Exponential decay in nonsimple thermoelasticity of type III

This paper deals with the model proposed for nonsimple materials with heat conduction of type III. We analyze first the general system of equations, determine the behavior of its solutions with respect to the time, and show that the semigroup associated with the system is not analytic. Two limiting cases of the model are studied later. Copyright © 2015 John Wiley & Sons, Ltd.     

Reflection of plasma waves from a plane boundary

We state and analytically solve the linearized problem of the reflection of plasma waves from the half-space boundary in gas plasma for the first time. We consider mirror and diffusion boundary conditions and find the reflection coefficient as a function of the original problem parameters. We analyze the long-wave limit. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 498–510, March, 2007.     

Energy decay rates for a Timoshenko-type system of thermoelasticity of type III with constant delay

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi theory. We establish the stability of the system for the case of equal and nonequal speeds of wave propagation.     

Energy decay result in a Timoshenko‐type system of thermoelasticity of type III with weak damping

We investigate in this paper a thermoelastic system where the oscillations are defined by the Timoshenko model and the heat conduction is given by Green and Naghdi theories. We introduce 2 new stability numbers κ₁ , κ₂, and we prove a general decay result, from which the exponential and polynomial decays are only special cases.     

A stability result for the vibrations given by the standard linear model with thermoelasticity of type III

The aim of this article is to establish the well-posedness as well as an exponential stability result for the standard linear solid vibrating systems of thermoelasticity type III. Numerical experiments using finite differences are given to confirm our analytical result.     

Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III

This paper is concerned with the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III. We first obtain the well-posedness of the system by using semigroup method. We then investigate the asymptotic behaviour of the system through the perturbed energy method. We prove that the energy of system decays exponentially in the case of equal wave speeds and decays polynomially in the case of nonequal wave speeds. Under the case of nonequal wave speeds, we also investigate the lack of exponential stability of the system.     

Multidimensional stability of planar waves for delayed reaction-diffusion equation with nonlocal diffusion

In this paper, we consider the multidimensional stability of planar waves for a class of nonlocal dispersal equation in $n$--dimensional space with time delay. We prove that all noncritical planar waves are exponentially stable in $L^{\infty}(\RR^n )$ in the form of $\ee^{-\mu_{\tau} t}$ for some constant $\mu_{\tau} =\mu(\tau)>0$( $\tau >0$ is the time delay) by using comparison principle and Fourier transform. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the critical planar waves, we prove that they are asymptotically stable by establishing some estimates in weighted $L^1(\RR^n)$ space and $H^k(\RR^n) (k \geq [\frac{n+1}{2}])$ space.     

A novel model of nonlocal thermoelasticity with time derivatives of higher order

The objective of this work is to introduce a new system of differential equations describing the nonlocal thermoelasticity theory with higher time derivatives and two-phase lags. In order to obtain this model, we used the nonlocal continuum theory proposed by Eringen and the methodology of the Taylor series expansion of higher-order time derivatives. Some generalized thermoelasticity theories follow as limited cases. This model is used to study the thermoelastic interaction in a nonlocal medium. The medium is exposed to an applied magnetic field and a periodic time heat source with a constant strength. Some comparisons have been displayed in figures to estimate the influences of the nonlocal parameter and magnetic field as well as the parameters of higher-order on all the field quantities.     

Some basic boundary value problems of the plane thermoelasticity with microtemperatures

The present paper is devoted to the two‐dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains. This paper describes the use of the LaTeX2? mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.     

Traveling waves in nonlocal diffusion systems with delays and partial quasi-monotonicity

This paper is devoted to the study of a three-dimensional delayed system with nonlocal diffusion and partial quasi-monotonicity. By developing a new definition of upper-lower solutions and a new cross iteration scheme, we establish some existence results of traveling wave solutions. The results are applied to a nonlocal diffusion model which takes the three-species Lotka-Volterra model as its special case.     

Exponential decay in one-dimensional Type II/III thermoelasticity with two porosities

In this paper, we consider the theory of thermoelasticity with a double porosity structure in the context of the Green–Naghdi Types II and III heat conduction models. For the Type II, the problem is given by four hyperbolic equations, and it is conservative (there is no energy dissipation). We introduce in the system a couple of dissipation mechanisms in order to obtain the exponential decay of the solutions. To be precise, we introduce a pair of the following damping mechanisms: viscoelasticity, viscoporosities, and thermal dissipation. We prove that the system is exponentially stable in three different scenarios: viscoporosity in one structure jointly with thermal dissipation, viscoporosity in each structure, and viscoporosity in one structure jointly with viscoelasticity. However, if viscoelasticity and thermal dissipation are considered together, undamped solutions can be obtained     

Reflection of magneto-thermoelastic waves with two relaxation times and temperature dependent elastic moduli

The model of the equations of generalized magneto-thermoelasticity with two relaxation times in an isotropic elastic medium under the effect of reference temperature on the modulus of elasticity is established. The modulus of elasticity is taken as a linear function of reference temperature. Reflection of magneto-thermoelastic waves under generalized thermoelasticity theory is employed to study the reflection of plane harmonic waves from a semi-infinite elastic solid in a vacuum. The expressions for the reflection coefficients, which are the relations of the amplitudes of the reflected waves to the amplitude of the incident waves, are obtained. Similarly, the reflection coefficients ratios variations with the angle of incident under different conditions are shown graphically. A comparison is made with the results predicted by the coupled theory and with the case where the modulus of elasticity is independent of temperature.     

Existence of the generalized exponential attractor for coupled suspension bridge equations with double nonlocal terms

We investigate the long-time dynamical behavior of coupled suspension bridge equations with double nonlocal terms by using the quasi-stable methods. We first establish the well-posedness of the solutions by means of the monotone operator theory. Secondly, the dissipation of solution semigroup is obtained, and then, the asymptotic smoothness of solution semigroup is verified by the energy reconstruction method; ultimately, we prove the existence of global attractor. Finally, we show the existence of the generalized exponential attractor.     

Energy decay in a Timoshenko-type system of thermoelasticity of type III

In this paper we consider a one-dimensional linear thermoelastic system of Timoshenko type, where the heat conduction is given by Green and Naghdi theories. We prove the exponential stability by using the energy method.     

Localization of plane waves in low temperature regime

Discrete ühling-Uhlenbeck models together with the introduction of an orientation-free parameter are adopted to study the possible localization of plane waves propagating in dilute hard-sphere gases under very-low temperature environment based on the acoustical analog. The curves of the scattering amplitude vs. the disorder and Pauli-blocking measure we obtained here resemble qualitatively those proposed before.     

Traveling waves of a nonlocal diffusion SIRS epidemic model with a class of nonlinear incidence rates and time delay

In this paper, we study the traveling waves of a delayed SIRS epidemic model with nonlocal diffusion and a class of nonlinear incidence rates. When the basic reproduction ratio $\mathscr{R}_0>1$, by using the Schauder''s fixed point theorem associated with upper-lower solutions, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of traveling wave solutions connecting the disease-free steady state and the endemic steady state. When $\mathscr{R}_0<1$, the nonexistence of traveling waves is obtained by the comparison principle.     

Homogenization of a fully coupled thermoelasticity problem for a highly heterogeneous medium with a priori known phase transformations

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two‐phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing an a priori known interface movement because of phase transformations. After transforming the moving geometry to an ? ‐periodic, fixed reference domain, we establish the well‐posedness of the model and derive a number of ? ‐independent a priori estimates. Via a two‐scale convergence argument, we then show that the ? ‐dependent solutions converge to solutions of a corresponding upscaled model with distributed time‐dependent microstructures. Copyright © 2017 John Wiley & Sons, Ltd.     

A uniform stability result for thermoelasticity of type III with boundary distributed delay

In this paper we consider a thermoelastic system of type III with boundary distributed delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Green and Naghdi's theory is still strong enough to uniformly stabilize the system even in the presence of time delay.