Información del autor
Autor Rodríguez Bernal, Aníbal |
Documentos disponibles escritos por este autor (73)
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Arrieta Algarra, José María ; Cholewa, Jan W. ; Dlotko, Tomasz ; Rodríguez Bernal, Aníbal | Elsevier | 2004In this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an-unbounded domain. The functional setting is based on standard Lebesgue and Sobolev-Lebesgue spaces. We show how the reaction and diff[...]texto impreso
Arrieta Algarra, José María ; Pardo San Gil, Rosa ; Rodríguez Bernal, Aníbal | Elsevier | 2015-12-05We analyze the asymptotic behavior of positive solutions of parabolic equations with a class of degenerate logistic nonlinearities of the type lambda u - n(x)u(rho). An important characteristic of this work is that the region where the logistic [...]texto impreso
In this paper we analyze the long time behavior of a phase field model by showing the existence of global compact attractors in the strong norm of high order Sobolev spaces.texto impreso
We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Gamma of the boundary and this neighborhood shrinks to Gamma as a parameter epsilo[...]texto impreso
In this paper, we establish the global fast dynamics for the time-dependent Ginzburg-Landau equations of superconductivity. We show the squeezing property and the existence of finite-dimensional exponential attractors for the system. In addition[...]texto impreso
The title of the paper says it all. The author considers a reaction-diffusion equation with nonlinear boundary conditions on a bounded domain ??R N . The initial value u 0 is allowed to be a function in L r (?) , with 1texto impreso
In this paper, we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in R-n. We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in L-2(R-n[...]texto impreso
Arrieta Algarra, José María ; Carvalho, Alexandre N. ; Rodríguez Bernal, Aníbal | Taylor & Francis | 2000The authors study the asymptotic behavior of solutions to a semilinear parabolic problem u t ?div(a(x)?u)+c(x)u=f(x,u) for u=u(x,t), t> 0, x????R N , a(x)> m> 0; u(x,0)=u 0 with nonlinear boundary conditions of the form u=0 on ? 0 , and a(x[...]texto impreso
Arrieta Algarra, José María ; Pardo San Gil, Rosa ; Rodríguez Bernal, Aníbal | Cambridge University Press | 2007-04We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation prod[...]texto impreso
Arrieta Algarra, José María ; Rodríguez Bernal, Aníbal ; Souplet, Philippe | Scuola Normale Superiore | 2004We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions u: either the space derivative u., blows up in finite time (with u i[...]texto impreso
The Cauchy problem for the time-dependent Ginzburg-Landau equations of superconductivity in R-d (d = 2, 3) is investigated in this paper. When d = 2, we show that the Cauchy problem for this model is well posed in L-2. When d = 3, we establish t[...]texto impreso
We study the possible continuation of solutions of a nonlinear parabolic problem after the blow-up time. The nonlinearity in the equation is dissipative and blow-up is caused by the nonlinear boundary condition of the form ?u/??=|u|q-1u, where q[...]texto impreso
The dynamics of a closed thermosyphon are considered. Using an explicit construction, obtained through an inertial manifold, exact low-dimensional models are derived. The behavior of solutions is analyzed for different ranges of the relevant par[...]texto impreso
Arrieta Algarra, José María ; Carvalho, Alexandre N. ; Langa, José A. ; Rodríguez Bernal, Aníbal | Springer | 2012-09In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous d[...]texto impreso
Due to the lack of the maximum principle the analysis of higher order parabolic problems in RN is still not as complete as the one of the second-order reaction-diffusion equations. While the critical exponents and then a dissipative mechanism in[...]texto impreso
Arrieta Algarra, José María ; Carvalho, Alexandre N. ; Rodríguez Bernal, Aníbal | Elsevier | 1998-08We prove existence, uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions. We study these problems with initial data in L-q(Ohm), W-1,W-q(Ohm), 1texto impreso
Rodríguez Bernal, Aníbal ; Van Vleck, Erik S. | Society for Industrial and Applied Mathematics | 1998-08The dynamics of a closed loop thermosyphon are considered. The model assumes a prescribed heat flux along the loop wall and the contribution of axial diffusion. The well-posedness of the model which consists of a coupled ODE and PDE is shown for[...]texto impreso
It is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems. In this paper we investigate the dissipative mechanism for some semilinear fourth-order parabolic equ[...]texto impreso
Arrieta Algarra, José María ; Cholewa, Jan W. ; Dlotko, Tomasz ; Rodríguez Bernal, Aníbal | Wiley-Blackwell | 2007The Cauchy problem for a semilinear second order parabolic equation u(t) = Delta u + f (x, u, del u), (t, x) epsilon R+ x R-N, is considered within the semigroup approach in locally uniform spaces W-U(s,p) (R-N). Global solvability, dissipativen[...]texto impreso
Jiménez Casas, Ángela ; Rodríguez Bernal, Aníbal | American Institute of Mathematical Sciences | 2011We obtain dynamic boundary conditions as a limit of parabolic problems with null flux where the time derivative concentrates near the boundary.texto impreso
We obtain nonhomogeneous dynamic boundary conditions as a singular limit of a parabolic problem with null flux and potentials and reaction terms concentrating at the boundary.texto impreso
Arrieta Algarra, José María ; Rodríguez Bernal, Aníbal ; Valero , José | World Scientific Publishing | 2006We study the nonlinear dynamics of a reaction-diffusion equation where the nonlinearity presents a discontinuity. We prove the upper semicontinuity of solutions and the global attractor with respect to smooth approximations of the nonlinear term[...]texto impreso
In this Note we study the asymptotic behavior of reaction diffusion equations with nonlinear boundary conditions. We obtain balance conditions between the reaction term and the nonlinear flux term which imply boundedness of solutions or blow-up [...]texto impreso
Arrieta Algarra, José María ; Pardo San Gil, Rosa María ; Rodríguez Bernal, Aníbal | Elsevier | 2009We consider a parabolic equation ut??u+u=0 with nonlinear boundary conditions , where as |s|??. In [J.M. Arrieta, R. Pardo, A. Rodríguez-Bernal, Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity[...]texto impreso
Rodríguez Bernal, Aníbal ; Vidal López, Alejandro ; Langa, J.A. ; Robinson, James C. ; Suárez, A. | American Institute of Mathematical Sciences | 2007The goal of this work is to study the forward dynamics of positive solutions for the nonautonomous logistic equation ut ? _u = _u ? b(t)up, with p > 1, b(t) > 0, for all t 2 R, limt!1 b(t) = 0. While the pullback asymptotic behaviour for this [...]texto impreso
Rodríguez Bernal, Aníbal ; Vidal López, Alejandro | American Institute of Mathematical Sciences | 2007We 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 ti[...]texto impreso
The authors find a growth condition on the nonlinear term f(x, u) of a nonlinear heat equation which ensures the existence of maximal and minimal equilibria that bound asymptotically all solutions to that nonlinear heat equation.texto impreso
We consider a reaction diffusion equation u(t) = Delta u + f(x, u) in R-N with initial data in the locally uniform space (L) over dot(U)(q)(R-N), q is an element of [1, infinity), and with dissipative nonlinearities satisfying sf(x, s) N/2. U[...]texto impreso
In this well-written paper, the authors consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. \par In the first part of the paper, some notions concerning d[...]texto impreso
We show the existence of two special equilibria, the extremal ones, for a wide class of reaction–diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and[...]texto impreso
We analyse the dynamics of a fluid transporting a soluble substance in the interior of a closed loop of arbitrary geometry and subjected to the action of gravity and natural convection. After obtaining the governing equations and analysing the w[...]texto impreso
Arrieta Algarra, José María ; Jiménez Casas, Ángela ; Rodríguez Bernal, Aníbal | World Scientific Publ Co Pte Ltd | 2005We analyze the limit of solutions of an elliptic problem, with zero flux boundary conditions when the reaction terms are concentrated in a neighborhood of the boundary and that shrinks to the boundary as a parameter goes to zero. We prove that t[...]texto impreso
Arrieta Algarra, José María ; Jiménez Casas, Ángela ; Rodríguez Bernal, Aníbal | Universidad Autónoma Madrid | 2008We analyze the limit of the solutions of an elliptic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Gamma of the boundary and this neighborhood shrinks to Gamma as a parameter goes to zero. We prov[...]texto impreso
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Arrieta Algarra, José María ; Pardo San Gil, Rosa María ; Rodríguez Bernal, Aníbal | World Scientific Publishing | 2010Summary: "We consider an elliptic equation ??u+u=0 with nonlinear boundary conditions ?u/?n=?u+g(?,x,u) , where (g(?,x,s))/s?0 as |s|?? . In [Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 2, 225--252; MR2360769 (2009d:35194); J. Differentia[...]texto impreso
In this paper we consider some fourth order linear and semilinear equations in R-N and make a detailed study of the solvability of the Cauchy problem. For the linear equation we consider some weakly integrable potential terms, and for any 1texto impreso
Linear 2m-th order uniformly elliptic operators are shown to generate semigroups of bounded linear operators with suitable smoothing properties in scales of locally uniform Bessel's and Lebesgue's spaces.texto impreso
The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ?N, graphs, manifolds, multi-structures and some fractal sets. For this, [...]texto impreso
Arrieta Algarra, José María ; Rodríguez Bernal, Aníbal ; Cholewa, Jan W. ; Dlotko, Tomasz | World Scientific | 2004We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharp L-p - L-q-type estimates in uniform spaces for heat and Schrodinger semigroups and analyze the regularizing effect and the exponential type of these semigroup[...]texto impreso
We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable[...]texto impreso
In this work we analyze the existence of solutions that blow-up in finite time for a reaction-diffusion equation ut??u=f(x,u) in a smooth domain ? with nonlinear boundary conditions ?u?n=g(x,u). We show that, if locally around some point of the [...]texto impreso
Arrieta Algarra, José M. ; Pardo, Rosa ; Rodríguez Bernal, Aníbal | Department of Mathematics Texas State University | 2014We analyze the behavior of positive solutions of elliptic equations with a degenerate logistic nonlinearity and Dirichlet boundary conditions. Our results concern existence and strong localization in the spatial region in which the logistic nonl[...]texto impreso
We analyze singular perturbations in elliptic equations, subjected to various boundary conditions, in which the diffusion is going to infinity in localized regions inside the domain and therefore solutions undergo a localized spatial homogenizat[...]texto impreso
We study the asymptotic behaviour in large diffusivity of inertial manifolds governing the long time dynamics of a semilinear evolution system of reaction and diffusion equations. A priori, we review both local and global dynamics of the system [...]texto impreso
Arrieta Algarra, José María ; Rodríguez Bernal, Aníbal | World Scientific Publ. Co. Pte. Ltd. | 2004-10In this paper we show that several known critical exponents for nonlinear parabolic problems axe optimal in the sense that supercritical problems are ill posed in a strong sense. We also give an answer to an open problem proposed by Brezis and C[...]texto impreso
The authors consider a reaction-diffusion equation in a bounded smooth domain ??R n with nonlinear flux terms on the boundary of ? . They derive suitable conditions on the nonlinear terms of the problem which imply its dissipativity. The autho[...]texto impreso
Cholewa, Jan W. ; Rodríguez Bernal, Aníbal | Institute of Mathematics, Academy of Sciences of the Czech Republic | 2014We consider the Cahn-Hilliard equation in H1(RN ) with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as |u| ? ? and logistic type nonlinearities. In both situations we prove the H2(RN )-bound[...]texto impreso
We show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in L-q(Omega) for each 1texto impreso
In this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping. We show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions. Conditions are give[...]texto impreso
We prove that compact attractors of nonlinear parabolic problems with general potentials have finite fractal and Haussdorf dimension. The linear potentials belong to the space of locally uniform functions in and, unlike other references, they a[...]texto impreso
In this paper we study in detail the geometrical structure of global pullback and forwards attractors associated to non-autonomous Lotka-Volterra systems in all the three cases of competition, symbiosis or prey-predator. In particular, under som[...]texto impreso
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Arrieta Algarra, José María ; Carvalho, Alexandre N. ; Rodríguez Bernal, Aníbal | Elsevier | 1999-08-10We prove existence, uniqueness and regularity of solutions For heat equations with nonlinear boundary conditions. We study these problems with initial data in L-q(Omega), W-1,W-q(Omega), 1texto impreso
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Rodríguez Bernal, Aníbal ; Langa, José A. ; Robinson, James C. ; Suárez, Antonio | Society for Industrial and Applied Mathematics | 2009Lotka–Volterra systems are the canonical ecological models used to analyze population dynamics of competition, symbiosis, or prey-predator behavior involving different interacting species in a fixed habitat. Much of the work on these models has [...]texto impreso
We study linear perturbations of analytic semigroups defined on a scale of Banach spaces. Fitting the action of the linear perturbation between two spaces of the scale determines the spaces of existence and regularity of solutions for the pertur[...]texto impreso
Let $\Omega$ be a bounded domain in a Euclidean space, with a smooth boundary. The paper deals with the linear non-autonomous model equation $$ u_t-\Delta u=C(t,x) \quad (x\in \Omega,\ t> 0), $$ where $C(x,t)$ is a given function. Besides, vario[...]texto impreso
We analyse the dynamics of the non-autonomous nonlinear reaction–diffusion equation ut ?_u = f (t,x,u), subject to appropriate boundary conditions, proving the existence of two bounding complete trajectories, one maximal and one minimal. Our mai[...]texto impreso
In this paper, we study approximate inertial manifolds for nonlinear evolution partial differential equations which possess symmetry. The relationship between symmetry and dimensions of approximate inertial manifolds is established. We demonstra[...]texto impreso
We solve second order parabolic equations with nonsmooth coefficients and initial data in suitable uniform spaces. We also show the smoothing effect of the corresponding analytic semigroup depending on the integrability properties of the coeffic[...]texto impreso
In this paper we show that dissipative reaction-diffusion equations in unbounded domains posses extremal semistable ground states equilibria, which bound asymptotically the global dynamics. Uniqueness of such positive ground state and their appr[...]texto impreso
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 [...]texto impreso
We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion ? of the boundary and this neighborhood shrinks to ? as a parameter ? goes to zero[...]texto impreso
In this paper we consider linear parabolic problems when some reaction and potential terms are concentrated in a neighborhood of a portion I" of the boundary. This neighborhood shrinks to I" as a parameter epsilon goes to zero. Then we derive th[...]texto impreso
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We solve some fourth order parabolic equations, obtained from perturbations of the parabolic bi-Laplacian equation, with special focus on smoothing estimates. Several classes of initial data are considered including data in Lebesgue and Bessel-L[...]texto impreso
Starting form basic principles, we obtain mathematical models that describe the traffic of material objects in a network represented by a graph. We analyze existence, uniqueness, and positivity of solutions for some implicit models. Also, some l[...]texto impreso
In this paper we survey some recent results on the behavior of solutions of parabolic equations subjected to nonlinear boundary conditions. The results range from local existence and regularity of solutions, to global existence, dissipativeness [...]texto impreso
We prove the existence of nonconstant stable stationary solutions of an evolution problem with a nonlinear reaction acting on the boundary. These solutions present layers at certain points of the boundary. We also study the behavior of these sol[...]texto impreso
Arrieta Algarra, José María ; Rodríguez Bernal, Aníbal ; Rossi, Julio D. | Cambridge University Press | 2008In this paper we prove that the best constant in the Sobolev trace embedding H1() ,! Lq(@) in a bounded smooth domain can be obtained as the limit as " ! 0 of the best constant of the usual Sobolev embedding H1() ,! Lq(!", dx/") where !" = {x 2 [...]texto impreso
In this paper we address the well posedness of the linear heat equation under general periodic boundary conditions in several settings depending on the properties of the initial data. We develop an Lq theory not based on separation of variables [...]texto impreso
Langa, J.A. ; Rodríguez Bernal, Aníbal ; Suárez, Antonio | Sociedad Española de Matemática Aplicada | 2010In this paper we study in detail the pullback and forwards attractions to non-autonomous competition Lotka-Volterra system. In particular, under some conditions on the parameters, we prove the existence of a unique non-degenerate global solution[...]texto impreso
Arrieta Algarra, José María ; Carvalho, Alexandre N. ; Rodríguez Bernal, Aníbal | Elsevier | 2000-11-20The motivations to study the problem considered in this paper come from the theory of composite materials, where the heat diffusion properties can change from one part of the domain to another. Mathematically, this leads to a nonlinear second-or[...]