Related papers: Inertial manifolds on squeezed domains
We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad\text{in }\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad\text{in }\quad \Omega, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminus\Omega,…
Given $\rho\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s…
The paper investigates the existence and upper semicontinuity of uniform attractors of the perturbed non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearity: $u_{tt}-\Delta u_{t}-(1+\epsilon\|\nabla…
We formulate the theory for a self-propelled domain in an excitable reaction-diffusion system in two dimensions where the domain deforms from a circular shape when the propagation velocity is increased. In the singular limit where the width…
We study a monostable reaction-diffusion equation of the form $u_t=du_{xx}+f(u)$ over a semi-infinite spatial domain $[g(t),\infty)$, with $x=g(t)$ the free boundary whose evolution is governed by equations derived from a ``preferred…
In this paper we show that there exists a family of domains $\Omega_{\varepsilon}\subseteq\mathbb{R}^N$ with $N\ge2$, such that the $stable$ solution of the problem \[ \begin{cases} -\Delta u= g(u)&\hbox{in }\Omega_\varepsilon\\…
We study a one dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equation \begin{equation*} \partial_t u = \varepsilon \partial_x^2 u -\partial_x…
We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For…
In this paper a reaction-diffusion type equation is the starting point for setting up a genuine thermodynamic reduction, i.e. involving a finite number of parameters or collective variables, of the initial system. This program is carried…
In this paper we consider the problem $$(P)\qquad \{{array}{rclll} u_t-\D u^m&=&|\n u|^q +\,f(x,t),&\quad u\ge 0 \hbox{in} \Omega_T\equiv \Omega\times (0,T), u(x,t)&=&0 &\quad \hbox{on} \partial\Omega\times (0,T) u(x,0)&=&u_0(x),&\quad x\in…
In this paper, we are interested in the properties of solution of the nonlocal equation $$\begin{cases}u_t+(-\Delta)^su=f(u),\quad t>0, \ x\in\mathbb{R}\\ u(0,x)=u_0(x),\quad x\in\mathbb{R}\end{cases}$$ where $0\le u_0<1$ is a Heaviside…
The reversible reactions like A+B <-> C in the many-component diffusive system affect the diffusive properties of the constituents. The effective conjugation of irreversible processes of different dimensionality takes place due to the…
We consider a semi-discrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain $\Omega \subset \mathbb{R}^2$, such that the curve…
This work is devoted to infinite-energy solutions of semi-linear wave equations in unbounded smooth domains of $\mathbb{R}^3$ with fractional damping of the form $(-\Delta_x+1)^\frac{1}{2}\partial_t u$. The work extends previously known…
The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…
For 3D reaction--diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic…
This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess…
We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…
In this paper we consider a bistable reaction-diffusion equation in unbounded domains and we investigate the existence of propagation phenomena, possibly partial, in some direction or, on the contrary, of blocking phenomena. We start by…
We study dynamics of interfaces in solutions of the equation $\epsilon \Box u + \frac 1 \epsilon f_\epsilon(u)=0$, for $f_\epsilon$ of the form $f_\epsilon(u) = (u^2-1)(2u- \epsilon\kappa)$, for $\kappa\in {\mathbb R}$, as well as more…