Related papers: Sharp interface limit of the Fisher-KPP equation
This letter is concerned with asymptotic analysis of a PDE model for motility of a eukaryotic cell on a substrate. This model was introduced in [1], where it was shown numerically that it successfully reproduces experimentally observed…
This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher-KPP reaction-diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the…
We study the Cauchy problem in the hyperbolic space for the heat equation with a Fisher-KPP type forcing term. Depending on the relative strength of diffusion, measured by the infimum of the spectrum of the Laplace-Beltrami operator, as…
In this work, we study a matrix-valued Allen-Cahn equation with a Saint Venant-Kirchhoff potential $F(\mathbf{A})=\frac{1}{4}\|\mathbf{A}\mathbf{A}^\top-\mathbf{I}\|^2$. Our approach employs the modulated energy method together with weak…
In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to that of…
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down…
This paper investigates the existence of almost periodic traveling fronts for Fisher-KPP lattice equations in one-dimensional almost periodic media. By the Lyapunov exponent of the linearized operator near the unstable steady state, we give…
We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian $(-\Delta)^{\alpha/2}$ on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp…
This paper studies the sharp interface limit for a mass conserving Allen-Cahn equation added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise…
A family of travelling wave solutions to the Fisher-KPP equation with speeds $c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in the…
We study the motion of phase interfaces in a diffusive lattice equation with bistable nonlinearity and derive a free boundary problem with hysteresis to describe the macroscopic evolution in the parabolic scaling limit. The first part of…
We study an $\ep$-dependent stochastic Allen--Cahn equation with a mild random noise on a bounded domain in $\mathbb{R}^n$, $n\geq 2$. Here $\ep$ is a small positive parameter that represents formally the thickness of the solution…
We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottsch\"afer and Wayne [28] which establish stability of fronts without identifying a precise decay rate. We…
We consider one dimensional porous media equations in spatially periodic environment. We will construct a periodic traveling sharp wave whose profile tends to a positive steady state at left infinity and takes zero on the right half line,…
We combine random walks, growth and decay, and convection, in a Monte Carlo simulation to model 1D interface dynamics with fluctuations. The continuum limit corresponds to the deterministic Fisher equation with convection. We find…
We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed $\epsilon^{-1}$ ($\epsilon\textgreater{}0$), and proliferate according to a reaction…
We investigate the sharp material interface limit of the Darcy-Boussinesq model for convection in layered porous media with diffused material interfaces, which allow a gradual transition of material parameters between different layers. We…
We consider the Fisher-KPP reaction-diffusion equation in the whole space. We prove that if a solution has, to main order and for all times (positive and negative), the same exponential decay as a planar traveling wave with speed larger…
We identify the $\Gamma$-limit of a nanoparticle-polymer model as the number of particles goes to infinity and as the size of the particles and the phase transition thickness of the polymer phases approach zero. The limiting energy consists…
We consider the large time behaviour of solutions to the porous medium equation with a Fisher-KPP type reaction term and nonnegative, compactly supported initial function in $L^\infty(\mathbb{R}^N)\setminus\{0\}$: \begin{equation}…