Monostable Equations Research Articles

Spreading in space–time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed

This is Part 2 of our work aimed at classifying the long-time behavior of the solution to a free boundary problem with monostable reaction term in space–time periodic media. In Part 1 (see [2]) we have established a theory on the existence and uniqueness of solutions to this free boundary problem with continuous initial functions, as well as a spreading-vanishing dichotomy. We are now able to develop the methods of Weinberger [15,16] and others [6–10] to prove the existence of asymptotic spreading speed when spreading happens, without knowing a priori the existence of the corresponding semi-wave solutions of the free boundary problem. This is a completely different approach from earlier works on the free boundary model, where the spreading speed is determined by firstly showing the existence of a corresponding semi-wave. Such a semi-wave appears difficult to obtain by the earlier approaches in the case of space–time periodic media considered in our work here.

Existence of fast positive semi-wavefront solutions to monostable integro-differential equations with delay

Existence of fast positive semi-wavefront solutions to monostable integro-differential equations with delay

Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\mathbb{R}^d}K(y)g(v(t-h,x-y))dy, x \in \mathbb{R}^d,\ t >0;$ where $h>0$ and $d\in\mathbb{Z}_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d = 1$ which yields to the global stability in Sobolev spaces with bounded weights. Here $g$ is not assumed to be monotone and the kernel $K$ is not assumed to be symmetric, therefore non-monotone semi-wavefronts and backward semi-wavefronts appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.

Dynamical stochastic resonance for non‐uniform illumination image enhancement

Images taken under poor illumination conditions have low contrast and dark tones. General dark image enhancement algorithms cannot effectively enhance these images without introducing over-enhancement, detail loss, and noise amplification. In this study, a simple and fast enhancement technique of non-uniform illumination images is proposed based on dynamical stochastic resonance (DSR). The low-contrast images are enhanced through the nonlinear iteration by solving monostable Langevin equation. Iteration parameters are dynamically adjusted according to the intensity distribution of the original images, which ensure the balance of visibility and naturalness in the entire areas. A threshold is defined to automatically confirm the optimal outputs. The enhanced image is obtained by fusing the DSR result, original component, and illumination compensation component. The computational time, no-reference perceptual quality assessment, and lightness order error are measured to evaluate the performance of experimental results. The subjective and objective comparison with state-of-the-art methods shows that our method performs well to enhance the non-uniform illumination images with a low-computational complexity.

Entire solutions in nonlocal monostable equations: Asymmetric case

This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., \begin{document}$u_{t}=J*u-u+f(u)$\end{document} . Here the kernel \begin{document}$J$\end{document} is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds \begin{document}$c^{*}$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} , where \begin{document}$c^*$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel \begin{document}$J$\end{document} is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.

When fast diffusion and reactive growth both induce accelerating invasions

We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [ 3 ] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.

Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems

The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habitats with nonlocal dispersal. It is assumed that the trivial solution ${\bf u} = {\bf 0}$ of such a system is unstable and the system has a stable space-time periodic positive solution ${\bf u^*}(t,x)$. We first show that in any direction $ξ∈ \mathbb{S}^{N-1}$, such a system has a finite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed $c^{*}(ξ)$ in the direction of $ξ$. Next, we show that for any $c>c^{*}(ξ)$, there are space-time periodic traveling wave solutions of the form ${\bf{u}}(t,x) = {\bf{Φ}}(x-ctξ,t,ctξ)$ connecting ${\bf u^*}$ and ${\bf 0}$, and propagating in the direction of $ξ$ with speed $c$, where $Φ(x,t,y)$ is periodic in $t$ and $y$, and there is no such solution for $c<c^{*}(ξ)$. We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.

Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations

This paper is concerned with the multidimensional stability of traveling fronts for the combustion and non-KPP monostable equations. Our study contains two parts: in the first part, we first show that the two-dimensional V-shaped traveling fronts are asymptotically stable in $$\mathbb {R}^{n+2}$$ with $$n\ge 1$$ under any (possibly large) initial perturbations that decay at space infinity, and then, we prove that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which implies that even very small perturbations to the V-shaped traveling front can lead to permanent oscillation. In the second part, we establish the multidimensional stability of planar traveling front in $$\mathbb {R}^{n+1}$$ with $$n\ge 1$$ .

Planar Traveling Waves of Mono-Stable Reaction-Diffusion Equations

This paper is concerned with planar traveling wavefronts of mono-stable reaction-diffusion equations in \mathbb{R}^n ( n\geq2 ). We show that the large time behavior of the disturbed fronts can be controlled by two functions, which are the solutions of the specified nonlinear parabolic equations in \mathbb{R}^{n-1} , and the planar traveling fronts are asymptotically stable in L^\infty(\mathbb{R}^n) under ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases.

Thin Front Limit of an Integro-differential Fisher-KPP Equation with Fat-Tailed Kernels

We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling for the size of mutations, we derive a Hamilton-Jacobi limit.

Propagation of monostable traveling fronts in discrete periodic media with delay

This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.

Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations

This paper is concerned with the global stability of V-shaped traveling fronts in reaction-diffusion equations with combustion and degenerate monostable nonlinearity. The existence of such curved fronts has been recently proved by [ 39 ]. In this paper, by constructing new subsolutions, we show the asymptotic stability of V-shaped traveling fronts.

Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions

We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity may involve a weak Allee effect, which tends to slow down the process. We study the balance between these three effects (nonlin-ear diffusion, initial tail, KPP nonlinearity/Allee effect), revealing the separation between no acceleration and acceleration . In most of the cases where acceleration occurs, we also give an accurate estimate of the position of the level sets.

Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line

ABSTRACTWe consider the accelerated propagation of solutions to equations with a nonlocal linear dispersion on the real line and monostable nonlinearities (both local or nonlocal, however, not degenerated at 0), in the case when either of the dispersion kernel or the initial condition has regularly heavy tails at both , perhaps different. We show that, in such case, the propagation to the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may give different orders of the acceleration. Our approach is based, in particular, on the extension of the theory of sub-exponential distributions, which we introduced early.

Propagation phenomena in monostable integro-differential equations: Acceleration or not?

We consider the homogeneous integro-differential equation ∂tu=J⁎u−u+f(u) with a monostable nonlinearity f. Our interest is twofold: we investigate the existence/nonexistence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel J is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed [7,10,11,22,26,27]. On the other hand, when the dispersion kernel J has heavy tails and the nonlinearity f is nondegenerate, i.e. f′(0)>0, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating [14,20,27]. For a general monostable nonlinearity, a dichotomy between these two types of propagation behaviour is still not known.The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of f, i.e. f′(0)=0. First, for algebraic decaying kernels, we prove the exact separation between existence and nonexistence of travelling waves. This in turn provides the exact separation between nonacceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.

Spreading in space–time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions

We aim to classify the long-time behavior of the solution to a free boundary problem with monostable reaction term in space–time periodic media. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. In time-periodic and space homogeneous environment, as well as in space-periodic and time autonomous environment, such a problem has been studied recently in [11,12]. In both cases, a spreading–vanishing dichotomy has been established, and when spreading happens, the asymptotic spreading speed is proved to exist by making use of the corresponding semi-wave solutions. The approaches in [11,12] seem difficult to apply to the current situation where the environment is periodic in both space and time. Here we take a different approach, based on the methods developed by Weinberger [31,32] and others [16,22–24,26], which yield the existence of the spreading speed without using traveling wave solutions. In Part 1 of this work, we establish the existence and uniqueness of classical solutions for the free boundary problem with continuous initial data, extending the existing theory which was established only for C2 initial data. This will enable us to develop Weinberger's method in Part 2 to determine the spreading speed without knowing a priori the existence of the corresponding semi-wave solutions. In Part 1 here, we also establish a spreading–vanishing dichotomy.

Longtime Behavior of Solutions of a SIS Epidemiological Model

This paper investigates the longtime behavior of solutions of a susceptible-infectious-susceptible (SIS) epidemiological model, proposed to explain an epidemiological phenomenon that pathogen spread does not necessarily keep pace with its host invasion. When the rate of transmission of disease is small, the infected population will vanish. When the rate is of moderate size, a local infection of disease will spread with a speed that is smaller than that of the host population. For large disease transmission rates, the infection can keep pace with the host invasion, provided that the host does not propagate too fast. This paper also modifies a method originated from [F. Hamel et al., Netw. Heterog. Media, 8 (2013), pp. 275--289] to prove the logarithmic lag of the front of the infected population from that of the minimum speed traveling wave of a reduced equation obtained by setting the host population to its capacity. Our analysis can be used to study monostable scalar equations in inhomogeneous media.

Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations

In this work we consider the global asymptotic stability of pushed traveling fronts for one-dimensional monostable reaction-diffusion equations with monotone delayed reactions. Pushed traveling front is a special type of critical wave front which converges to zero more rapidly than the near non-critical wave fronts. Recently, Trofimchuk et al. [ 16 ] proved the existence and uniqueness of pushed traveling fronts of the considered equation when the reaction term lost the sub-tangency condition. In this article, using the comparison method via a pair of super-and sub-solution and squeezing technique, we prove that the pushed traveling fronts are globally exponentially stable. This also gives an affirmative answer to an open problem presented by Solar and Trofimchuk [ 14 ].

Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations

In this paper, we mainly discuss the existence and asymptotic stability of traveling fronts for the nonlocal evolution equations. With the monostable assumption, we obtain that there exists a constant \begin{document}$c^*>0$\end{document} , such that the equation has no traveling fronts for \begin{document}$0 and a traveling front for each c ≥ c*. For \begin{document}$c>c^*$\end{document} , we will further show that the traveling front is globally asymptotic stable and is unique up to translation. If we applied to some differential equations or integro-differential equations, our results recover and/or complement a number of existing ones.

Impulsive Steering Between Coexisting Stable Periodic Solutions With an Application to Vibrating Plates

Single-degree-of-freedom (single-DOF) nonlinear mechanical systems under periodic excitation may possess multiple coexisting stable periodic solutions. Depending on the application, one of these stable periodic solutions is desired. In energy-harvesting applications, the large-amplitude periodic solutions are preferred, and in vibration reduction problems, the small-amplitude periodic solutions are desired. We propose a method to design an impulsive force that will bring the system from an undesired to a desired stable periodic solution, which requires only limited information about the applied force. We illustrate our method for a single-degree-of-freedom model of a rectangular plate with geometric nonlinearity, which takes the form of a monostable forced Duffing equation with hardening nonlinearity.