SPACE-TIME NONLOCAL MODEL FOR HEAT-CONDUCTION

We consider a space-time nonlocal heat conduction model with balance laws in the form of integral equations (so-called strong nonlocality). The model identifies two internal parameters-the time tau and the space h scales of nonlocality. In going from the strong nonlocal model to its approximations of various accuracy in the form of partial differential equations, which correspond to weak nonlocality, we introduce two limiting relations between tau and h as tau,h-->0. In the diffusion limit, which preserves the thermal diffusivity a=h(2)/tau=const as tau,h-->0, the strong nonlocal model gives a hierarchy of parabolic equations with an infinite speed of heat waves. In the wave limit, which preserves the ratio v=h/tau=const as tau,h-->0, a hierarchy of hyperbolic equations has been obtained. The hyperbolic equations imply a finite speed of heat waves. These results suggest that for diffusion (low-k) and propagative (high-k) regimes distinct models are responsible for the space-time evolution of the temperature and heat flux. The connection with phonon hydrodynamic theory and applications to other problems are discussed.