In this paper we investigate several qualitative properties of the solutions of the dual-phase-lag heat equation and the three-phase-lag heat equation. In the first case we assume that the parameter tTtT depends on the spatial position. We prove that when 2tT-tq2tT-tq is strictly positive the solutions are exponentially stable. When this property is satisfied in a proper sub-domain, but 2tT-tq=02tT-tq=0 for all the points in the case of the one-dimensional problem we also prove the exponential stability of solutions. A critical case corresponds to the situation when 2tT-tq=02tT-tq=0 in the whole domain. It is known that the solutions are not exponentially stable. We here obtain the polynomial stability for this case. Last section of the paper is devoted to the three-phase-lag case when tTtT and t¿¿ depend on the spatial variable. We here consider the case when t¿¿=¿¿tq and tTtT is a positive constant, and obtain the analyticity of the semigroup of solutions. Exponential stability and impossibility of localization are consequences of the analyticity of the semigroup.
We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful in order to study the number of 2-periodic orbits that can bifurcate from the fixed point. As an application we study the 2-cyclicity of some natural families of polynomial maps.
In this paper, we use the Green–Naghdi theory of thermomechanics of continua to derive a linear strain gradient theory of Cosserat thermoelastic bodies. The theory is capable of predicting a finite speed of heat propagation and leads to a symmetric conductivity tensor. The constitutive equations for isotropic chiral thermoelastic materials are presented. In this case, in contrast with the classical Cosserat thermoelasticity, a thermal field produces a microrotation of the particles. The thermal field is influenced by the displacement and microrotation fields even in the equilibrium theory. Existence and uniqueness results are established. The theory is used to study the effects of a concentrated heat source in an unbounded homogeneous and isotropic chiral solid.
In , a study of the existence and uniqueness of solution of partial overdetermined boundary value problems for finite networks was performed. These problems involve Schrodinger operators and the novel feature is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. In the present work, we study the resolvent kernels associated with overdetermined partial boundary value problems on finite network and we express them in terms of the well-known Green operator and the Dirichlet-to-Robin map. Moreover, we analyze their main properties and we compute them in the case of a generalized cylinder. The obtained expression involve polynomials that can be seen as a generalization of Chebyshev polynomials, and indeed when the conductances along axes are constant the expressions for the overdetermined partial resolvent kernels are given in terms of second kind Chebyshev polynomials. (C) 2015 Elsevier Inc. All rights reserved.
We prove that any planar birational integrable map, which preserves a fibration given by genus 0 curves has a Lie symmetry and some associated invariant measures. The obtained results allow to study in a systematic way the global dynamics of these maps. Using this approach, the dynamics of several maps is described. In particular we are able to give, for particular examples, the explicit expression of the rotation number function, and the set of periods
of the considered maps.
In this study, we define a class of non-self-adjoint boundary value problems on finite networks associated with Schrodinger operators. The novel feature of this study is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. We show that overdetermined partial boundary value problems are crucial for solving inverse boundary value problems on finite networks since they provide the theoretical foundations for the recovery algorithm. We analyze the uniqueness and the existence of solution for overdetermined partial boundary value problems based on the nonsingularity of partial Dirichlet-to-Neumann maps. These maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed. We also execute full conductance recovery for spider networks. (C) 2014 Elsevier Inc. All rights reserved.
This paper deals with the linear theory of isotropic micropolar thermoviscoelastic materials. When the dissipation is positive definite, we present two uniqueness theorems. The first one requires the extra assumption that some coupling terms vanish; in this case, the instability of solutions is also proved. When the internal energy and the dissipation are both positive definite, we prove the well-posedness of the problem and the analyticity of the solutions. Exponential decay and impossibility of localization are corollaries of the analyticity.
Ezquerro, J.A.; Grau, M.; Hernández, M.A.; Noguera, M. Journal of mathematical analysis and applications Vol. 398, num. 1, p. 100-112 DOI: 10.1016/j.jmaa.2012.08.040 Data de publicació: 2013-02-01 Article en revista
This note is concerned with the linear (and linearised) type III thermoelastic theory
proposed by Green and Naghdi. First, the continuous dependence of the solutions upon
the initial data and supply terms is established for noncentrosymmetric bodies. Then a
uniqueness result for centrosymmetric materials is established.
Pamplona, P.; Muñoz Rivera, J.; Quintanilla, R. Journal of mathematical analysis and applications Vol. 394, num. 2, p. 645-655 DOI: 10.1016/j.jmaa.2012.04.024 Data de publicació: 2012-10-15 Article en revista
In this note we study the analyticity of the solutions to the one-dimensional porous-elasticity problem with temperatures and microtemperatures when viscoelasticity and porous viscosity effects are also present. We show the lack of analyticity when the porous dissipation is weak, and the analyticity when it is strong.
In this note we study the analyticity of the solutions to the one-dimensional porouselasticity
problem with temperatures and microtemperatures when viscoelasticity and
porous viscosity effects are also present. We show the lack of analyticity when the porous
dissipation is weak, and the analyticity when it is strong
Lazaro, J. Tomás; Gasull, A.; Torregrosa, J. Journal of mathematical analysis and applications Vol. 387, num. 2, p. 631-644 DOI: 10.1016/j.jmaa.2011.09.019 Data de publicació: 2012-03 Article en revista
Olm, Josep M.; Ros, X.; Martinez-seara, Tere Journal of mathematical analysis and applications Vol. 381, num. 2, p. 582-589 DOI: 10.1016/j.jmaa.2011.02.084 Data de publicació: 2011-09 Article en revista
The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation xmaps tox−1, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.
Pamplona, P.; Muñoz, J.; Quintanilla, R. Journal of mathematical analysis and applications Vol. 379, num. 2, p. 682-705 DOI: 10.1016/j.jmaa.2011.01.045 Data de publicació: 2011-07-15 Article en revista
In this paper we study the asymptotic behavior to an one-dimensional porous-elasticity problem with history. We show the lack of exponential stability when the porous dissipation or the elastic dissipation is absent. And we show the lack of analyticity and exponential stability when the porous viscosity and the elastic dissipation are present.
The decay of solutions in nonsimple elasticity with memory is addressed, analyzing how
the decay rate is influenced by the different dissipation mechanisms appearing in the
equations. In particular, a first order dissipation is shown to guarantee the asymptotic
stability of the related solution semigroup, but is not strong enough to entail exponential
stability. The latter occurs for a dissipation mechanism of the second order, that is, the
same order as the one of the leading operator.
In the first part of this paper we present a linear theory of thermoelastic bodies
with microstructure and microtemperatures which permits the transmission of heat as
thermal waves at finite speed. The theory is based on the entropy balance postulated
by Green and Naghdi [A.E. Green, P.M. Naghdi, A re-examination of the basic postulates
of thermomechanics, Proc. R. Soc. London A 432 (1991) 171–194]. We consider bodies
with microstructure whose microelements can stretch and contract independently of their
translations. Then we establish existence and uniqueness results in the context of the
The fixed points of a T-indistinguishability operator are characterized as its generators in the sense of the Representation Theorem of Valverde. The geometric description of the set of fixed points of a reflexive and symmetric fuzzy relation based on this characterization gives a way to explicitly calculate it.