Metastability and Stability of Patterns in a Convolution Model for Phase Transitions
TLDR
In this article, a spatial pattern is generated, and it persists on a time scale OðeceÞ but is eventually destroyed: typically uðx; tÞ! 1 uðex; t ðuÞ du 1⁄4 0, for x on 1¼20; 1 as t! 1: It is possible that uð x; t ǫ converges to a or a nonconstant steady state, but this is an unlikely event because all these steady states are unstable.About:
This article is published in Journal of Differential Equations.The article was published on 2002-08-10 and is currently open access. It has received 83 citations till now.read more
Citations
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The evolution of dispersal
TL;DR: It is shown that, as in the case of reaction-diffusion models, for fixed spread slower rates of diffusion are always optimal and fixing the dispersal rate and varying the spread while assuming a constant cost of dispersal leads to more complicated results.
Book ChapterDOI
Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions
TL;DR: In this article, an overview is given of some nonlinear parabolic-like evolution problems which are off the classical beaten track, but have increased in importance during the past decade.
Journal ArticleDOI
Asymptotic behavior for nonlocal diffusion equations
TL;DR: In this paper, the authors studied the asymptotic behavior of nonlocal diffusion models with Dirichlet or Neumann boundary conditions and showed that the long time behavior of the solutions is determined by the Fourier transform of J near the origin, which is linked to the behavior of J at infinity.
Journal ArticleDOI
How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems
TL;DR: In this paper, a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary is presented, and the limit of this family of non-local diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero is studied.
Journal ArticleDOI
Boundary fluxes for nonlocal diffusion
TL;DR: In this paper, a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary is studied, and the authors prove existence, uniqueness and a comparison principle for the problem.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Journal ArticleDOI
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening
Samuel M. Allen,John W. Cahn +1 more
A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening
TL;DR: In this paper, a microscopic diffusional theory for the motion of a curved antiphase boundary is presented, where the interfacial velocity is linearly proportional to the mean curvature of the boundary, but unlike earlier theories the constant of proportionality does not include the specific surface free energy.
Journal ArticleDOI
Multidimensional Nonlinear Diffusion Arising in Population Genetics
Journal ArticleDOI
The approach of solutions of nonlinear diffusion equations to travelling front solutions
TL;DR: In this paper, the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut-uxx-∞;(u)=O, x∈( ∞, ∞), in the case ∞(0)=∞(1)=0,