Is a nonlocal diffusion strategy convenient for biological populations in competition?

Is Medical Application Software a New Strategy for Oncologists?

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

Asymptotic Expansions for Nonlocal Diffusion

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form ut = J ∗ u − u in the whole R with an initial condition u(x, 0) = u0(x). Under suitable hypotheses on J (involving its Fourier transform) and u0, it is proved an expansion of the form ∥∥u(u)− ∑ |α|≤k (−1)|α| α! ( ∫ u0(x)x dx ) ∂Kt ∥∥ Lq(Rd) ≤ Ct−A, where Kt is the regular part of the fundamental solution and...

A nonlocal continuum model for biological aggregation.

We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short-range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady-s...

A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions

We develop and analyze an optimization–based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the ...