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
40 results
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Now showing 1 - 10 of 40
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 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 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 A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities(Communications on pure and applied analysis, 2014) Rodríguez Bernal, AníbalWe show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in L-q(Omega) for each 1 <= q <= infinity. In particular, we do not assume restriction on the growth of the nonlinearites required by the standar local existence theory.Item Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems(Discrete and Continuous Dynamical Systems. Series A., 2007) Rodríguez Bernal, Aníbal; Vidal López, AlejandroWe 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.Item Attractors of parabolic problems with nonlinear boundary conditions uniform bounds(Communications in Partial Differential Equations, 2000) Arrieta Algarra, José María; Carvalho, Alexandre N.; Rodríguez Bernal, AníbalThe 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∈Ω⊂⊂R N , a(x)>m>0; u(x,0)=u 0 with nonlinear boundary conditions of the form u=0 on Γ 0 , and a(x)∂ n u+b(x)u=g(x,u) on Γ 1 , where Γ i are components of ∂Ω . Under smoothness and growth conditions which ensure the local classical well-posedness of the problem, they indicate some sign conditions under which the solutions are globally defined in time, and somewhat more strong dissipativeness conditions under which they possess a global attractor that captures the asymptotic dynamics of the system. After that the authors study the dependence of the attractors on the diffusion. For a(x)=a ε (x) they show their upper semicontinuity on ε . Throughout the paper they also pay special attention to the dependence of the estimates obtained on the domain Ω and show that in certain instances the L ∞ bounds on the attractors do not depend on the shape of Ω but rather on |Ω| .Item Perturbation of Analytic Semigroups in Scales of Banach Spaces and Applications to Linear Parabolic Equations with Low Regularity Data(SEMA Journal, 2011) Rodríguez Bernal, AníbalWe study linear perturbations of analytic semigroups defined on a scale of Banach spaces. Fitting the action of the linear perturbation between two spaces of the scale determines the spaces of existence and regularity of solutions for the perturbed semigroup, within the original scale. Also continuity of the resulting perturbed semigroup with respect to the perturbation is analyzed. As the main tools we exploit the smoothing of the original semigroup on the scale and the variation of constants formula. These general results are applied to several situations for linear partial differential equations of parabolic type. The main attention is set on low regularity perturbations of linear diffusion equations in either bounded or unbounded domains. Different scales of spaces are considered such as Lebesgue or Bessel spaces. However, the application of the abstract results are not limited to such examples and many other situatons can be considered.Item Equilibria and global dynamics of a problem with bifurcation from infinity(Journal of Differential Equations, 2009) Arrieta Algarra, José María; Pardo San Gil, Rosa María; Rodríguez Bernal, AníbalWe consider a parabolic equation ut−Δu+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, Proc. Roy. Soc. Edinburgh Sect. A 137 (2) (2007) 225–252] the authors proved the existence of unbounded branches of equilibria for λ close to a Steklov eigenvalue of odd multiplicity. In this work, we characterize the stability of such equilibria and analyze several features of the bifurcating branches. We also investigate several question related to the global dynamical properties of the system for different values of the parameter, including the behavior of the attractor of the system when the parameter crosses the first Steklov eigenvalue and the existence of extremal equilibria. We include Appendix A where we prove a uniform antimaximum principle and several results related to the spectral behavior when the potential at the boundary is perturbed.Item Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity(International Journal of Bifurcation and Chaos, 2006) Arrieta Algarra, José María; Rodríguez Bernal, Aníbal; Valero , JoséWe 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. We also give a complete description of the set of fixed points and study their stability. Finally, we analyze the existence of heteroclinic connections between the fixed points, obtaining information on the fine structure of the global attractor.