We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of `sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar $et\ al$, we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front.
This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = Δv(t,x) - v(t,x) + \int_{\R^d}K(y)g(v(t-h,x-y))dy, x \in \R^d,\ t >0;$ where $h>0$ and $d\in\Z_+$. 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 includes the global stability on Sobolev spaces. Here $g$ is not assumed to be monotone and the kernel $K$ is not …
The wavefronts of a nonlinear nonlocal bistable reaction-diffusion equation, \begin{align*} \frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+u^2(1-J_σ*u)-du,\quad(t,x)\in(0,\infty)\times\mathbb R, \end{align*} with $J_σ(x)=(1/σ)= J(x/σ)$ and $ \int_{\mathbb R} J(x)dx=1 $ are investigated in this article. It is proven that there exists a $c_*(σ)$ such that for all $c\geq c_*(σ)$, a monotone wavefront $(c,ω)$ can be connected by the two positive equilibrium points. On the other hand, there exists a $c^*(σ)$ such that the model admits a semi-wavefront $(c^*(σ),ω)$ with $ω(-\infty)=0…
In this note, we present a monostable delayed reaction-diffusion equation with the unimodal birth function which admits only non-monotone wavefronts. Moreover, these fronts are either eventually monotone (in particular, such is the minimal wave) or slowly oscillating. Hence, for the Mackey-Glass type diffusive equations, we answer affirmatively the question about the existence of non-monotone non-oscillating wavefronts. As it was recently established by Hasik {\it et al.} and Ducrot {\it et al.}, the same question has a negative answer for the KPP-Fisher equation with a single delay.
We revisit Wu and Zou non-standard quasi-monotonicity approach for proving existence of monotone wavefronts in monostable reaction-diffusion equations with delays. This allows to solve the problem of existence of monotone wavefronts in a neutral KPP-Fisher equation. In addition, using some new ideas proposed recently by Solar et al., we establish the uniqueness (up to a translation) of these monotone wavefronts.
We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh-Nagumo type system and equations depending on the Lebesgue measu…
Bistable wavefronts in the Gurtin-MacCamy population model
Université Le Havre Normandie, Normandie Univ.（勒阿弗尔诺曼底大学）；Instituto de Matemáticas, Universidad de Talca（塔尔卡大学数学研究所）；Center for Applied Mathematics, Tianjin University（天津大学应用数学中心）
We study monotone numerical schemes for nonlocal Isaacs equations, the dynamic programming equations of stochastic differential games with jump-diffusion state processes. These equations are fully-nonlinear non-convex equations of order less than $2$. In this paper they are also allowed to be degenerate and have non-smooth solutions. The main contribution is a series of new a priori error estimates: The first results for nonlocal Isaacs equations, the first general results for degenerate non-convex equations of order greater than $1$, and the first results in the viscosity solution setting giv…
In this paper, we answer the question about the criteria of existence of monotone travelling fronts $u = ϕ(ν\cdot x+ct), ϕ(-\infty) =0, ϕ(+\infty) = κ,$ for the monostable (and, in general, non-quasi-monotone) delayed reaction-diffusion equations $u_t(t,x) - Δu(t,x) = f(u(t,x), u(t-h,x)).$ $C^{1,γ}$-smooth $f$ is supposed to satisfy $f(0,0) = f(κ,κ) =0$ together with other monostability restrictions. Our theory covers the two most important cases: Mackey-Glass type diffusive equations and KPP-Fisher type equations. The proofs are based on a variant of Hale-Lin functional-analytic approach to t…
In this work, we answer three fundamental questions concerning monostable travelling fronts for the scalar Kolmogorov ecological equation with diffusion and spatiotemporal interaction: these are the questions about their existence, uniqueness and geometric shape. In the particular case of the food-limited model, we give a rigorous proof of the existence of a peculiar, yet substantive non-linearly determined class of non-monotone and non-oscillating wavefronts. As regards to the scalar models coming from applications, this kind of wave solutions is analyzed here for the first time.
We consider traveling fronts to the nonlocal bistable equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We show that there is a traveling wave solution with monotone profile. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.
This paper concerns the semi-wavefronts (i.e. bounded solutions $u=ϕ(x ν+ct) >0,$ $ |ν|=1, $ satisfying $ϕ(-\infty)=0$) to the delayed KPP-Fisher equation $$u_t(t,x) = Δu(t,x) + u(t,x)(1-u(t-τ,x)), \ u \geq 0,\ x \in \R^m. \eqno(*)$$ First, we show that each semi-wavefront should be either monotone or slowly oscillating. Then a complete solution to the problem of existence of semi-wavefronts is provided. We prove next that the semi-wavefronts are in fact wavefronts (i.e. additionally $ϕ(+\infty)=1$) if $c \geq 2$ and $τ\leq 1$; our proof uses dynamical properties of some auxiliary one-dimensio…
We propose a new approach for proving uniqueness of semi-wavefronts in generally non-monotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.
$H$-convergence and $Γ$-convergence in the Riesz fractional setting: the nonlinear case
Scuola Internazionale Superiore di Studi Avanzati (SISSA)（高等高级研究所）；Università degli Studi dell’Aquila（阿奎拉大学）；Università del Salento（萨伦托大学）；Politecnico di Milano（米兰理工大学）；Università di Padova（帕多瓦大学）
We consider a nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure for the convolution is absolutely continuous. In order to show the main result, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semiflow generated by the equation does not have compactness with respect to the compact-open topology. At the end, we propose a discrete model that describes the measurement process.
We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = Δu(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a simi…
We study a nonlocal 4th order degenerate equation deriving from the epitaxial growth on crystalline materials. We first prove the global existence of evolution variational inequality solution with a general initial data using the gradient flow structure. Then with a monotone initial data, we prove the subdifferential of the associated convex functional is indeed single-valued, which gives higher regularities of the global solution. Particularly, higher regularites imply that the strict monotonicity maintains for all time, which provides rigorous justification for global-in time monotone soluti…
We are interested in nonlocal Eikonal Equations arising in the study of the dynamics of dislocations lines in crystals. For these nonlocal but also non monotone equations, only the existence and uniqueness of Lipschitz and local-in-time solutions were available in some particular cases. In this paper, we propose a definition of weak solutions for which we are able to prove the existence for all time. Then we discuss the uniqueness of such solutions in several situations, both in the monotone and non monotone case.
In this work we obtain a compactness result for the $H-$convergence of a family of nonlocal and nonlinear monotone elliptic-type problems by means of Tartar's method of oscillating test functions.
In this paper, we study a class of nonlocal dispersion equation with monostable nonlinearity in $n$-dimensional space u_t - J\ast u +u+d(u(t,x))= \int_{\mathbb{R}^n} f_β(y) b(u(t-τ,x-y)) dy, u(s,x)=u_0(s,x), s\in[-τ,0], \ x\in \mathbb{R}^n} \] where the nonlinear functions $d(u)$ and $b(u)$ possess the monostable characters like Fisher-KPP type, $f_β(x)$ is the heat kernel, and the kernel $J(x)$ satisfies ${\hat J}(ξ)=1-\mathcal{K}|ξ|^α+o(|ξ|^α)$ for $0<α\le 2$. After establishing the existence for both the planar traveling waves $ϕ(x\cdot{\bf e}+ct)$ for $c\ge c_*$ ($c_*$ is the critical wave…
We study the asymptotic stability of traveling fronts and front's velocity selection problem for the time-delayed monostable equation $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),\ x \in \mathbb{R},\ t >0$, considered with Lipschitz continuous reaction term $g: \mathbb{R}_+ \to \mathbb{R}_+$. We are also assuming that $g$ is $C^{1,α}$-smooth in some neighbourhood of the equilibria $0$ and $κ>0$ to $(*)$. In difference with the previous works, we do not impose any convexity or subtangency condition on the graph of $g$ so that equation $(*)$ can possess pushed traveling fronts. Our fir…
We study the asymptotic behaviour of solutions to the delayed monostable equation $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in R,\ t >0,$ with monotone reaction term $g: R_+ \to R_+$. Our basic assumption is that this equation possesses pushed traveling fronts. First we prove that the pushed wavefronts are nonlinearly stable with asymptotic phase. Moreover, combinations of these waves attract, uniformly on $R$, every solution of equation $(*)$ with the initial datum sufficiently rapidly decaying at one (or at the both) infinities of the real line. These results provide a sh…
In this paper,under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of monotone evolution systems without spatial translation invariance. Then we apply the developed theory to study traveling waves and spatio-temporal propagation patterns for time-delayed nonlocal equations, reaction-diffusion equations in a cylinder, and asymptotically homogeneous KPP-type equations. We also obtain the existence of steady state solutions and asymptotic spreading properties of solutions for a time-delayed reacti…
We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as so…
We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation $u_t=Δu+\lam/(1-u)^2(1+χ\int_Ω1/(1-u) dx)^2$ in $Ω\times (0,\infty)$, $u=0$ on $\1Ω\times (0,\infty)$, $u(x,0)=u_0$ in $Ω$, when $λ$ is large. We prove the compactness of the quenching set under a mild condition on the initial data. When $Ω=B_R$ and $u_0$ is radially symmetric and monotone decreasing in $0\le r\le R$, we prove that the point $x=0$ is the only possible quenching set. When $u_0$ also satisfies some strict concavity assumption, we prove that for any $β\in (2,3)$ the solution sat…
We describe recent results obtained by G. Barles, P. Cardaliaguet, R. Monneau and the author recently. They are concerned with nonlocal Eikonal equations arising in the study of the dynamics of dislocation lines in crystals. These equations are nonlocal but also non monotone. We use a notion of weak solution to provide solutions for all time. Then, we discuss the link between these weak solutions and the classical viscosity solutions, and state some uniqueness results in particular cases. A counter-example to uniqueness is given.
Seasonality frequently occurs in population models, and the corresponding seasonal patterns have been of great interest to scientists. This paper is concerned with traveling waves to a time-periodic bistable Lotka-Volterra competition system with nonlocal dispersal. We first establish the existence, uniqueness and stability of traveling wave solutions for this system. Then, by utilizing comparison principle and the stability property, the relationship among the bistable wave speed, the asymptotic propagation speeds of the associated monotone subsystems and the speed of upper/lower solutions is…
We consider the Whitham equation $u_t + 2u u_x+Lu_x = 0$, where L is the nonlocal Fourier multiplier operator given by the symbol $m(ξ) = \sqrt{\tanh ξ/ξ}$. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of $P$-periodic solutions, and give several qualitative properties of it, including its optimal $C^{1/2}$-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m(ξ)$, and a following study of…
We investigate the traveling front solutions of a nonlocal Lotka Volterra system to illustrate the outcome of the competition between two species. The existence of the front solution is obtained through a new monotone iteration scheme, the uniqueness of the front solution corresponding to each propagation speed is proved by sliding domain method adapted to nonlocal systems, and the asymptotic decay rate of the fronts with critical and noncritical wave speeds is derived by a new method, which is different from the single equation case. The results demonstrate that in the long run, two weakly co…
We study the equation $u_t +uu_x +u-K*u=0$ in the case of an arbitrary $K \geq 0$, which is a generalization of a model for radiating gas, in which $K(y)={1/2}e^{-|y|}$. Using a monotone iteration scheme argument we establish the existence of traveling waves, which gives a solution to an open question raised by Denis Serre.
The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of a…
We consider a Fisher-KPP-type equation, where both diffusion and nonlinear part are nonlocal, with anisotropic probability kernels. Under minimal conditions on the coefficients, we prove existence, uniqueness, and uniform space-time boundedness of the positive solution. We investigate existence, uniqueness, and asymptotic behavior of monotone traveling waves for the equation. We also describe the existence and main properties of the front of propagation.
Of concern is the a priori symmetry of traveling wave solutions for a general class of nonlocal dispersive equations \[ u_t +(u^2 +Lu)_x =0, \] where $L$ is a Fourier multiplier operator with symbol $m$. Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a class of symbols $m$ guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. Particularly, instead of considering waves with a unique crest and trough per period or a monotone structure near troughs as classically imposed in the water wave problem, we formu…
We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties. In addition, we investigate the main properties of these heteroclinic connections. We show that these minimizers are monotone, and therefore they satisfy a suitable Euler-Lagrange equation. We also prove that, differently from the classical cases …
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on $\mathbb{R}^d$. Using the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.
In this work we prove the uniqueness of solutions to the nonlocal linear equation $L φ- c(x)φ= 0$ in $\mathbb{R}$, where $L$ is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $L u = f(u)$ in $\mathbb{R}$ when the nonlinearity is of Allen-Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli…
We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We assume local conditions ensuring the existence of sub- and supersolutions. So we prove existence of two positive solutions, in both the coercive and noncoercive cases.
This paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model. It is shown that all supercritical solitary wave solutions are symmetric and monotone on either side of the crest. The proof is based on a priori decay estimates and the method of moving planes. Furthermore, a close relation between symmetric and traveling wave solutions is established.
A unified clasification of Liouville properties and nontrivial solution for fractional elliptic equations with general Hénon-type superquadratic and gradient growth
