Propagation dynamics for monotone evolution systems without spatial translation invariance
Introduction
This paper is devoted to the study of the propagation dynamics for nonlinear evolution equations admitting the comparison principle. Since the seminal works of Fisher [16] and KPP [22], there have been extensive investigations on traveling wave solutions and propagation phenomena for various evolution equations (see, e.g., [15], [32], [34] and references therein). A fundamental feature of propagation problem is the asymptotic spreading speed introduced by Aronson and Weinberger [1]. Under an abstract setting, Weinberger [37] established the theory of traveling waves and spreading speeds for monotone discrete-time systems with spatial translation invariance. This theory has been greatly developed in [13], [14], [17], [26], [27], [28], [29], [38], [41], [43] for more general monotone and some non-monotone semiflows so that it can be applied to a variety of discrete and continuous-time evolution systems in homogeneous or periodic media. By using the Harnack inequality up to boundary and the strict positivity of solutions, Berestycki et al. [5], [6] studied the asymptotic spreading speed for KPP equations in periodic or non-periodic spatial domains. Differently from these two approaches, the spreading speed and asymptotic propagation were obtained in [40] for the Dirichlet problem of monostable reaction-diffusion equations on the half line by employing the iterative properties of traveling wave maps. Note that the solution maps of such a Dirichlet problem have no spatial translation invariance and the Harnack inequality cannot be extended to the boundary.
With an increasing interest in impacts of climate changes (see, e.g., [2], [21], [35]), there have been quite a few works on traveling waves and asymptotic behavior for evolution equation models with a shifting environment, see [3], [4], [7], [8], [9], [10], [11], [12], [18], [20], [24], [25], [30], [36], [39], [44] and references therein. Another class of evolution equations consists of those in locally spatially inhomogeneous media (see [23]). We should point out that these evolution equations admit the comparison principle, but their solution maps no longer possess the spatial translation invariance. This motivated us to develop the theory of spreading speeds and traveling waves for the monotone semiflows without spatial translation invariance. As a starting point, we assume that the given monotone system has two limiting systems in certain translation sense, and then establish the existence of steady state solutions and asymptotic propagation properties for monotone semiflows without translation invariance.
In order to overcome the difficulty induced by the lack of translation invariance, we first introduce two limiting systems admitting the translation invariance under an abstract setting. Then for a special class of initial functions having compact supports, we obtain certain estimates of their orbits under translations for the limiting system with the upward convergence property, and further carry them to the given system without the translation invariance by comparison arguments (see Section 2). Combining these estimates with the asymptotic annihilation property of the other limiting system, we are able to characterize the propagation dynamics for the given system.
The rest of the paper is organized as follows. In Section 2, we present notations and preliminary results. In order to avoid using traveling wave mappings, we directly establish the links between the system without translation invariance and its limiting systems. In Section 3, we prove the existence of fixed points and asymptotic propagation properties for discrete-time semiflows. In Sections 4 and 5, we extend these results to continuous-time semiflows and a class of nonautonomous evolution systems without translation invariance, respectively. In Section 6, we apply the developed theory to two types of time-delayed nonlocal equations with a shifting habitat, a reaction-diffusion equation in a cylinder, the Dirichlet problem for a time-delayed equation on the half line, and a KPP-type equation in spatially inhomogeneous media. We expect that our developed theory and methods in this paper may be applied to other monotone evolution systems including cooperative and competitive models with spatio-temporal heterogeneity.
Section snippets
Preliminaries
Let , , , , , and be the sets of all integers, nonnegative integers, reals, nonnegative reals, N-dimensional real vectors, and N-dimensional nonnegative real vectors, respectively. We equip with the norm . Let be the normed vector space of all bounded and continuous functions from to with the norm . Let and .
For a given compact topological space M, let be
Discrete-time semiflows
In this section, we study the upward convergence, asymptotic annihilation, and the existence of fixed points for discrete-time maps.
Theorem 3.1 Assume that satisfies (UC) and either Q is a subhomogeneous map with (SP), or Q satisfies (ASH-UC-SP). If , then for any and , there holds .
Proof For any , we define
Continuous-time semiflows
In this section, we extend our results on spreading speeds and asymptotic behavior to a continuous-time semiflow on . A map is said to be a continuous-time semiflow on if for any vector , is continuous, , and for all , where for all .
In this section, we need the following assumption for some results.
- (SC)
For any with and , there holds in C uniformly for
Nonautonomous systems
In this section, we extend our results on spreading speeds and asymptotic behavior to a class of nonautonomous evolution systems. Assume that is a map such that for any vector , is continuous. For any given , we define a family of mappings with parameter .
By Theorem 4.1 and the definition of , we have the following result.
Theorem 5.1 Assume that there exist and such that is a continuous-time semiflow on , and
Applications
In this section, we apply the results obtained in Sections 4 and 5 to four classes of monotone evolution equations. We start with the definition of KPP property (see, e.g., [5], [6]). Definition 6.1 Let and be a continuously differentiable function. We say that F satisfies the KPP property with respect to , or has the KPP property if , , and , for all , for all .
For simplicity, we always assume in this section that f
Acknowledgements
T. Yi's research is supported by the National Natural Science Foundation of China (Grant No. 11971494), and X.-Q. Zhao's research is supported in part by the NSERC of Canada. This work was initiated during Dr. Yi's visit to Memorial University of Newfoundland, and he would like to thank the Department of Mathematics and Statistics there for its kind hospitality.
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2022, Journal des Mathematiques Pures et AppliqueesCitation Excerpt :It was shown that the spreading speed coincides with the spreading speed in the limiting homogeneous systems with KPP structure. In particular, [55] generalizes [33] in the context of (1.3). Asmptotically homogeneous environments