Arrieta Algarra, José María

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First Name
José María
Last Name
Arrieta Algarra
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Químicas
Análisis Matemático Matemática Aplicada
Matemática Aplicada
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Now showing 1 - 10 of 35
  • Publication
    Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation
    (World Scientific, 2007) Arrieta Algarra, José María; Bruschi, Simone M.
    We continue the analysis started in [3] and announced in [2], studying the behavior of solutions of nonlinear elliptic equations Delta u + f(x, u) = 0 in Omega(epsilon) with nonlinear boundary conditions of type partial derivative u/partial derivative n + g(x, u) = 0, when the boundary of the domain varies very rapidly. We show that if the oscillations are very rapid, in the sense that, roughly speaking, its period is much smaller than its amplitude and the function g is of a dissipative type, that is, it satisfies g(x, u)u >= b vertical bar u vertical bar(d+1), then the boundary condition in the limit problem is u = 0, that is, we obtain a homogeneus Dirichlet boundary condition. We show the convergence of solutions in H(1) and C(0) norms and the convergence of the eigenvalues and eigenfunctions of the linearizations around the solutions. Moreover, if a solution of the limit problem is hyperbolic (non degenerate) and some extra conditions in g are satisfied, then we show that there exists one and only one solution of the perturbed problem nearby.
  • Publication
    Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries
    (Elsevier, 2019-01-15) Arrieta Algarra, José María; Nogueira, Ariadne; Pereira, Marcone C.
    In this work we study the behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary. Our main result is concerned with the upper and lower semicontinuity of the set of solutions. We show that the solutions of our perturbed equation can be approximated with one of a one-dimensional equation, which also captures the effects of all relevant physical processes that take place in the original problem.
  • Publication
    Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions
    (Elsevier, 2000-11-20) Arrieta Algarra, José María; Carvalho, Alexandre N.; Rodríguez Bernal, Aníbal
    The 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-order parabolic equation for which the diffusion coefficient becomes large in a subdomain Ω 0 ⊂Ω . The equation is supplemented by a nonlinear boundary condition on ∂Ω and an initial condition. The authors determine the form of the limit problem (the so-called shadow system), which involves an evolution equation for the averages of the density over Ω 0 . The main results include global-in-time existence of solutions and upper semicontinuity of the associated global attractors when the system approaches the shadow system.
  • Publication
    Dynamics in Dumbbell domains. I: Continuity of the set of equilibria.
    (Elsevier, 2006) Arrieta Algarra, José María; Carvalho, Alexandre N.; Lozada-Cruz, Germán
    We analyze the dynamics of a reaction–diffusion equation with homogeneous Neumann boundary conditions in a dumbbell domain. We provide an appropriate functional setting to treat this problem and, as a first step, we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds.
  • Publication
    Critical nonlinearities at the boundary
    (Elsevier, 1998-08) Arrieta Algarra, José María; Carvalho, Alexandre N.; Rodríguez Bernal, Aníbal
    We 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 < q < infinity, or measures and with critically growing nonlinearities.
  • Publication
    Cascades of Hopf bifurcations from boundary delay
    (Elsevier, 2010) Arrieta Algarra, José María; Cónsul, Neus; Oliva, Sergio M.
    We consider a 1-dimensional reaction–diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u≡1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain.
  • Publication
    The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary
    (Cambridge University Press, 2008) Arrieta Algarra, José María; Rodríguez Bernal, Aníbal; Rossi, Julio D.
    In 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 : dist(x, @) < "} is a small neighborhood of the boundary. We also analyze symmetry properties of extremals of this last embedding when is a ball.
  • Publication
    Non well posedness of parabolic equations with supercritical nonlinearities
    (World Scientific Publ. Co. Pte. Ltd., 2004-10) Arrieta Algarra, José María; Rodríguez Bernal, Aníbal
    In 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 Cazenave in [9], concerning the behavior of the existence time for critical problems. Our results cover nonlinear heat equations including the case of nonlinear boundary conditions and weigthed spaces settings. In the latter case we show that in some cases the critical exponent is equal to one.
  • Publication
    Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions.
    (Taylor & Francis, 2004-07) Arrieta Algarra, José María; Rodríguez Bernal, Aníbal
    In this work we analyze the existence of solutions that blow-up in finite time for a reaction-diffusion equation ut−Δu=f(x,u) in a smooth domain Ω with nonlinear boundary conditions ∂u∂n=g(x,u). We show that, if locally around some point of the boundary, we have f(x,u)=−βup,β≥0, and g(x,u)=uq, then blow-up in finite time occurs if 2q>p+1 or if 2q=p+1 and β0 and p>1, we show that blow-up occurs only on the boundary.
  • Publication
    Dissipative parabolic equations in locally uniform spaces
    (Wiley-Blackwell, 2007) Arrieta Algarra, José María; Cholewa, Jan W.; Dlotko, Tomasz; Rodríguez Bernal, Aníbal
    The 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, dissipativeness and the existence of an attractor are established under the same assumptions as for problems in bounded domains. In particular, the condition sf (s, 0) < 0, |s| > s(0) > 0, together with gradient's "subquadratic" growth restriction, are shown to guarantee the existence of an attractor for the above mentioned equation. This result cannot be located in the previous references devoted to reaction-diffusion equations in the whole of R-N.