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We consider the Moore–Gibson–Thompson–Gurtin–Pipkin model u ttt + α u tt + β Δ 2 u t + γ Δ 2 u = - ϱ Δ θ θ t - ∫ 0 ∞ g (s) Δ θ (t - s) d s = ϱ Δ u tt + ϱ α Δ u t where g is a positive, convex, and summable memory kernel. The system is shown to generate a strongly continuous semigroup, whose stability properties depend of the structural parameters α , β , γ > 0 . In the subcritical regime α β > γ , we provide a necessary and sufficient condition on the memory kernel in order for exponential stability to occur. Such a condition is actually very general, allowing, for instance, any compactly supported g of the above kind. On the contrary, in the critical regime α β = γ exponential stability never takes place. Even more, there exist particular kernels, called resonant, for which the semigroup exhibits periodic trajectories. [ABSTRACT FROM AUTHOR]

We investigate the dissipative structure for the system of Moore–Gibson–Thompson (MGT) thermoelasticity in the whole space. To analyze the dissipative structure, it is very useful to rewrite the equations into a symmetric hyperbolic system and apply the so‐called stability condition. When we rewrite our system into the symmetric hyperbolic form in the multidimensional case, it is important to take the constraint conditions into account. Indeed, the stability condition with the constraint conditions guarantees the dissipative property for our system in some cases. In this paper, we introduce the stability condition with constraints for the general problem and apply this argument to the system of MGT thermoelasticity. Furthermore, we discuss the optimality of the decay estimates we obtain, together with the no regularity‐loss phenomenon. [ABSTRACT FROM AUTHOR]

We consider the isotropic initial boundary value problem for the heat equation on open regions with noncompact boundary and construct differential inequalities for a generalized heat flowmeasure defined over a spherical cross section. Under suitable assumptions, integration of the differential inequality leads to spatial growth and decay rate estimates formean-square cross-sectional measures of the time-weighted temperature spatial gradient. The estimates are then used to obtain similar results for the time-weighted temperature. In particular, when the base heat flow measure is positive, the timeweighted temperature becomes pointwise unbounded at large spatial distance. [ABSTRACT FROM AUTHOR]

We consider a binary mixture system with local Kelvin–Voigt damping in one of the components of the mixture. The well‐posedness and the polynomial stability are proved by the semigroup theory and the frequency domain approach. [ABSTRACT FROM AUTHOR]

In this paper, we investigate how the solutions vary when the relaxation parameter, the conductivity rate parameter, or the thermal conductivity parameter change in the case of the Moore-Gibson-Thompson heat equation. In fact, we prove that they can be controlled by a term depending upon the square of the variation of the parameter. These results concern the structural stability of the problem. We also compare the solutions of the MGT equation with the Maxwell-Cattaneo heat conduction equation and the type III heat equation (limit cases for the first two previous parameters) and we show how the difference between the solutions can be controlled by a term depending on the square of the limit parameter. This result gives a measure of the convergence between the solutions for the different theories. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a model of poro-thermoelasticity with microtemperatures, where the behavior of the body is influenced by the history of both temperature and microtemperatures. Mathematically, this translates into a system of partial integro-differential equations. Under suitable condition on the tensors appearing in the model, we prove that the resulting system is well posed. In the one-dimensional case, the exponential decay of the energy is proved. [ABSTRACT FROM AUTHOR]

We study antiplane shear deformations for isotropic and homogeneous strain gradient mixtures of the Kelvin‐Voigt type in a cylinder. Our aim is to analyze the behaviour of the solutions with respect to the time variable when a dissipative structural mechanism is considered. We study three different cases, each at a time. For each case we prove existence and uniqueness of solutions. We obtain the exponential decay of the solutions in the hyperviscosity and viscosity cases. Exponential decay is also expected when the dissipation is generated by the relative velocity (in the generic case, although a particular combination of the constitutive parameters leads to slow decay). These results are proved with the help of the theory of semigroups. We study antiplane shear deformations for isotropic and homogeneous strain gradient mixtures of the Kelvin‐Voigt type in a cylinder. Our aim is to analyze the behaviour of the solutions with respect to the time variable when a dissipative structural mechanism is considered. We study three different cases, each at a time. For each case we prove existence and uniqueness of solutions. We obtain the exponential decay of the solutions in the hyperviscosity and viscosity cases.... [ABSTRACT FROM AUTHOR]

In this paper we consider the following three-phase-lag thermoelastic plate: ρ u ¨ = − Δ (μ Δ u − m θ) , c (θ ˙ + τ q θ ¨) = m Δ (u ˙ + τ q u ¨) + Δ (k ∗ α + τ ν ∗ θ + k τ θ θ ˙). We obtain analyticity and exponential stability for the associated C 0 -semigroup when the coefficients satisfy τ ν ≥ k ∗ τ q . The analyticity of the semigroup still holds when τ ν < k ∗ τ q . However, the instability of solution was proved by the Routh–Hurwitz rule. [ABSTRACT FROM AUTHOR]

In this paper, we propose a system of equations governing the dual‐phase‐lag heat conduction with microtemperatures. Several conditions on the coefficients are imposed so that the energy of the system is positive definite and dissipative. On this base we prove the well‐posedness and exponential stability of the system by means of the semigroup theory and frequency domain method. [ABSTRACT FROM AUTHOR]

In this paper we discuss some thermoelastic and thermoviscoelastic models obtained from the Gurtin theory, based on the invariance of the entropy under time reversal. We derive differential systems where the temperature and the velocity are ruled by generalized versions of the Moore-Gibson-Thompson equation. In the one-dimensional case, we provide a complete analysis of the evolution, establishing an existence and uniqueness result valid for any choice of the constitutive parameters. This result turns out to be new also for the MGT equation itself. Then, under suitable assumptions on the parameters, corresponding to the subcritical regime of the system, we prove the exponential stability of the related semigroup. [ABSTRACT FROM AUTHOR]

In this paper, we consider a thermoelastic model where heat conduction is described by the history dependent version of the Moore–Gibson–Thompson equation, arising via the introduction of a relaxation parameter in the Green-Naghdi type III theory. The well-posedness of the resulting integro-differential system is discussed. In the one-dimensional case, the exponential decay of the energy is proved. [ABSTRACT FROM AUTHOR]

We provide some spatial estimates for the nonlinear partial differential equation governing anti-plane motions in a nonlinear viscoelastic theory of Kelvin–Voigt type when the viscosity is a function of the strain rate. The spatial estimates we prove are an alternative of Phragmen–Lindelöf type. These estimates are possible when a precise balance between the elastic and viscoelastic nonlinearities holds. [ABSTRACT FROM AUTHOR]

It is known that in the case that several constitutive tensors fail to be positive definite the system of the thermoelasticity could become unstable and, in certain cases, ill-posed in the sense of Hadamard. In this paper, we consider the Moore–Gibson–Thompson thermoelasticity in the case that some of the constitutive tensors fail to be positive and we will prove basic results concerning uniqueness and instability of solutions. We first consider the case of the heat conduction when dissipation condition holds, but some constitutive tensors can fail to be positive. In this case, we prove the uniqueness and instability by means of the logarithmic convexity argument. Second we study the thermoelastic system only assuming that the thermal conductivity tensor and the mass density are positive and we obtain the uniqueness of solutions by means of the Lagrange identities method. By the logarithmic convexity argument we prove later the instability of solutions whenever the elasticity tensor fails to be positive, but assuming that the conductivity rate is positive and the thermal dissipation condition hold. We also sketch similar results when conductivity rate and/or the thermal conductivity fail to be positive definite, but the elasticity tensor is positive definite and the dissipation condition holds. Last sections are devoted to considering the case when a third-order equation is proposed for the displacement (which comes from the viscoelasticiy). A similar study is sketched in these cases. [ABSTRACT FROM AUTHOR]

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 first part, the heat equation is based on the usual Fourier law. In the second one, it is based on the type Ⅲ heat conduction law. In both parts, we prove the existence of exponential attractors and, therefore, of finite-dimensional global attractors. [ABSTRACT FROM AUTHOR]

In this paper we consider the case of a functionally graded heat-conducting hollowed cylinder. Our purpose is to investigate the consequences of the material inhomogeneity on the decay of Saint-Venant end effects in the case of linear isotropic rigid solids. The mathematical issues involve the implications of spatial inhomogeneity on the decay rates of solutions to Dirichlet boundary-value problems. The rate of decay is characterized in terms of the smallest eigenvalue of a Sturm–Liouville problem. We first consider the case where the inhomogeneity depends on the radius of the cross-section, but later we also consider the case where the inhomogeneity also depends on the axial variable. The last section considers the case where the cross-section is increasing. Some tables and figures illustrate our estimates. [ABSTRACT FROM AUTHOR]

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 3‐phase‐lag theories, reflecting Saint‐Venant's principle. In a recent paper, 2 families of cases for high‐order partial differential equations were studied. Here, we investigate a third family of cases, which corresponds to the fact that a certain condition on the time derivative must be satisfied. We also study the spatial behavior of a thermoelastic problem. We obtain a Phragmén‐Lindelöf alternative for the solutions in both cases. The main tool to handle these problems is the use of an exponentially weighted Poincaré inequality. [ABSTRACT FROM AUTHOR]

This paper is devoted to the strain gradient theory of thermoelastic materials whose microelements possess microtemperatures. The work is motivated by an increasing use of materials which possess thermal variation at a microstructure level. In the first part of this paper we deduce the system of basic equations of the linear theory and formulate the boundary-initial-value problem. We establish existence, uniqueness, and continuous dependence results by the means of semigroup theory. Then, we study the one-dimensional problem and establish the analyticity of solutions. Exponential stability and impossibility of localization are consequences of this result. In the case of the anti-plane problem we derive uniqueness and instability results without assuming the positivity of the mechanical energy. Finally, we study equilibrium theory and investigate the effects of a concentrated heat source in an unbounded body. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study generalizations of the nonconserved and conserved Caginalp phase-field systems based on the Maxwell-Cattaneo law with two temperatures for heat conduction. In particular, we obtain well-posedness results and study the dissipativity of the associated solution operators. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this note, we investigate the spatial behavior of the solutions of the equation proposed to describe a theory for the heat conduction with two delay terms. We obtain an alternative of the Phragmén-Lindelöf type, which means that the solutions either decay or blow-up at infinity, both options in an exponential way. We also describe how to obtain an upper bound for the amplitude term. This is the first contribution on spatial behavior for partial differential equations involving two delay terms. We use energy arguments. The main point of the contribution is the use of an exponentially weighted energy function. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 Poincaré 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. [ABSTRACT FROM AUTHOR]

In this paper we investigate the spatial behavior of the solutions for a theory for the heat conduction with one delay term. We obtain a Phragméen-Lindelöf type alternative. That is, the solutions either decay in an exponential way or blow-up at infinity in an exponential way. We also show how to obtain an upper bound for the amplitude term. Later we point out how to extend the results to a thermoelastic problem. We finish the paper by considering the equation obtained by the Taylor approximation to the delay term. A Phragmén-Lindelöf type alternative is obtained for the forward and backward in time equations. [ABSTRACT FROM AUTHOR]

This article is concerned with the linear theory of thermoelasticity with microtemperatures. In a recent article we have used the logarithmic convexity method to investigate uniqueness, instability and structural stability. The results are restricted to the case when the constitutive coefficientsk1andk3have the same signs. Here we prove that these results also hold when the coefficientsk1andk3have opposite signs. [ABSTRACT FROM AUTHOR]

Our aim in this article is to study a phase-field system based on a three-phase-lag for the thermal flux vector. In particular, we prove the existence and uniqueness of solutions and then study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist. [ABSTRACT FROM AUTHOR]

For β ∈ ℝ, the authors consider the evolution system in the unknown variables u and α describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established. [ABSTRACT FROM AUTHOR]

The three-dual-phase-lag theory based on the constitutive law q(P, t + τq) = -(k∇T(P, t + τT) + k*∇ν(P, t + τν)), [image omitted] was proposed by Roy Choudhuri as an extension of the theory by Tzou where the recent theories of Greeen and Naghdi could be recover. Although it proposes a law that could be compatible with our intuition, when we adjoin it with the energy equation -∇q(x, t) = cT˙(x, t), we obtain an ill-posed problem, that is, a problem which has a sequence of eigenvalues such that their real part are positive (and go to infinity). A consequence of this fact is that the problem is always unstable and we cannot obtain continuous dependence on the initial data. In this note we combine this constitutive equation with two-temperature heat conduction theory and we show in this new context that the problem is well-posed. [ABSTRACT FROM AUTHOR]

Our aim in this article is to study generalizations of the Caginalp phase-field system based on a thermomechanical theory of deformable continua proposed by Green and Naghdi. In particular, we obtain well-posedness results and study the asymptotic and spatial behaviours of the solutions. [ABSTRACT FROM AUTHOR]

In this paper we investigate the thermomechanical theory of thermoelasticity without energy dissipation. We consider the case where the assumptions on the constitutive tensor are different from the usual ones in the current literature. These assumptions are suitable because they correspond to situations where the material is prestressed and has initial entropy flux. When we restrict our attention to the one-dimensional problem, we first establish a condition on the coefficients to guarantee the hyperbolicity of the system. Then we prove that the problem is well posed and stable. When the hyperbolicity condition is not satisfied we prove that the problem is ill-posed. We also prove a similar result when the dimension is greater than one whenever the domain satisfies a certain condition. Though we cannot expect in general the stability of solutions in dimension greater than one, we prove the uniqueness and stability of radial solutions in the general case. To obtain the results, we need the use of a conservation law which has not previously been considered in the literature. Similar results are not known for other thermoelastic theories. [ABSTRACT FROM AUTHOR]

The dual-phase lag theory based on the law q(x, t + τq) = -K ∇ T(x, t + τT) was proposed from an intuitive point of view by Tzou. This equation shows that the temperature gradient established across a material volume at the position x at time t + τT results in a heat flux to flow at a different instant of time t + τq. Though it proposes a law which could be compatible with our intuition, when we adjoin it with the energy equation - ∇ q(x, t) = c [image omitted](x, t), we may obtain an ill posed problem, that is, a problem which has a sequence of eigenvalues such that its real part is positive (and goes to infinity). A consequence of this fact is that the problem may be unstable and we cannot obtain continuous dependence on the initial data.1 In this note we combine this constitutive equation with a two temperatures heat conduction theory and we show in this new context the problem is well posed. We also show that whenever the constitutive constants satisfy suitable relations, the real part of a point spectrum of the problem is negative. We also discuss a 2-temperature thermoelasticity and a higher-order theory corresponding to the higher order Taylor expansions of the fields involved. [ABSTRACT FROM AUTHOR]

A theory of thermoelasticity without energy dissipation has recently been initiated. Its evolution equations are hyperbolic. In this article we study the spatial behavior of solutions of the linear problem. We prove spatial estimates that are similar to those obtained in other theories of a hyperbolic type. [ABSTRACT FROM AUTHOR]

The authors present a note on this issue of the journal including fundamental contributions to the field of applied mathematics. They particularly thank Robin Knops for his fundamental contributions on the subject. Also, the editorial board of the journal is thankful to Nigel Scott, for having given them the opportunity to edit this special issue. They further thank Mike Hayes for writing the biographical notes.