In the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a ...

We develop the theory of nonlinear localized modes (intrinsic localized modes or discrete breathers) in two-dimensional (2D) photonic crystal waveguides. We consider different geometries of the waveguides created by an array of nonlinear dielectric rods embedded into an otherwise perfect linear 2D photonic crystal, and demonstrate that the effective interaction in such waveguides is nonlocal, b...

Nonlocal effects occur in many nonequilibrium interfaces, due to diverse physical mechanisms like diffusive, ballistic, or anomalous transport, with examples from flame fronts to thin films. While dimensional analysis describes stable nonlocal interfaces, we show the morphologically unstable condition to be nontrivial. This is the case for a family of stochastic equations of experimental releva...

We consider nonlinear modes of the nonlinear Schrödinger equation with nonlocal nonlinearities and and PT -symmetric parabolic potential. We show that there exists a set of continuous families of nonlinear modes and study their linear stability in the limit of small nonlinearity. It is demonstrated that either PT symmetry or the nonlocality can be used to manage the stability of the small-ampli...

In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, establish the existence at least one weak solution for parameter-dependent nonlinearity multiple solutions perturbed nonlinearity. Our argument is based on variational approximation approach.

[1] Models for the overall topographic evolution of erosional and depositional systems can be grouped into two broad classes. The first class is local models in which the sediment flux at a point is expressed as a linear or nonlinear function of local hydrogeomorphic measures (e.g., water discharge and slope). The second class is nonlocal models, where the sediment flux at a point is expressed ...

We demonstrate the existence of stable three-dimensional spatiotemporal solitons (STSs) in media with a nonlocal cubic nonlinearity. Fundamental (nonspinning) STSs forming one-parameter families are stable if their propagation constant exceeds a certain critical value that is inversely proportional to the range of nonlocality of nonlinear response. All spinning three-dimensional STSs are found ...

In this paper, we investigate the local existence and the finite-time blow-up of solutions of semilinear parabolic system with nonlocal in time nonlinearity. In addition, we also give the blow-up rate and necessary conditions for local and global existence.

In this paper, we study a class of nonlocal semilinear elliptic problems with inhomogeneous strong Allee effect. By means of variational approach, we prove that the problem has at least two positive solutions for large λ under suitable hypotheses about nonlinearity. We also prove some nonexistence results. In particular, we give a positive answer to the conjecture of Liu-Wang-Shi.

This article concerns the bistable traveling wavefronts of a nonlocal dispersal equation with convolution type bistable nonlinearity. Applying a homotopy method, we establish the existence of traveling wavefronts. If the wave speed does not vanish, i.e. c 6= 0, then the uniqueness (up to translation) and the globally asymptotical stability of traveling wavefronts are proved by the comparison pr...