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.
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.
In this paper, we study the energy decay rate for a mixed type II and type III thermoelastic system. The system consist of a wave equation and a heat equation of type II in another part of the domain, coupled in certain pattern. When the damping coefficient function satisfies certain conditions at the interface, a polynomial type decay rate is obtained. This result is proved by verifying the frequency domain conditions.
Leseduarte, M. C.; Magaña, A.; Quintanilla, R. Discrete and continuous dynamical systems. Series B Vol. 13, num. 2, p. 375-391 DOI: 10.3934/dcdsb.2010.13.375 Data de publicació: 2010-03 Article en revista
In this paper we give a mechanism to compute the families of
classical hamiltonians of two degrees of freedom with an invariant
plane and normal variational equations of Hill-Schr\"odinger type
selected in a suitable way. In particular we deeply study the case
of these equations with polynomial or trigonometrical potentials,
analyzing their integrability in the Picard-Vessiot sense using
Kovacic's algorithm and introducing an algebraic method
(algebrization) that transforms equations with transcendental
coefficients in equations with rational coefficients without
changing the Galoisian structure of the equation. We compute all
Galois groups of Hill-Schr\"odinger type equations with polynomial
and trigonometric (Mathieu equation) potentials, obtaining
Galoisian obstructions to integrability of hamiltonian systems by
means of meromorphic or rational first integrals via Morales-Ramis
A two-dimensional thermal convection problem in a circular annulus subject to a constant inward radial gravity and heated from the inside is considered. A branch of spatio-temporal symmetric periodic orbits that are known only numerically shows a multi-critical codimension-two point with a triple +1-Floquet multiplier. The weakly nonlinear analysis of the dynamics near such point is performed by deriving a system of amplitude equations using a perturbation technique, which is an extension of the Lindstedt-Poincaré method, and solvability conditions. The results obtained using the amplitude equation are compared with those from the original system of partial differential equations showing a very good agreement.