Nanoscale heat transfer involves a highly complex process, as has been witnessed in the last years in which remarkable novel phenomena related to very short time and spatial scales, such as enhancement of thermal conductivity in nanofluids, granular materials, thin layers, and composite systems among others, have been reported [

1–

5]. The traditional approach to deal with these phenomena has been to use the Fourier heat transfer equation. This methodology has proven to be extensively useful in the analysis of heat transport in a great variety of physical systems, however, when applied to highly heterogeneous systems or when the time and space scale are very short, they show serious inconsistencies [

6,

7]. In order to understand the nanoscale heat transfer, a great diversity of novel theoretical approaches have been developed [

3,

5,

7,

8]. In particular, when analyzing two-phase systems, one of the simplest heat conduction models that considers the microstructure is known as the two-equation model [

9,

10], which has been developed writing the Fourier law of heat conduction [

11] for each phase and performing a volume averaging procedure [

9]. This model takes into account the porosity of the component phases as well as their interface effects by means of two coefficients [

12]. Besides, it has been shown that the two-equation model is equivalent to the one-equation model known as the dual-phase lagging (DPL) model, in which the microstructural effects are taken into account by means of two time delays [

3,

10,

13–

15]. DPL model have been proposed to surmount the well-known drawbacks of the Fourier law and the Cattaneo equation of heat conduction [

7], and establishes that either the temperature gradient may precede the heat flux or the heat flux may precede the temperature gradient. Mathematically, this is written in the form

where
is the position vector, *t* is the time,
is the heat flux vector, *T*[K] is the absolute temperature, *k*[W.m^{-1}.K^{-1}] is the thermal conductivity, ͌_{
q
}is the phase lag of the heat flux, and ͌_{
T
}is the phase lag of the temperature gradient. For the case of ͌_{
q
}>͌_{
T
}, the heat flux (effect) established across the material is a result of the temperature gradient (cause); while for ͌_{
q
}<͌_{
T
}, the heat flux (cause) induces the temperature gradient (effect). Notice that when ͌_{
q
}= ͌_{
T
}, the response between the temperature gradient and the heat flux is instantaneous and Equation 1 reduces to Fourier law except for a trivial shift in the time scale. In addition, note that for ͌_{
T
}= 0; the DPL model reduces to the single-phase lagging (SPL) model [3]. The time delay ͌_{
q
}is interpreted as the relaxation time due to the fast-transient effects of thermal inertia, while the phase lag ͌_{
T
}represents the time required for the thermal activation in micro-scale [3]. For the case of composite materials, the phase lag ͌_{
q
}takes into account the time delay due to contact thermal resistance among the particles, while ͌_{
T
}is interpreted as the time required to establish the temperature gradient through the particles [12, 16]. The lagging behavior in the transient process is caused by the finite time required for the microscopic interactions to take place. This time of response has been claimed to be in the range of a few nanoseconds in metals and up to the order of several seconds in granular matter [3]. In this last case, due to the low-conducting pores among the grains and their interface thermal resistance.

The thermal conductivity is an intrinsic property of each material which measures its ability for the transfer of heat and is determined by the kinetic properties of the energy carriers and the material microstructure [6, 17]. Under the framework of Boltzmann kinetic theory [3, 6], it can be shown that the thermal conductivity is directly proportional to the group velocity and mean free path of the energy carriers (electrons and phonons). These parameters depend strongly on the material temperature, due to the multiple scattering processes involved among energy carriers and defects, such as impurities, dislocations, and grain boundaries, [6, 18]. Thus, in general; thermal conductivity exhibits complicated temperature dependence. However, in many cases of practical interest, the thermal conductivity can be considered independent of the temperature for a considerable range of operating temperatures [3, 6, 11]. Based on this fact and to keep our mathematical approach tractable, we assume that thermal conductivity is a temperature-independent parameter.

Phase lags represent the time parameters required by the material to start up the heat flux and temperature gradient, after a thermal excitation has been imposed; larger phase lags are expected in material with smaller thermal conductivities, as is the case of granular matter [3]. Materials, in which the temperature gradient phase lag dominates, show a strong attenuation of the neat heat flux. In this case, the behavior is dominated by parabolic terms of the heat transport equation. In contrast, materials in which the heat flux phase lag is dominant show a slight attenuation of the heat flux, implying that a hyperbolic Cattaneo-Vernotte heat propagation is present. For a further discussion of the relationship between thermal conductivity and phase lags, Tzou's book [3] is recommended.

It is convenient to take into account that the heat flux and temperature gradient shown in Equation 1 are the local responses within the medium. They must not be confused with the global quantities specified in the boundary conditions. When a heat flux (as a laser source) is applied to the boundary of a solid medium, the temperature gradient established within the medium can still precede the heat flux. The application of the heat flux at the boundary does not guarantee the precedence of the heat flux vector to the temperature gradient at all. In fact, whether the heat flux vector precedes the temperature gradient or not depends on the combined effects of the thermal loading and thermal properties of the materials, as was explained by Tzou [3]. In this way, the DPL model should provide a comprehensive treatment of the heterogeneous nature of composite media [3, 13].

It has been shown that under the DPL model and in absence of internal heat sources, the temperature satisfies the following differential-difference equation [

19–

22]:

where *α*[m^{2}.s^{-1}] is the thermal diffusivity of the medium, and *͌* = *͌*_{
q
}*-͌*_{
T
} is the difference of the phase lags. Equation 2 shows explicitly that the DPL and SPL models, both in their exact form, are entirely equivalent, when *͌*> 0(*͌*_{
q
}*-͌*_{
T
} )[19].

The solutions of Equation 2 for some geometries have been explored [19–22]. In the time domain, Jordan et al. [19] and Quintanilla and Jordan [22] have shown that the SPL model, in its exact form, can lead to instabilities with respect to specific initial values. Additionally, in the frequency domain, using a modulated heat source, Ordonez-Miranda and Alvarado-Gil [21] have shown that the if the DPL model is valid, its applicability must be restricted to frequency-interval strips, which are determined only by the difference of the time delays *͌* = *͌*_{
q
}*-͌*_{
T
} . These studies have pointed out that the usefulness of the Cattaneo-Vernotte and DPL exact models is limited.

In this study, by means of the method of separation of variables, the solution of Equation 2 is obtained in a bounded domain. It is shown that, for any kind of homogeneous boundary conditions, its solutions go to infinity in the long time domain. This explosive characteristic of the temperature predicted by Equation 2 indicates that the DPL model, in its exact form, cannot be considered as a valid model of heat conduction.