Related papers: Spreading speeds for one-dimensional monostable re…
Many monostable reaction-diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present…
We study a porous medium equation with fractional potential pressure: $$ \partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-\Delta)^{-s}u, $$ for $m>1$, $0<s<1$ and $u(x,t)\ge 0$. To be specific, the problem is posed for $x\in…
We establish the existence of semi-wavefronts solutions for a non-local delayed reaction-diffusion equation with monostable nonlinearity. The existence result is proved for all speeds $c\geq c_\star$, where the determination of $c_\star$ is…
In this paper, we study the spreading speeds and traveling wave solutions of the PDE $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) + u(1-u),\ \ x\in\mathbb{R}^N 0=\Delta v-v+u, \ \ x\in\mathbb{R}^N, \end{cases} $$ where…
In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating dissipation (monotone and non monotone) and singular nonlinear diffusions are considered. In…
We study the single particle velocity distribution for a granular fluid of inelastic hard spheres or disks, using the Enskog-Boltzmann equation, both for the homogeneous cooling of a freely evolving system and for the stationary state of a…
Much has been studied on the spreading speed and traveling wave solutions for cooperative reaction-diffusion systems. In this paper, we shall establish the spreading speed for a large class of non-cooperative reaction-diffusion systems and…
In this paper we focus on three problems about the spreading speeds of nonlocal dispersal Fisher-KPP equations. First, we study the signs of spreading speeds and find that they are determined by the asymmetry level of the nonlocal dispersal…
This paper is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equations with Kolmogrov-Petrovsky-Piskunov (KPP) type nonlinearities in general periodic domains or in infinite cylinders with oscillating…
The dynamics of a point charged particle which is driven by a uniform external electric field and moves in a medium of elastic scatterers is investigated. Using rudimentary approaches, we reproduce, in one dimension, the known results that…
We study reaction-diffusion equations of various types in the half-space. For bistable reactions with Dirichlet boundary conditions, we prove conditional uniqueness: there is a unique nonzero bounded steady state which exceeds the bistable…
The goal of this paper is to find the homogenized equation of a heterogenous Fisher-KPP model in a periodic medium. The solutions of this model are pulsating travelling fronts whose \emph{speeds} are superior to a parametric minimal speed…
This paper is concerned with spreading properties of space-time heterogeneous Fisher--KPP equations in one space dimension. We focus on the case of everywhere favorable environment with three different zones, a left half-line with slow or…
This paper is concerned with the propagating speeds of transition fronts in $R^N$ for spatially periodic bistable reaction-diffusion equations. The notion of transition fronts generalizes the standard notions of traveling fronts. Under the…
We consider reaction-diffusion equations of porous medium type, with different kind of reaction terms, and nonnegative bounded initial data. For all the reaction terms under consideration there are initial data for which the solution…
We prove that if the initial condition of the Swift-Hohenberg equation $\partial_t u(x,t)=\bigl(\epsilon^2-(1+\partial_ x^2)^2\bigr) u(x,t) -u^3(x,t)$ is bounded in modulus by $Ce^{-\beta x}$ as $x\to+\infty $, the solution cannot propagate…
The work proposes and studies a one-dimensional model, which involves nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation and the general…
We consider reaction-diffusion equations of the form \begin{equation*} u_t - d u_{xx} = f(t,u), \quad t>0,\ \ x \in [g(t), h(t)], \end{equation*} where $f(t,u)$ is periodic in $t$ and monostable in $u$, and the interval $[g(t), h(t)]$…
We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix $a^\varepsilon$ that is periodic with characteristic length…
We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on…