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Asymptotic properties of solutions of odd-order nonlinear dispersion equations are studied. The global in time similarity solutions, which lead to eigenfunctions of the rescaled ODEs, are constructed.
This study handles spatial three-dimensional solution of the nonlinear diffusion equation without particular initial conditions. The functional behavior of the equation and the concentration have been studied in new ways. An auxiliary…
We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite…
In this article, we study the long-time asymptotic properties of a non-linear and non-local equation of diffusive type which describes the rock-paper-scissors game in an interconnected population.We fully characterize the self-similar…
In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous…
In this work, we introduce a new difference equation which is discrete analogue of Diffusion differential equation and analyze some essential spectral properties, Diffusion difference operator is self-adjoint, eigenvalues of this problem…
A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can…
Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase surface, based on the elegant Gaussian modes of optical…
We present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations,…
We study the diffusion equation with an appropriate change of variables. This equation is in general a partial differential equation (PDE). With the self-similar and related Ansat\"atze we transform the PDE of diffusion to an ordinary…
We study the long-time dynamics of the nonlinear processes modeled by diffusion-transport partial differential equations in non-divergence form with drifts. The solutions are subject to some inhomogeneous Dirichlet boundary condition.…
The silo discharge process is studied by molecular dynamics simulations. The development of the velocity profile and the probability density function for the displacements in the horizontal and vertical axis are obtained. The PDFs obtained…
Several nonlinear stochastic differential equations have been proposed in connection with self-organized critical phenomena. Due to the threshold condition involved in its dynamic evolution an infinite number of nonlinearities arises in a…
A nonlinear inequality is formulated in the paper. An estimate of the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be…
We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general…
This paper investigates the well-posedness and small-noise asymptotics of a class of stochastic partial differential equations defined on a bounded domain of $\mathbb{R}^d$, where the diffusion coefficient depends nonlinearly and…
In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic string theory. These equations are of much interest…
We consider the slow nonlinear diffusion equation subject to a constant absorption rate and construct local self-similar solutions for reversing (and anti-reversing) interfaces, where an initially advancing (receding) interface gives way to…
Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular,…
The question is studied whether weak solutions of linear partial integrodifferential equations approach a constant spatial profile after rescaling, as time goes to infinity. The possible limits and corresponding scaling functions are…