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Existence, Uniqueness, And Asymptotic Stability Of Traveling Waves In Nonlocal Evolution Equations
 Adv. Differential Equations
, 1997
"... . The existence, uniqueness, and global exponential stability of traveling wave solutions of a class of nonlinear and nonlocal evolution equations are established. It is assumed that there are two stable equilibria so that a tr aveling wave is a solution that connects them. A basic assumption is the ..."
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Cited by 123 (2 self)
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. The existence, uniqueness, and global exponential stability of traveling wave solutions of a class of nonlinear and nonlocal evolution equations are established. It is assumed that there are two stable equilibria so that a tr aveling wave is a solution that connects them. A basic assumption is the comparison principle: a smaller initial value produces a smaller solution. When applied to differential equations or integrodifferential equations, the result recovers and/or complements a number of existing ones. Key Words. Traveling wave, asymptotic stability, comparison principle, nonlocal evolution. AMS subject classifications (1991). 35B05, 35B40, 35A05, 35R99. 1 Introduction. In this paper, we are concerned with a one space dimensional evolution equation u t (x; t) = A[u(\Delta; t)](x); x 2 IR; t ? 0; (1.1) where A is a nonlinear operator which is independent of the time t, maps functions of space variable \Delta to functions of x, and, via (1.1), generates a semigroup on the Ban...
The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems
 J. Dynam. Differential Equations
, 1997
"... We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solution ..."
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Cited by 74 (5 self)
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We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c 6= 0. Convergence results for solutions are obtained at the singular perturbation limit c ! 0. 1 Introduction We are interested in lattice differential equations, namely infinite systems of ordinary differential equations indexed by points on a spatial lattice, such as the Ddimensional integer lattice Z D . Our focus in this paper is the global structure of the set of traveling wave solutions for such systems. This entails results on existence and uniqueness, and on continuous (or smooth) dependence of traveling waves and their speeds on parameters, as well as some delicate convergence results in the singular perturbation case c ! 0 of the wav...
Some Nonclassical Trends in Parabolic and Paraboliclike Evolutions
"... : An overview will be given of some nonlinear paraboliclike evolution problems which are off the classical beaten track, but have increased in importance during the past decade. The emphasis is on problems which are nonlocal, patternforming (including exhibiting propagative phenomena), and/or lea ..."
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Cited by 65 (0 self)
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: An overview will be given of some nonlinear paraboliclike evolution problems which are off the classical beaten track, but have increased in importance during the past decade. The emphasis is on problems which are nonlocal, patternforming (including exhibiting propagative phenomena), and/or lead in some singular limit to free boundary problems. In all cases they have been proposed as models for phenomena in the natural sciences. Also emphasized are the relationships among the various classes of phenomena. 1
Asymptotic behavior for nonlocal diffusion equations
 J. Math. Pures Appl
, 2006
"... Abstract. We study the asymptotic behavior for nonlocal diffusion models of the form ut = J ∗ u − u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier t ..."
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Cited by 64 (17 self)
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Abstract. We study the asymptotic behavior for nonlocal diffusion models of the form ut = J ∗ u − u in the whole RN or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In RN we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity. If Ĵ(ξ) = 1 − Aξα + o(ξα) (0 < α 6 2), the asymptotic behavior is the same as the one for solutions of the evolution given by the α/2 fractional power of the laplacian. In particular when the nonlocal diffusion is given by a compactly supported kernel the asymptotic behavior is the same as the one for the heat equation, which is a local model. Concerning the Dirichlet problem for the nonlocal model we prove that the asymptotic behavior is given by an exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile an eigenfunction of the first eigenvalue. Finally, we analyze the Neumann problem and find an exponential convergence to the mean value of the initial condition. 1.
Traveling Waves in Lattice Dynamical Systems
"... In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODE's) and in coupled map lattices (CML's). Instead of employing the moving coordinate approach as for partial differentia ..."
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Cited by 60 (2 self)
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In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODE's) and in coupled map lattices (CML's). Instead of employing the moving coordinate approach as for partial differential equations, we construct a local coordinate system around a traveling wave solution of a lattice ODE, analogous to the local coordinate system around a periodic solution of an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution of this equation. We prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. We also show the existence of traveling waves in CML's which arise as timediscretizations of lattice ODE's. Finally, we show that these results apply to the discrete Nagumo equation. 1 Introduction This paper is concer...
PDE Methods for Nonlocal Models
, 2003
"... We develop partial differential equation (PDE) methods to study the dynamics of pattern formation in partial integrodifferential equations (PIDEs) defined on a spatially extended domain.Our primary focus is on scalar equations in two spatial dimensions.These models arise in a variety of neuronal m ..."
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Cited by 58 (5 self)
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We develop partial differential equation (PDE) methods to study the dynamics of pattern formation in partial integrodifferential equations (PIDEs) defined on a spatially extended domain.Our primary focus is on scalar equations in two spatial dimensions.These models arise in a variety of neuronal modeling problems and also occur in material science.We first derive a PDE which is equivalent to the PIDE.We then find circularly symmetric solutions of the resultant PDE; the linearization of the PDE around these solutions provides a criterion for their stability.When a solution is unstable, our analysis predicts the exact number of peaks that form to comprise a multipeak solution of the full PDE.We illustrate our results with specific numerical examples and discuss other systems for which this technique can be used.
HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS
, 2006
"... Abstract. We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes ..."
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Cited by 40 (17 self)
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Abstract. We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions. 1.
Boundary fluxes for nonlocal diffusion
"... Abstract. We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the ..."
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Cited by 35 (14 self)
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Abstract. We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition. 1.
On a nonlocal equation arising in population dynamics
 Proc. Roy. Soc. Edinburgh
, 2007
"... We study a onedimensional nonlocal variant of Fisher’s equation describing the spatial spread of a mutant in a given population, and its generalization to the socalled monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modelled by a con ..."
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Cited by 34 (3 self)
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We study a onedimensional nonlocal variant of Fisher’s equation describing the spatial spread of a mutant in a given population, and its generalization to the socalled monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modelled by a convolution operator. We prove that as in the classical (local) problem, there exist travellingwave solutions of arbitrary speed beyond a critical value and also characterize the asymptotic behaviour of such solutions at infinity. Our proofs rely on an appropriate version of the maximum principle, qualitative properties of solutions and approximation schemes leading to singular limits. 1 1
A nonlocal pLaplacian evolution equation with nonhomogeneous Dirichlet boundary conditions
 SIAM Journal of Mathematical Analysis
, 2009
"... Abstract. In this paper we study the nonlocal p−Laplacian type diffusion equation, ut(t, x) = Ω J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. If p> 1, this is the nonlocal analogous problem to the well known local p−Laplacian evolution equation ut = div(∇up−2∇u) with homogeneous Neuman ..."
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Cited by 33 (12 self)
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Abstract. In this paper we study the nonlocal p−Laplacian type diffusion equation, ut(t, x) = Ω J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. If p> 1, this is the nonlocal analogous problem to the well known local p−Laplacian evolution equation ut = div(∇up−2∇u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞(0, T;Lp(Ω)) to the solution of the p−laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behaviour of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition. 1.