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
85 results
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Now showing 1 - 10 of 85
Item Parabolic problems with nonlinear boundary conditions and critical nonlinearities(Journal of Differential Equations, 1999) Arrieta Algarra, José María; Carvalho, Alexandre N.; Rodríguez Bernal, AníbalWe 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), 1 < q < infinity or measures and with critically growing non-linearities.Item Non well posedness of parabolic equations with supercritical nonlinearities(Communications in contemporary mathematics, 2004) Arrieta Algarra, José María; Rodríguez Bernal, AníbalIn 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.Item Dissipative parabolic equations in locally uniform spaces(Mathematische Nachrichten, 2007) Arrieta Algarra, José María; Cholewa, Jan W.; Dlotko, Tomasz; Rodríguez Bernal, AníbalThe 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.Item On the loss of mass for the heat equation in an exterior domain with general boundary conditions(São Paulo Journal of Mathematical Sciences, 2023) Rodríguez Bernal, Aníbal; Domínguez de Tena, JoaquínIn this work, we study the decay of mass for solutions to the heat equation in exterior domains, i.e., domains which are the complement of a compact set in RN . Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann conditions. We determine the exact amount of mass loss and identify criteria for complete mass decay, in which the dimension of the space plays a key role. Furthermore, the paper provides explicit mass decay rates.Item Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions.(Communications in Partial Differential Equations, 2004) Arrieta Algarra, José María; Rodríguez Bernal, AníbalIn 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.Item Finite-dimensional asymptotic behavior in a thermosyphon including the Soret effect(Mathematical Methods in the Applied Sciences, 1999) Rodríguez Bernal, Aníbal; Jiménez Casas, ÁngelaWe 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 well posedness of the system we prove the existence of a global attractor. Finally, using inertial manifold techniques, we obtain an explicit reduced system of ODE's that describes the asymptotic behaviour of the full systemItem Dynamics of reaction diffusion equations under nonlinear boundary conditions(Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 2000) Rodríguez Bernal, Aníbal; Tajdine, AnasIn 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 in finite time.Item Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations(Discrete and Continuous Dynamical Systems. Series A., 2009) Rodríguez Bernal, AníbalLet $\Omega$ 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, various boundary conditions are imposed. The author suggests sharp qualitative and quantitative conditions to guarantee that the exponential type of the considered equation is modified by a linear perturbation. No assumption (periodic, almost periodic, quasi periodic etc.) is made on the time behavior of the coefficients of the equation or the perturbation. The obtained results are then applied to the investigation of the asymptotic behavior, both forwards and backwards, of solutions of certain nonautonomous nonlinear equations.Item Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations(Journal of Differential Equations, 2010) Rodríguez Bernal, Aníbal; Cholewa, Jan W.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 dissipative systems in ordered space are recalled. Then follow results on the existence of extremal solutions and global attractors and finally on the inclusion of the global attractor in an order interval formed by the minimal and the maximal equilibria. \par In the second part of the paper, they then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, they exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in $\Bbb R^N$ with nonlinearities depending on the gradient of the solution. \par The authors consider as well systems of reaction-diffusion equations in $\Bbb R^N$ and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in $\Bbb R^N$. They further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation.Item Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary(Revista Matemática Iberoamericana, 2008) Arrieta Algarra, José María; Jiménez Casas, Ángela; Rodríguez Bernal, AníbalWe 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 prove that this family of solutions converges in certain Sobolev spaces and also in the sup norm, to the solution of an elliptic problem where the reaction term and the concentrating potential are transformed into a flux condition and a potential on Gamma.