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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[...]![]()
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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 [...]![]()
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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.![]()
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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[...]![]()
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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[...]![]()
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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 1![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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), 1![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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.![]()
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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.![]()
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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[...]![]()
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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 [...]![]()
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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[...]![]()
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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 [...]![]()
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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[...]![]()
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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.![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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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 1![]()
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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.![]()
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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, [...]![]()
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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[...]![]()
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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[...]![]()
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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 [...]![]()
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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[...]![]()
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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[...]![]()
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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 [...]![]()
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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[...]![]()
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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[...]![]()
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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[...]![]()
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We show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in L-q(Omega) for each 1![]()
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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[...]![]()
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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[...]