Vidal López, Alejandro

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Vidal López
Universidad Complutense de Madrid
Faculty / Institute
Matemática Aplicada
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Now showing 1 - 8 of 8
  • Publication
    Minimal periods of semilinear evolution equations with Lipschitz nonlinearity
    (Elsevier, 2006-01-15) Robinson, James C.; Vidal López, Alejandro
    It is known that any periodic orbit of a Lipschitz ordinary differential equation must have period at least 2π/L, where L is the Lipschitz constant of f. In this paper, we prove a similar result for the semilinear evolution equation du/dt=-Au+f(u): for each α with 0 α 1/2 there exists a constant Kα such that if L is the Lipschitz constant of f as a map from D(Aα) into H then any periodic orbit has period at least KαL-1/(1-α). As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier–Stokes equations with periodic boundary conditions.
  • Publication
    Soluciones extremales para problemas parabólicos de evolución no lineales y aplicaciones
    (Universidad Complutense de Madrid, Servicio de Publicaciones, 2006) Vidal López, Alejandro; Rodríguez Bernal, Aníbal
    En la segunda parte, se estudian problemas no autónomos. Ahora, las soluciones extremales son trayectorias completas (soluciones definidas para todo tiempo) y el concepto de atracción considerado es el de atracción pullback: lo importante es el estado actual de aquello que comenzó a evolucionar hace mucho tiempo. Estas trayectorias acotan al atractor pullback del sistema. Como aplicación se consideran ecuaciones logísticas (no autónomas) obteniendo resultados a problemas. En la última parte, se aplican los resultados obtenidos a problemas de perturbación singular. Concretamente dos problemas de difusión alta en la frontera: uno donde la zona de difusión alta tiene contacto con la frontera Dirichlet y Robin; otro donde esta zona es un entorno de la frontera Robin. Se analiza el problema límite y el comportamiento de las soluciones extremales así como el comportamiento del atractor.
  • Publication
    Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations.
    (American Institute of Mathematical Sciences, 2007) Rodríguez Bernal, Aníbal; Vidal López, Alejandro; Langa, J.A.; Robinson, James C.; Suárez, A.
    The 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 equation is now well understood, several different possibilities are realised in the forward asymptotic regime.
  • Publication
    Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems
    (American Institute of Mathematical Sciences, 2007) Rodríguez Bernal, Aníbal; Vidal López, Alejandro
    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.
  • Publication
    Semistable extremal ground states for nonlinear evolution equations in unbounded domains
    (Elsevier, 2008) Rodríguez Bernal, Aníbal; Vidal López, Alejandro
    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 approximation by extremal equilibria in bounded domains is also studied. The results are then applied to the important case of logistic equations. (C) 2007 Elsevier Inc. All rights reserved.
  • Publication
    Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems
    (Elsevier, 2007) Rodríguez Bernal, Aníbal; Vidal López, Alejandro; Robinson, James C.
    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 main assumption is that the nonlinear term satisfies a bound of the form f (t,x,u)u _ C(t, x)|u|2 + D(t, x)|u|, where the linear evolution operator associated with _ + C(t, x) is exponentially stable. As an important step in our argument we give a detailed analysis of the exponential stability properties of the evolution operator for the non-autonomous linear problem ut − _u = C(t, x)u between different Lp spaces.
  • Publication
    Extremal equilibria and asymptotic behavior of parabolic nonlinear reaction-diffusion equations
    (BIRKHAUSER VERLAG AG, 2005) Rodríguez Bernal, Aníbal; Vidal López, Alejandro; Chipot, M.; Escher, J.
    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.
  • Publication
    Extremal equilibria for reaction-diffusion equations in bounded domains and applications
    (Elsevier, 2008) Rodríguez Bernal, Aníbal; Vidal López, Alejandro
    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 so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation