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We give a short and self-contained proof for rates of convergence of the Allen--Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based ...

This paper is concerned with the Cauchy problem $u_{t}=u_{xx}+f(t,u),x\in \mathbb{R} ,t>0$ with initial function $u(0,\cdot)=u_0 (\cdot)\in L^{\infty} (\mathbb{R})$, where $f$ is a rather general nonlinearity that is periodic in $t$, and satisfies $f(\...

We investigate the stability of time-periodic solutions of semilinear parabolic problems with Neumann boundary conditions, posed on a domain of a Riemannian manifold. On the domain we consider metrics that vary periodically in time. The discussion is based ...

Simulation of electrophysiology and intracellular ${Ca}^{2+}$ dynamics in cardiomyocytes comprises fast stochastic dynamics in tiny subcompartments, partial differential equations (PDEs) with stochastic source terms for concentration fields, and the ...

The paper treats a reaction-diffusion equation with hysteretic nonlinearity on a one-dimensional lattice. It arises as a result of the spatial discretization of the corresponding continuous model with so-called nontransverse initial data and exhibits a ...

We discuss a new notion of distance on the space of finite and nonnegative measures on $\Omega \subset {\mathbb R}^d$, which we call the Hellinger--Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich--Wasserstein ...

We consider a reaction-diffusion system where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish local well-posedness and ...

We consider a reaction-diffusion equation with a nonlinear term of the Fisher-KPP-type, depending on time $t$ and admitting two limits as $t\to\pm\infty$. We derive the set of admissible asymptotic past and future speeds of transition fronts for such an ...

We consider parameter identification for the classic Gierer--Meinhardt reaction-diffusion system. The original Gierer--Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp. 30--39] was formulated with constant parameters and has been used ...

The goal of this article is to study the boundary layers of reaction–diffusion equations in a circle and provide some numerical applications which utilize the so-called boundary layer elements. Via the boundary layer analysis, we obtain the valid ...

For a singularly perturbed parabolic equation termed in applications as the reaction-diffusion-advection equation, stationary solutions with internal transition layers (contrast structures) are studied. An arbitrary-order asymptotic approximation of ...

We study the asymptotic (long-time/long-range) exponential behavior of the solutions to reaction-diffusion equations with negative square root--type reaction terms. In contrast to the classical cases, for example, KPP-type nonlinearities, the behavior is not ...

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reaction waves. This type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the ...

We present the derivation of a CTMC with levels model of diffusion in cylindrical coordinates from the partial differential equation for Fick's law. The resulting model abstracts both molar concentration, by discrete levels, and spatial location, by ...

The anatomical variability of the human brain folds remains an unclear and challenging issue. Several hypotheses coexist for explaining the rapid development of cortical sulci and it is clear that understanding their variability would improve the ...

Multilevel additive Schwarz methods are analyzed and studied numerically for the anisotropic cardiac Bidomain model in three dimensions. This is the most complete model to date of the bioelectrical activity of the heart tissue, consisting of a ...

We present a multigrid algorithm for the solution of source identification inverse problems constrained by variable-coefficient linear parabolic partial differential equations. We consider problems in which the inversion variable is a function of space ...

In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reaction-diffusion equation. Our goal is to analyze and accurately estimate ...

The paper introduces and studies numerical methods that are fully adaptive in both three-dimensional (3D) space and time to challenging multiscale cardiac reaction-diffusion models. In these methods, temporal adaptivity comes via stepsize control in ...

In this paper, we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions. The idea is to embed the domain into a box and use a smoothing term to encode the boundary ...