## Person: Rodríguez Bernal, Aníbal

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##### First Name

Aníbal

##### Last Name

Rodríguez Bernal

##### Affiliation

Universidad Complutense de Madrid

##### Faculty / Institute

Ciencias Matemáticas

##### Department

Análisis Matemático Matemática Aplicada

##### Area

Matemática Aplicada

##### Identifiers

82 results

## Search Results

Now showing 1 - 10 of 82

Publication Cauchy problem for the time-dependent Ginzburg-Landau model of superconductivity(Cambridge University Press, 2000) Rodríguez Bernal, Aníbal; Wang, BixiangThe 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 the existence result of solutions for L-3 initial data and the uniqueness result for L-4 initial data.Publication Localization phenomena in a degenerate logistic equation(Department of Mathematics Texas State University, 2014) Arrieta Algarra, José M.; Pardo, Rosa; Rodríguez Bernal, AníbalWe 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 nonlinearity cancels. This type of nonlinearity has applications in the nonlinear Schrodinger equation and the study of Bose–Einstein condensates. In this context, our analysis explains the fact that the ground state presents a strong localization in the spatial region in which the nonlinearity cancels.Publication Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem(American Institute of Mathematical Sciences, 2005-05) Rodríguez Bernal, Aníbal; Willie, RobertWe 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(Omega). In the parabolic case we prove convergence in the functional space L-infinity((0, T), L-2(Omega)) boolean AND L-2((0, T), H-1(Omega)).Publication Critical and supercritical higher order parabolic problems in R-N(Elsevier, 2014-07) Cholewa, Jan W.; Rodríguez Bernal, AníbalDue 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 the subcritical case have already been satisfactorily described (see Cholewa and Rodriguez-Bernal (2012)), for problems in the critical or supercritical regime the questions concerning well or illposedness, as well as possible dissipative properties of the solutions, have not yet been satisfactorily answered. This article is devoted to the analysis of the higher order parabolic problems in R-N in the latter case. Focusing on the critical and supercritical regimes we give sufficient "good"-sign conditions proving that the problem is then globally well posed in L-2(R-N) and even possesses a compact global attractor. On the other hand, for supercritically growing "bad"-signed nonlinearities we show that the problem is ill-posed.Publication Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary(Elsevier, 2011) Rodríguez Bernal, Aníbal; Jiménez Casas, ÁngelaWe 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 Γ of the boundary and this neighborhood shrinks to Γ as a parameter ε goes to zero. We prove that the family of attractors is upper continuous at the ε=0.Publication Linear higher order parabolic problems in locally uniform Lebesgue's spaces(Elsevier, 2017) Cholawa, Jan W.; Rodríguez Bernal, AníbalLinear 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.Publication Linear Non-Autonomous Heat Flow in $$L_0^1({{\mathbb {R}}}^{d})$$ and Applications to Elliptic Equations in $${{\mathbb {R}}}^{d}$$(Springer, 2022-10-11) Robinson, James C.; Rodríguez Bernal, AníbalWe study solutions of the equation ut−Δu+λu=f, for initial data that is ‘large at infinity’ as treated in our previous papers on the unforced heat equation. When f=0 we characterise those (u0,λ) for which solutions converge to 0 as t→∞, as not every λ>0 is able to achieve that for all initial data. When f≠0 we give conditions to guarantee that the solution is given by the usual ‘variation of constants formula’ u(t)=e−λtS(t)u0+∫t0e−λ(t−s)S(t−s)f(s)ds, where S(⋅) is the heat semigroup. We use these results to treat the elliptic problem −Δu+λu=f when f is allowed to be ‘large at infinity’, giving conditions under which a solution exists that is given by convolution with the usual Green’s function for the problem. Many of our results are sharp when u0,f≥0.Publication Parabolic singular limit of a wave equation with localized interior damping(World Scientific Publ. Co. Pte. Ltd., 2001-05) Rodríguez Bernal, Aníbal; Zuazua Iriondo, EnriquePublication Reduction of dimension of approximate intertial manifolds by symmetry(Cambridge University Press, 1999-10) Rodríguez Bernal, Aníbal; Wang, BixiangIn 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 demonstrate that symmetry can reduce the dimensions of an approximate inertial manifold. Applications for concrete evolution equations are given.Publication Asymptotic behaviour of a parabolic problem with terms concentrated in the boundary(Elsevier, 2009) Rodríguez Bernal, Aníbal; Jiménez Casas, ÁngelaWe 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 epsilon goes to zero. We prove that this family of attractors is upper continuous at epsilon = 0.