A kinetic model for the elastoplastic dynamics of a jammed material is proposed, which takes the form of a nonlocal--Boltzmann-like--kinetic equation for the stress distribution function. Coarse graining this equation yields a nonlocal constitutive law for the flow, exhibiting as a key dynamic quantity the local rate of plastic events. This quantity, interpreted as a local fluidity, is spatiall...

The long time behavior of a logistic-type equation modeling the motion of cells is investigated. The equation we consider takes into account birth and death processes using a simple logistic effect as well as a nonlocal motion of cells using a nonlocal Darcy's law with regular kernel. Using the periodic framework we first investigate the well-posedness of the problem before deriving some inform...

Nonlocal problems is a major research area in many branches of modern physics, biotechnology, chemistry and engineering, which arises when it is impossible to determine the boundary values of unknown function and its derivatives. Increasingly often, there arise problems with nonlocal integral boundary conditions, especially in particle diffusion [1] and heat conduction [2, 3]. Partial different...

We study the existence of stationary solutions for a nonlocal version of the FisherKolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is a recent study by Berestycki et al. [Nonlinearity 22 (2009), pp. 2813–2844] where the nonlocal FKPP equation has been studied and it was shown for the spatial domain R and sufficiently small nonlocality that there are only two bounded non-neg...

Paraxial light in a Cole-Cole nonlocal medium: integrable regimes and singularities. ABSTRACT Nonlocal nonlinear Schrödinger-type equation is derived as a model to describe paraxial light propagation in nonlinear media with different 'degrees' of nonlocality. High frequency limit of this equation is studied under specific assumptions of Cole-Cole dispersion law and a slow dependence along propa...

The classical phase-field model represents a coupling of an Allen-Cahn type nonlinear equation with a standard diffusion equation and has been proposed to describe uniform phase separation in a pure substance. This research considers an extension of that model which incorporates a more accurate approximation to the diffuse interface between states of matter through the use of a nonlocal operato...

We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace, ∞-Laplace, mean curvature of graph, and even strongly degenerate operators. Our main results are non-trivial comparison, uniqueness, and existence results fo...