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The random walk process in a nonhomogeneous medium, characterised by a L\'evy stable distribution of jump length, is discussed. The width depends on a position: either before the jump or after that. In the latter case, the density slope is…
Accurate prediction of mixing transition induced by interfacial instabilities is vital for engineering applications, but has remained a great challenge for decades. For engineering practices, Reynolds-averaged Navier-Stokes simulation…
We extend the derivation of the time-dependent Hartree-Fock equation recently obtained in [2] to fermions with a relativistic dispersion law. The main new ingredient is the propagation of semiclassical commutator bounds along the…
We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a L\'evy process and can reach resources in a…
Stability of the kink-like and soliton-like travelling wave solutions to the generalized convection-reaction-diffusion equation is studied by means of the qualitative methods and numerical simulation.
We examine the validity of the kinetic description of wave turbulence for a model quadratic equation. We focus on the space-inhomogeneous case, which had not been treated earlier; the space-homogeneous case is a simple variant. We determine…
We address the propagation into an unstable state of a localised disturbance in a forward-backward diffusion pseudo-parabolic equation. Three asymptotic regimes are distinguished as t tends to infinity, the first being a regime ahead of the…
The lateral diffusion coefficient of a Brownian particle on a two-dimensional random surface is studied in the quenched limit for which the surface configuration is time-independent. We start with the stochastic equation of motion for a…
We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. It is a new…
We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an $\alpha$-entropy…
We study the evolution of a wave packet in a nonlinear Schr\"odinger lattice equation subject to a dc bias. In the absence of nonlinearity all normal modes are spatially localized giving rise to a Stark ladder with an equidistant eigenvalue…
According to classical non-relativistic Schr\"odinger equation, any local perturbation of wave function instantaneously affects all infinite region, because this equation is of parabolic type, and its solutions demonstrate infinite speed of…
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and…
We propose a novel method for establishing the convergence rates of solutions to reaction-diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in [2]. It turns out…
We study the traveling wave solutions to a reaction diffusion system modeling the public goods game with altruistic behaviors. The existence of the waves is derived through monotone iteration of a pair of classical upper- and lower…
We study a non-local parabolic Lotka-Volterra type equation describing a population structured by a space variable x 2 Rd and a phenotypical trait 2 . Considering diffusion, mutations and space-local competition between the individuals, we…
We investigate spreading properties of solutions of a large class of two-component reaction-diffusion systems, including prey-predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts…
In this paper, we discuss existence and finite speed of propagation for the solutions to an initial-boundary value problem for a family of fractional thin-film equations in a bounded domain in $\mathbb{R}^d$. The nonlocal operator we…
Several experiments on tagged molecules or particles in living systems suggest that they move anomalously slow - their mean squared displacement (MSD) increase slower than linearly with time. Leading models aimed at understanding these…
We consider an evolution equation whose time-diffusion is of fractional type and we provide decay estimates in time for the $L^s$-norm of the solutions in a bounded domain. The spatial operator that we take into account is very general and…