scholarly journals Nonlocal Diffusion Equations with Integrable Kernels

2020 ◽  
Vol 67 (08) ◽  
pp. 1
Author(s):  
Julio D. Rossi
Keyword(s):  
Diffusion Equations ◽  
Nonlocal Diffusion

Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

2021 ◽  
pp. 1-18
Author(s):  
Yuming Qin ◽  
Bin Yang
Keyword(s):  
Weak Solution ◽  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Force Term ◽  
Pullback Attractors ◽  
Critical Exponential Growth ◽  
Nonclassical Diffusion Equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions

2018 ◽  
Vol 39 (2) ◽  
pp. 607-625 ◽  
Author(s):  
Qiang Du ◽  
Yunzhe Tao ◽  
Xiaochuan Tian ◽  
Jiang Yang
Keyword(s):  
Green's Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Optimal Order ◽  
Numerical Approximations ◽  
Splitting Technique ◽  
Singular Measures ◽  
Nonlocal Models

AbstractNonlocal diffusion equations and their numerical approximations have attracted much attention in the literature as nonlocal modeling becomes popular in various applications. This paper continues the study of robust discretization schemes for the numerical solution of nonlocal models. In particular, we present quadrature-based finite difference approximations of some linear nonlocal diffusion equations in multidimensions. These approximations are able to preserve various nice properties of the nonlocal continuum models such as the maximum principle and they are shown to be asymptotically compatible in the sense that as the nonlocality vanishes, the numerical solutions can give consistent local limits. The approximation errors are proved to be of optimal order in both nonlocal and asymptotically local settings. The numerical schemes involve a unique design of quadrature weights that reflect the multidimensional nature and require technical estimates on nonconventional divided differences for their numerical analysis. We also study numerical approximations of nonlocal Green’s functions associated with nonlocal models. Unlike their local counterparts, nonlocal Green’s functions might become singular measures that are not well defined pointwise. We demonstrate how to combine a splitting technique with the asymptotically compatible schemes to provide effective numerical approximations of these singular measures.

Asymptotically compatible schemes for space-time nonlocal diffusion equations

2017 ◽  
Vol 102 ◽  
pp. 361-371 ◽  
Author(s):  
An Chen ◽  
Qiang Du ◽  
Changpin Li ◽  
Zhi Zhou
Keyword(s):  
Space Time ◽  
Diffusion Equations ◽  
Nonlocal Diffusion

scholarly journals Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread

2020 ◽  
Vol 18 (04) ◽  
pp. 585-614 ◽  
Author(s):  
Yuan-Hang Su ◽  
Wan-Tong Li ◽  
Fei-Ying Yang
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Nonlocal Dispersal ◽  
Asymptotic Behaviors ◽  
Cost Parameter ◽  
The Cost

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator [Formula: see text] and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalized principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter [Formula: see text]. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter [Formula: see text] is in a different range. In particular, for the case [Formula: see text], we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition.

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