Parabolic singular limit of a wave equation with localized boundary damping
Aníbal Rodríguez-Bernal Enrique Zuazua
We will consider the family of wave equations with boundary damping

$\qquad\qquad \qquad\qquad \epsilon u_{t t} -\Delta u + \lambda u =f $ on $\Omega \times (0,T)$

$(P_{\epsilon, \lambda, \Gamma_0})\qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $

$\qquad\qquad u=0 $ on $\Gamma_0 \times (0,T)$

where $0< \epsilon \leq \epsilon_0$, $\Omega \subset \mathbb R^N$ is a regular open connected set, $\lambda \geq 0$ and $\Gamma = \Gamma_0\cup \Gamma_1$ is a partition of the boundary of $\Omega$. We will also consider the case where $\Gamma_0$ is empty (see below for more precise assumptions on $\lambda$, $\Omega$ and $\Gamma_0$, $\Gamma_1$).
For this problem the corresponding formal singular perturbation at $\epsilon =0$ is

$\qquad\qquad \qquad\qquad -\Delta u + \lambda u =f$ on $\Omega \times (0,T) $

$(P_{0, \lambda, \Gamma_0}) \qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $

$\qquad\qquad u=0 $ on $ \Gamma_0 \times (0,T)$

We are here concerned with the well possedness of both problems for the non--homogeneous case, i.e. $f=f(t,x)$, $g=g(t,x)$, and with the convergence, as $\epsilon$ approaches $0$, of the solutions of $(P_{\epsilon, \lambda, \Gamma_0})$ to solutions of $(P_{0, \lambda, \Gamma_0})$.

keywords: boundary damping singular perturbation. Wave equation
Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations
José A. Langa James C. Robinson Aníbal Rodríguez-Bernal A. Suárez A. Vidal-López
The goal of this work is to study the forward dynamics of positive solutions for the non-autonomous logistic equation $u_{t}-\Delta u=\lambda u-b(t)u^{p}$, with $p>1$, $b(t)>0$, for all $t\in \mathbb{R}$, $\lim_{t\to \infty }b(t)=0$. While the pullback asymptotic behaviour for this equation is now well understood, several different possibilities are realized in the forward asymptotic regime.
keywords: Unbounded forwards attractor Non-autonomous reaction diffusion equations.
Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems
Aníbal Rodríguez-Bernal Alejandro Vidal–López
We give conditions for the existence of a unique positive complete trajectories for non-autonomous reaction-diffusion equations. Also, attraction properties of the unique complete trajectory is obtained in a pullback sense and also forward in time. As an example, a non-autonomous logistic equation is considered.
keywords: forward behaviour. pull--back attraction Logistic equations
Dynamic boundary conditions as limit of singularly perturbed parabolic problems
Ángela Jiménez-Casas Aníbal Rodríguez-Bernal
We obtain dynamic boundary conditions as a limit of parabolic problems with null flux where the time derivative concentrates near the boundary.
keywords: flux null Dynamic boundary conditions parabolic problems concentrating integrals
A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities
Aníbal Rodríguez-Bernal Alejandro Vidal-López
We show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in $L^q(\Omega)$ for each $1\leq q < \infty$. In particular, we do not assume restriction on the growth of the nonlinearites required by the standar local existence theory.
keywords: asymptotic behavior. Singular initial data global existence critical exponents strong solutions reaction--diffusion
Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem
Aníbal Rodríguez-Bernal Robert Willie
We make precise the sense in which spatial homogenization to a constant function in space is attained in a linear parabolic problem when large diffusion in all parts of the domain is assumed. Also interaction between diffusion and boundary flux terms is considered. Our starting point is a detailed analysis of the large diffusion effects on the associated elliptic and eigenvalue problems. Here convergence is shown in the energy space $H^1(\Omega)$ and in the space of continuous functions $C(\overline{\Omega})$. In the parabolic case we prove convergence in the functional space $L^\infty ((0,T),L^2(\Omega)) \bigcap L^2 ((0,T),H^1(\Omega)).
keywords: convergence of solutions. linear elliptic problem eigenvalue problem linear parabolic problem non homogeneous boundary conditions large diffusion analytic semigroups
Linear model of traffic flow in an isolated network
Ángela Jiménez-Casas Aníbal Rodríguez-Bernal
We obtain a mathematical linear model which describes automatic operation of the traffic of material objects in a network. Existence and global solutions is obtained for such model. A related model which used outdated information is shown to collapse in finite time.
keywords: traffic flow Networks delay integral systems.

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