Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay

Producción CientíficaThis paper provides a dynamical frame to study non- autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type II over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.MINECO/FEDER grant MTM2015-66330-PEuropean Commission under project H2020-MSCA-ITN-2014 643073 CRITIC

Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson systems

Producción CientíficaUsing techniques of non-autonomous dynamical systems, we completely characterize the persistence properties of an almost periodic Nicholson system in terms of some numerically computable exponents. Although similar results hold for a class of cooperative and sublinear models, in the general nonautonomous setting one has to consider persistence as a collective property of the family of systems over the hull: the reason is that uniform persistence is not a robust property in models given by almost periodic differential equations.MINECO / FEDER grant MTM2015-66330-

Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems

Producción CientíficaWe determine sufficient conditions for uniform and strict persistence in the case of skew-product semiflows generated by solutions of non-autonomous families of cooperative systems of ODEs or delay FDEs in terms of the principal spectrums of some associated linear skew-product semiflows which admit a continuous separation. Our conditions are also necessary in the linear case. We apply our results to a noncooperative almost periodic Nicholson system with a patch structure, whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution.MINECO/FEDER MTM2015-6633

Attractors in almost periodic Nicholson systems and some numerical simulations

Producción CientíficaThe existence of a global attractor is proved for the skew-product semiflow induced by almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which exponentially attracts every other positive solution. Besides, some numerical simulations are included to illustrate our results in some concrete Nicholson systems.Ana M. Sanz: MICIIN/FEDER under project PID2021-125446NB-I00 and by Universidad de Valladolid under project PIP-TCESC-2020Víctor M. Villarragut: MICIIN/FEDER under project PID2021- 125446NB-I0

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

In this paper we obtain a detailed description of the global and cocycle attractors for the skew-product semiflows induced by the mild solutions of a family of scalar linear-dissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, basically two different types of attractors can appear, depending on whether the linear coefficient in the equations determines a bounded or an unbounded associated real cocycle. In the first case (the one for periodic equations), the structure of the attractor is simple, whereas in the second case (which happens in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations in which the attractor is chaotic in measure in the sense of Li-Yorke. Besides, we obtain a non-autonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a linear-dissipative version of the Chafee-Infante equation.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalEuropean CommissionJunta de Andalucí

Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications

The paper concerns a class of n-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.Ministerio de Economía, Industria y Competitividad /FEDER (MTM2015-66330)Fundaçao para a Ciencia e a Tecnologia under project UID/MAT/-04561/201

Forwards attraction properties in scalar non-autonomous linear-dissipative parabolic PDEs. The case of null upper Lyapunov exponent

Producción CientíficaAs it is well-known, the forwards and pullback dynamics are in general unrelated. In this paper we present an in-depth study of whether the pullback attractor is also a forwards attractor for the processes involved with the skew-product semiflow induced by a family of scalar non-autonomous reaction-diffusion equations which are linear in a neighbourhood of zero and have null upper Lyapunov exponent. Besides, the notion of Li-Yorke chaotic pullback attractor for a process is introduced, and we prove that this chaotic behaviour might occur for almost all the processes. When the problems are additionally sublinear, more cases of forwards attraction are found, which had not been previously considered even in the case of linear-dissipative ODEs.Este trabajo forma parte de los proyectos de investigación siguientes: FEDER Ministerio de Economía y Competitividad grant MTM2015-66330-P y European Commission project H2020-MSCA-ITN-2014 643073 CRITICS} (Rafael Obaya y Ana M. Sanz). Junta de Andalucía Proyecto de Excelencia FQM-1492 y FEDER Ministerio de Economía y Competitividad grant MTM2015-63723-P (José A. Langa)

Topologies of continuity for Carathéodory parabolic PDEs from a dynamical perspective

Producción CientíficaSystems of non-autonomous parabolic partial differential equations over a bounded domain with nonlinear term of Carathéodory type are considered. Appropriate topologies on sets of Lipschitz Carathéodory maps are defined in order to have a continuous dependence of the mild solutions with respect to the variation of both the nonlinear term and the initial conditions, under different assumptions on the bound-maps of the nonlinearities.All authors were partly supported by MICIIN/FEDER under project RTI2018-096523-B-I00 and by the University of Valladolid under project PIP-TCESC-2020. I.P. Longo was also partly supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska Curie grant agreement No 754462, by the European Union’s Horizon 2020 – Societal Challenges grant agreement No 820970 and by TUM International Graduate School of Science and Engineering (IGSSE

The exponential ordering for nonautonomous delay systems with applications to compartmental Nicholson systems

Producción CientíficaThe exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.The first three authors were partly supported by MICIIN/FEDER project RTI2018- 096523-B-I00 and by Universidad de Valladolid under project PIP-TCESC-2020. The fourth author was partly supported by MICINN/FEDER under projects RTI2018-096523-B-I00 and PGC2018-097565-B-I0

Utilización de derivados de quinazolinas para enfermedades neurodegenerativas

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