DCDS
Parabolic singular limit of a wave equation with localized boundary damping
Aníbal Rodríguez-Bernal Enrique Zuazua
Discrete & Continuous Dynamical Systems - A 1995, 1(3): 303-346 doi: 10.3934/dcds.1995.1.303
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
DCDS
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
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 483-497 doi: 10.3934/dcds.2007.18.483
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.
DCDS
Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems
Aníbal Rodríguez-Bernal Alejandro Vidal–López
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 537-567 doi: 10.3934/dcds.2007.18.537
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
PROC
Dynamic boundary conditions as limit of singularly perturbed parabolic problems
Ángela Jiménez-Casas Aníbal Rodríguez-Bernal
Conference Publications 2011, 2011(Special): 737-746 doi: 10.3934/proc.2011.2011.737
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
CPAA
A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities
Aníbal Rodríguez-Bernal Alejandro Vidal-López
Communications on Pure & Applied Analysis 2014, 13(2): 635-644 doi: 10.3934/cpaa.2014.13.635
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
DCDS-B
Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem
Aníbal Rodríguez-Bernal Robert Willie
Discrete & Continuous Dynamical Systems - B 2005, 5(2): 385-410 doi: 10.3934/dcdsb.2005.5.385
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
PROC
Linear model of traffic flow in an isolated network
Ángela Jiménez-Casas Aníbal Rodríguez-Bernal
Conference Publications 2015, 2015(special): 670-677 doi: 10.3934/proc.2015.0670
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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