DSpace Collection:
http://hdl.handle.net/2117/3339
2015-05-27T06:06:58ZA generalization of the Allen–Cahn equation
http://hdl.handle.net/2117/27478
Title: A generalization of the Allen–Cahn equation
Authors: Miranville, Alain; Quintanilla de Latorre, Ramón
Abstract: Our aim in this paper is to study generalizations of the Allen–Cahn equation based on a modification of the Ginzburg–Landau free energy proposed in S. Torabi et al. (2009, A new phase-field model for strongly anisotropic systems. Proc. R. Soc. A, 465, 1337–1359). In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence to the Allen–Cahn equation, when the Willmore regularization goes to zero. We finally study the spatial behaviour of solutions in a semi-infinite cylinder, assuming that such solutions exist.
Description: This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record: Miranville, A.; Quintanilla, R. A generalization of the Allen–Cahn equation. "IMA Journal of Applied Mathematics", 01 Abril 2015, vol. 80, núm. 2, p. 410-430 is available online at:http://imamat.oxfordjournals.org/content/80/2/410.2015-04-21T13:31:16ZLower bounds of end effects for a nonhomogeneous isotropic linear elastic solid in anti-plane shear
http://hdl.handle.net/2117/26875
Title: Lower bounds of end effects for a nonhomogeneous isotropic linear elastic solid in anti-plane shear
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: In this paper we give lower bounds for the spatial decay of the solutions for anti-plane shear deformations in the case of
isotropic inhomogeneous elastic materials. We first consider the case when the shear modulus only depends on the lateral
direction. By means of the logarithmic convexity arguments we obtain the required estimates. Some pictures illustrate
our results. We also study the general inhomogeneity. We give some lower bounds whenever shear modulus
satisfies several requirements.2015-03-19T18:59:40ZSpatial stability in linear thermoelasticity
http://hdl.handle.net/2117/26874
Title: Spatial stability in linear thermoelasticity
Authors: Knops, Robin J.; Quintanilla de Latorre, Ramón
Abstract: Uniqueness and spatial stability are investigated for smooth solutions to boundary value
problems in non-classical linearised and linear thermoelasticity subject to certain conditions
on material coefficients. Uniqueness is derived for standard boundary conditions
on bounded regions using a generalisation of Kirchhoff’s method. Spatial stability is discussed
for the semi-infinite prismatic cylinder in the absence of specified axial asymptotic
behaviour. Alternative growth and decay estimates are established principally for the
cross-sectional energy flux that is shown to satisfy a first order differential inequality.
Uniqueness in the class of solutions with bounded energy follows as a corollary.
Separate discussion is required for the linearised and linear theories. Although the general
approach is similar for both theories, the argument must be considerably modified for
the treatment of the linear theory.2015-03-19T18:33:43ZOn the asymptotic spatial behaviour of the solutions of the nerve system
http://hdl.handle.net/2117/26512
Title: On the asymptotic spatial behaviour of the solutions of the nerve system
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: In this paper we investigate the asymptotic spatial behavior of the solutions for several models for the nerve fibers.
First, our analysis deals with the coupling of two parabolic equations. We prove that, under suitable assumptions on the coefficients
and the nonlinear function, the decay is similar to the one corresponding to the heat equation. A limit case of this system
corresponds to the coupling of a parabolic equation with an ordinary differential equation. In this situation, we see that for suitable
boundary conditions the solution ceases to exist for a finite value of the spatial variable. Next two sections correspond to
the coupling of a hyperbolic/parabolic and hyperbolic/ordinary differential problems. For the first one we obtain that the decay
is like an exponential of a second degree polynomial in the spatial variable. In the second one, we prove a similar behaviour to
the one corresponding to the wave equation. In these two sections we use in a relevant way an exponentially weighted Poincaré
inequality which has been revealed very useful in several thermal and mechanical problems. This kind of results have relevance
to understand the propagation of perturbations for nerve models.2015-02-25T14:52:52ZForeword to special issue on ‘‘Qualitative Methods in Engineering Science’’
http://hdl.handle.net/2117/26438
Title: Foreword to special issue on ‘‘Qualitative Methods in Engineering Science’’
Authors: Quintanilla de Latorre, Ramón; Fu, Yibin2015-02-19T16:17:32ZPhase-lag heat conduction: decay rates for limit problems and well-posedness
http://hdl.handle.net/2117/26420
Title: Phase-lag heat conduction: decay rates for limit problems and well-posedness
Authors: Borgmeyer, Karin; Quintanilla de Latorre, Ramón; Racke, Reinhard
Abstract: In two recent papers, the authors have studied conditions on the relaxation parameters in order to guarantee the stability or instability of solutions for the Taylor approximations to dual-phase-lag and three-phase-lag heat conduction equations. However, for several limit cases relating to the parameters, the kind of stability was unclear. Here, we analyze these limit cases and clarify whether we can expect exponential or slow decay for the solutions. Moreover, rather general well-posedness results for three-phase-lag models are presented. Finally, the exponential stability expected by spectral analysis is rigorously proved exemplarily.
Description: The final publication is available at Springer via http://dx.doi.org/10.1007/s00028-014-0242-6.2015-02-18T17:18:03ZHölder stability in type III thermoelastodynamics
http://hdl.handle.net/2117/26416
Title: Hölder stability in type III thermoelastodynamics
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We here assume that the mass density is positive and the thermal conductivity tensor is positive definite. However, we do not assume the positivity of any other tensor. In this situation, we obtain Holder continuous dependence results on the supply terms. We also sketch how to prove the continuous dependence on the initial data.
Description: Electronic version of an article published as "Archive of applied mechanics", vol. 84 nº 9-11, October 2014, p. 1465-1476. DOI No 10.1007/s00419-014-0827-0.2015-02-18T15:34:37ZSpatial behavior in phase-lag heat conduction
http://hdl.handle.net/2117/26266
Title: Spatial behavior in phase-lag heat conduction
Authors: Quintanilla de Latorre, Ramón; Racke, Reinhard
Abstract: In this paper, we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual-phase-lag and three-phase-lag theories, reflecting Saint-Venant's principle. Depending on the relative order of derivation, with respect to the time, we propose different arguments. One is inspired by the arguments for parabolic problems and the other is inspired by the arguments for hyperbolic problems. In the first case, we obtain a Phragmén-Lindelöf alternative for the solutions. In the second case, we obtain an estimate for the decay as well as a domain of influence result. The main tool to manage these problems is the use of an exponentially weighted Poincaré inequality.2015-02-09T13:45:15ZAnalysis of the equations governing the motion of a degrading elastic solid due to diffusion of a fluid
http://hdl.handle.net/2117/24474
Title: Analysis of the equations governing the motion of a degrading elastic solid due to diffusion of a fluid
Authors: Quintanilla de Latorre, Ramón; Rajagopal, Kumbakonam
Abstract: Degradation of solids bearing load due to the infusion of moisture when exposed to the environment
can leads to a decrease in their load carrying capacity and can also lead to the failure of the body from
performing its intended task. In this short paper, we study some qualitative properties of the solution
to systems of equations that describe the degradation in a linearized elastic solid due to the diffusion
of a fluid. The model that is considered allows for the material properties of the solid to depend on the
concentration of the diffusing fluid. While the load carrying capacity of a solid could decrease or increase
due to the infusion of a fluid, we consider the case when degradation takes place. We are able to obtain
results concerning the uniqueness of solutions to the problem under consideration. We also consider
special anti-plane and quasi-static deformations of the body.
Description: Electronic version of an article published as "IMA Journal of applied mathematics", vol. 79, no 5, 2014, p. 778-789. DOI: 10.1093/imamat/hxt050 <http://imamat.oxfordjournals.org/content/79/5/778.abstract>2014-10-24T16:30:54ZOn a theory of thermoelastic materials with double porosity structure
http://hdl.handle.net/2117/24346
Title: On a theory of thermoelastic materials with double porosity structure
Authors: Iesan, Dorin; Quintanilla de Latorre, Ramón
Abstract: In this article, we use the Nunziato–Cowin theory of materials with voids to derive a theory of thermoelastic solids, which have a double porosity structure. The new theory is not based on Darcy's law. In the case of equilibrium, in contrast with the classical theory of elastic materials with double porosity, the porosity structure of the body is influenced by the displacement field. We prove the uniqueness of solutions by means of the logarithmic convexity arguments as well as the instability of solutions whenever the internal energy is not positive definite. Later, we use semigroup arguments to prove the existence of solutions in the case that the internal energy is positive. The deformation of an elastic space with a spherical cavity is investigated.
Description: Electronic version of an article published as "Journal of thermal stresses", vol. 37, no 9, 2014, p. 1017-1036. DOI: 10.1080/01495739.2014.914776 <http://www.tandfonline.com/doi/abs/10.1080/01495739.2014.914776#.VDf2nRbQo5h>2014-10-10T16:05:00ZOn the backward in time problem for the thermoelasticity with two temperatures
http://hdl.handle.net/2117/23165
Title: On the backward in time problem for the thermoelasticity with two temperatures
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: This paper is devoted to the study of the existence, uniqueness, continuous dependence and spatial behaviour of the solutions for the backward in time problem determined by the Type III with two temperatures thermoelastodynamic theory. We first show the existence, uniqueness and continuous dependence of the solutions. Instability of the solutions for the Type II with two temperatures theory is proved later. For the one-dimensional Type III with two temperatures theory, the exponential instability is also pointed-out. We also analyze the spatial behaviour of the solutions. By means of the exponentially weighted Poincare inequality, we are able to obtain a function that defines a measure on the solutions and, therefore, we obtain the usual exponential type alternative for the solutions of the problem defined in a semi-infinite cylinder.
Description: Electronic version of an article published as "Discrete and continuous dynamical series", vol. 19, no 3, May 2014, p. 679-695. DOI 10.3934/dcdsb.2014.19.679.2014-06-05T13:19:17ZOn the uniqueness and analyticity of solutions in micropolar thermoviscoelasticity
http://hdl.handle.net/2117/21495
Title: On the uniqueness and analyticity of solutions in micropolar thermoviscoelasticity
Authors: Magaña Nieto, Antonio; Quintanilla de Latorre, Ramón
Abstract: 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.2014-02-10T14:06:59ZOn the spatial behavior in type III thermoelastodynamics
http://hdl.handle.net/2117/21453
Title: On the spatial behavior in type III thermoelastodynamics
Authors: Leseduarte Milán, María Carme; Quintanilla de Latorre, Ramón
Abstract: This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We investigate the spatial behavior of the solutions when we assume the positivity of the elasticity tensor, the thermal conductivity tensor, the mass density and the heat capacity. However, we do not assume (a priori) the positivity of the internal energy. We first obtain a Phragmén-Lindelöf alternative of exponential type for the solutions. Later, we prove that the decay can be controlled by the exponential of a second-degree polynomial. This is similar to other thermoelastic situations.2014-02-04T16:20:58ZSpatial decay for several phase-field models
http://hdl.handle.net/2117/20931
Title: Spatial decay for several phase-field models
Authors: Miranville, Alain; Quintanilla de Latorre, Ramón
Abstract: In this paper, we study the spatial behavior of three phase-field models. First, we consider the Cahn-Hilliard equation and
we obtain the exponential decay of solutions under suitable assumptions on the data. Then, for the classical isothermal
phase-field equation (i.e., the Allen-Cahn equation), we prove the nonexistence and the fast decay of solutions and, for
the nonisothermal case governed by the Fourier law, we obtain a Phragm ́
en-Lindel ̈
of alternative of exponential type,
respectively.2013-12-05T15:30:33ZDecay of solutions for a mixture of thermoelastic one dimensional solids
http://hdl.handle.net/2117/19560
Title: Decay of solutions for a mixture of thermoelastic one dimensional solids
Authors: Muñoz Rivera, Jaime E.; Naso, Maria-Grazia; Quintanilla de Latorre, Ramón
Abstract: We study a PDE system modeling thermomechanical deformations for a mixture of thermoelastic solids. In particular we investigate the asymptotic behavior of the solutions. First, we identify conditions on the constitutive coefficients to guarantee that the imaginary axis is contained in the resolvent. Subsequently, we find the necessary and sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate.2013-06-17T16:52:48Z