Propagation of singularities for Cauchy problems of semilinear thermoelastic systems with microtemperatures

Abstract

The propagation of singularities of solutions to the Cauchy problem of a semilinear thermoelastic system with microtemperatures in one space variable is studied. First, by using a diagonalization argument of phase space analysis, the coupled thermoelastic system with microtemperatures will be decoupled microlocally. Second, using a classical bootstrap argument, the property of finite propagation speed of singularities for the semilinear thermoelastic system is obtained. Finally, it is also shown that the microlocal weak singularities propagate along the null bicharacteristics of the hyperbolic operators of the coupled semilinear system.

Some Non-Standard Problems Related with the Mathematical Model of Thermoelasticity with Microtemperatures

“…We have to outline that, by substituting the relations (1) and (2) into relations (3)-(5), we obtain the following system of linear partial differential equations for = u T w x t u r ss + + u s sr − T r + f r = ü r kT ss − T 0us s + 1 w s s + S = aT 0Ṫ 6 w r ss + 4 + 5 w s sr − 3 T r − 2 w r − M r = bẇ r (11) in B × − 0 . It has to be observed that, in the case when 1 3 = 0, the three equations in (11) decouple with respect to one to the other and so we have to study separately the differential system associated with the classical theory of elasticity and with the heat equation and with the differential system of microtemperatures.…”

“…It has to be observed that, in the case when 1 3 = 0, the three equations in (11) decouple with respect to one to the other and so we have to study separately the differential system associated with the classical theory of elasticity and with the heat equation and with the differential system of microtemperatures. For this reason, throughout this article we assume that 1 3 …”

“…Thus, Svanadze [4,5] and Scalia and Svanadze [6,7] and Scalia et al [8] studied the fundamental solutions and proves some existence and uniqueness theorems for equilibrium solutions and steady-state vibrations by means of the potential method, while Ieşan and Scalia [9] considered the plane strain in a homogeneous and isotropic body with microtemperatures. The behavior of shock waves and higher-order discontinuities that propagate in a thermoelastic body with inner structure and microtemperatures are studied by Ieşan [10] and the propagation of singularities of solutions to the Cauchy problem of a semilinear thermoelastic system with microtemperatures in one-space variable was studied by Yang and Huang in [11]. Some basic theorems were established by Aouadi [12] and Svanadze and Tracina [13] in the linear theory of microstretch thermoelasticity for isotropic solids with microtemperatures.…”

On the theory of thermoelasticity with microtemperatures

“…Thus, Svanadze [21,22], Scalia and Svanadze [23,24] and Scalia et al [25] study the fundamental solutions and proves some existence and uniqueness theorems for equilibrium solutions and steady state vibrations by means of the potential method, while Ieşan and Scalia [26] consider the plane strain in a homogeneous and isotropic body with microtemperatures. The behavior of shock waves and higher-order discontinuities which propagate in a thermoelastic body with inner structure and microtemperatures are studied by Ieşan [27] and the propagation of singularities of solutions to the Cauchy problem of a semilinear thermoelastic system with microtemperatures in one space variable is studied by Yang and Huang in [28]. Some basic theorems are established by Aouadi [29] and Svanadze and Tracina [30] in the linear theory of microstretch thermoelasticity for isotropic solids with microtemperatures.…”

Propagation of Rayleigh Wave in a Thermoelastic Solid Half-Space with Microtemperatures

“…Ieşan and Quintanilla [6] presented the theory of thermoelastic bodies with inner structure and microtemperatures, which permits the transmission of heat as thermal waves at finite speed. Several papers based on the theory of thermoelasticity with microtemperatures have been published such as Iesan [7], Svanadze [8,9], Casas and Quintanilla [10], Scalia and Svanadze [11,12], Aoudai [13], Iesan [14], Scalia et al [15], Ieşan and Scalia [16], Yang and Huang [17], Quintanilla [18], Svanadze and Tracinà [19], Chirita et al [20], and Steeb et al [21].…”