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139
Finite Scale Microstructures in Nonlocal Elasticity
, 2000
"... In this paper we develop a simple onedimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen’s model [9] of an elastic “bar” with nonconvex energy by including both oscillationinhibiting and oscillationforcing terms in the en ..."
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Cited by 32 (9 self)
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In this paper we develop a simple onedimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen’s model [9] of an elastic “bar” with nonconvex energy by including both oscillationinhibiting and oscillationforcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the longrange elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known longrange interaction. In this limit the augmented Ericksen’s problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.
The Neumann problem for nonlocal nonlinear diffusion equations
 J. Evol. Equations
"... Abstract. We study nonlocal diffusion models of the form (γ(u))t(t, x) = Ω J(x − y)(u(t, y) − u(t, x)) dy. Here Ω is a bounded smooth domain and γ is a maximal monotone graph in R2. This is a nonlocal diffusion problem analogous with the usual Laplacian with Neumann boundary conditions. We prove ex ..."
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Cited by 28 (7 self)
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Abstract. We study nonlocal diffusion models of the form (γ(u))t(t, x) = Ω J(x − y)(u(t, y) − u(t, x)) dy. Here Ω is a bounded smooth domain and γ is a maximal monotone graph in R2. This is a nonlocal diffusion problem analogous with the usual Laplacian with Neumann boundary conditions. We prove existence and uniqueness of solutions with initial conditions in L1(Ω). Moreover, when γ is a continuous function we find the asymptotic behaviour of the solutions, they converge as t → ∞ to the mean value of the initial condition. 1.
Exponential dichotomies and WienerHopf factorizations for mixedtype functional differential equations
, 2001
"... We study linear systems of functional differential equations of mixed type, both autonomous and (asymptotically hyperbolic) nonautonomous. Such equations arise naturally in various contexts, for example, in lattice differential equations. We obtain a decomposition of the state space into stable and ..."
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Cited by 27 (0 self)
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We study linear systems of functional differential equations of mixed type, both autonomous and (asymptotically hyperbolic) nonautonomous. Such equations arise naturally in various contexts, for example, in lattice differential equations. We obtain a decomposition of the state space into stable and unstable subspaces with associated semigroups or evolutionary processes. In the autonomous case we additionally obtain representations of the semigroups in terms of retarded and advanced equations. We also obtain a factorization of the characteristic function, analogous to a WienerHopf factorization, with which we define an integer invariant for the system. Finally, we study the boundary value problem on intervals of long but finite length in the spirit of the finite section method.
Nonlocal anisotropic dispersal with monostable nonlinearity
 J. Differential Equations
"... ABSTRACT. We study the travelling wave problem J ⋆ u − u − cu ′ + f(u) = 0 in R, u(−∞) = 0, u(+∞) = 1 with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for c 6 = 0. For c = 0 we show ex ..."
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Cited by 25 (3 self)
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ABSTRACT. We study the travelling wave problem J ⋆ u − u − cu ′ + f(u) = 0 in R, u(−∞) = 0, u(+∞) = 1 with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for c 6 = 0. For c = 0 we show examples of nonuniqueness. 1. Introduction and
A nonlocal convectiondiffusion equation
 J. Functional Analysis
"... Abstract. In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut = J ∗u−u+G ∗ (f(u)) − f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial c ..."
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Cited by 23 (11 self)
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Abstract. In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut = J ∗u−u+G ∗ (f(u)) − f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convectiondiffusion equation ut = ∆u+ b · ∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convectiondiffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t → ∞ when f(u) = uq−1u with q> 1. We find the decay rate and the first order term in the asymptotic regime. 1.
Chmaj Homoclinic Solutions of an Integral Equation: Existence and Stability
 J. Diff. Eq
, 1999
"... We study existence and stability of homoclinic type solutions of a bistable integral equation. These are stationary solutions of an integrodifferential equation, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. 1 ..."
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Cited by 23 (2 self)
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We study existence and stability of homoclinic type solutions of a bistable integral equation. These are stationary solutions of an integrodifferential equation, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. 1
Decay estimates for nonlocal problems via energy estimates
 Journal de Mathematiques Pures et Applique’es
"... Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x ..."
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Cited by 18 (6 self)
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Abstract. In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut(x, t) = Rd G(x − y)(u(y, t) − u(x, t)) dy. For example, we will consider equations like, ut(x, t) =
The limit as p→∞ in a nonlocal p−Laplacian evolution equation. A nonlocal approximation of a model for sandpiles
"... Abstract. In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p→ ∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = RN J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. We prove existence and uniqueness of a limit solution that verif ..."
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Cited by 17 (8 self)
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Abstract. In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p→ ∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = RN J(x − y)u(t, y) − u(t, x)p−2(u(t, y) − u(t, x)) dy. We prove existence and uniqueness of a limit solution that verifies an equation governed by the subdifferetial of a convex energy functional associated to the indicator function of the set K = {u ∈ L2(RN) : u(x)−u(y)  ≤ 1, when x − y ∈ supp(J)}. We also find some explicit examples of solutions to the limit equation. 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;L2(Ω)) to the limit solution of the local evolutions of the p−laplacian, vt = ∆pv. This last limit problem has been proposed as a model to describe the formation of a sandpile. Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large. Finally, we give an interpretation of the limit problem in terms of MongeKantorovich mass transport theory. 1.
Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions
 Israel J. Math
"... Abstract. We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled non local problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation. 1. ..."
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Cited by 16 (2 self)
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Abstract. We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled non local problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation. 1.
Spatial Patterns, Spatial Chaos, And Traveling Waves In Lattice Differential Equations
 in: Stochastic and Spatial Structures of Dynamical Systems, eds. S.J. van Strien and S.M. Verduyn Lunel, NorthHolland
, 1996
"... We survey recent results in the theory of lattice differential equations. Such equations yield continuoustime, usually infinitedimensional, dynamical systems, which possess a discrete spatial structure modeled on a lattice. The systems we consider, generally over a higherdimensional lattice such ..."
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Cited by 16 (6 self)
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We survey recent results in the theory of lattice differential equations. Such equations yield continuoustime, usually infinitedimensional, dynamical systems, which possess a discrete spatial structure modeled on a lattice. The systems we consider, generally over a higherdimensional lattice such as ZZ D ` IR D , are the simplest nontrivial ones which incorporate both local nonlinear dynamics and short range interactions. Of particular interest are stable equilibria, and the regular patterns, or lack thereof, that are displayed. Traveling wave solutions in such systems are also discussed. 1 Introduction By a lattice differential equation or LDE we mean a system of ordinary differential equations, often of infinite order, in which the state vector u = fu j g j2 is coordinatized by a set , the lattice, which possesses some underlying spatial structure. Typical choices of ` IR D are the Ddimensional integer lattices ZZ D , the hexagonal lattice in the plane, and the crystall...