5. Weakly Nonlocal and Nonlinear Heat Transport

The thermo-mechanical behavior of miniaturized systems, the characteristic lengths of which is of the order of few nanometers, is strongly influenced by memory, nonlocal, and nonlinear effects [1, 18, 27, 50]. In one-dimensional steady-state situations, in modeling the heat transport along nanowires or thin layers, some of these effects may be incorporated into a size-dependent effective thermal conductivity \(\lambda _{ \text{eff}}\) [2, 43], and a Fourier law (FL)-type equation may still be used with \(\lambda _{ \text{eff}}\) as the thermal conductivity, instead of the bulk value \(\lambda\). However, in fast perturbations, or under strong heat gradients, or in axial geometries an effective thermal conductivity is not enough to overcome the different problems related to the FL, as for instance, the infinite speed of propagation of thermal disturbances, or some genuinely nonlinear effects in steady states [9, 17, 25, 28, 30, 38]. Therefore, in modeling heat conduction, it is necessary to go beyond FL by introducing more general heat-transport equations, and analyze more general geometries than those considered in Chaps. 3 and 4. In Chap. 2 the nonlinear heat-transport equation (2.​16) has been introduced. Here we will analyze some consequences of it.