A note on the spatial behavior for the generalized Tricomi equation

AbstractThis short note is devoted to the study of the spatial decay estimates for the solutions of the generalized Tricomi equation. The relevance of this kind of study is that we obtain the decay for an equation which can be elliptic, parabolic and hyperbolic depending on the different points of the region. This equation is relevant in the study of fluids as well as for the anti-plane deformations of prestressed functionally graded linear elastic solids

On the asymptotic spatial behaviour of the solutions of the nerve system

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.Peer ReviewedPostprint (author’s final draft

Phragmén-Lindelöf alternative for the Laplace equation with dynamic boundary conditions

This paper investigates the spatial behavior of the solutions of the Laplace equation on a semi-infinite cylinder when dynamical nonlinear boundary conditions are imposed on its lateral side. We prove a Phragmén-Lindelöf alternative for the solutions. To be precise, we see that the solutions increase in an exponential way or they decay as a polynomial. To give a complete description of the decay in this last case we also obtain an upper bound for the amplitude term by means of the boundary conditions. In the last section we sketch how to generalize the results to a system of two elliptic equations related with the heat conduction in mixtures.Peer ReviewedPostprint (author's final draft

Fast spatial behavior in higher order in time equations and systems

In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered beforePeer ReviewedPostprint (published version

Fast spatial behavior in higher order in time equations and systems

Financiado para publicación en acceso aberto: Universidade de Vigo/CISUGIn this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before.Agencia Estatal de Investigación | Ref. PGC2018-096696-B-I00Agencia Estatal de Investigación | Ref. PID2019-105118GB-I0

On the spatial behavior in two-temperature generalized thermoelastic theories

The final publication is available at link.springer.com via https://doi.org/10.1007/s00033-017-0857-xThis paper investigates the spatial behavior of the solutions of two generalized thermoelastic theories with two temperatures. To be more precise, we focus on the Green–Lindsay theory with two temperatures and the Lord–Shulman theory with two temperatures. We prove that a Phragmén–Lindelöf alternative of exponential type can be obtained in both cases. We also describe how to obtain a bound on the amplitude term by means of the boundary conditions for the Green–Lindsay theory with two temperatures.Peer ReviewedPostprint (author's final draft

Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature

The main goal of this paper is to study the asymptotic behavior of a coupled Cahn-Hilliard/Allen-Cahn system with temperature. The work is divided into two parts: In the rst part, the heat equation is based on the usual Fourier law. In the second one, it's based on the type III heat conduction law. In both parts, we prove the existence of exponential attractors and, therefore, of nite-dimensional global attractorsPeer ReviewedPostprint (author's final draft