: The behavior of solutions of some nonlinear diffusion equations for large time. J. Math. Kyoto Univ. 18–3, 453–508 (1978)
Weinberger, H.F.: Long-time behavior of a class of biology models. SIAM J. Math. Anal. 13, 353–396 (1982)
Weinberger, H.F.: On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45, 511–548 (2002)
Xin, J.: Front propagation in heterogeneous media. SIAM Rev. 42, 161–230 (2000)
Zlatoš, A.: Transition fronts in inhomogeneous Fisher–KPP reaction–diffusion equations. J. Math. Pures Appl. 98(1), 89–102 (2012). 98
Zou, X., Wu, J.: Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method. Proc. Am. Math. Soc. 125(9), 2589–2598 (1997)
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Wenxian Shen Partially supported by the NSF Grant DMS–1645673.
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Salako, R.B., Shen, W. Long Time Behavior of Random and Nonautonomous Fisher–KPP Equations: Part I—Stability of Equilibria and Spreading Speeds. J Dyn Diff Equat 33, 1035–1070 (2021). https://doi.org/10.1007/s10884-020-09847-2
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DOI: https://doi.org/10.1007/s10884-020-09847-2
Keywords
- Random Fisher–KPP equation
- Nonautonomous Fisher–KPP equation
- Spreading speed
- Take-over property
- Ergodic metric dynamical system
- Subadditive ergodic theorem