Nonlinear Hyperbolic Rigid Heat Conductor of the Coleman Type

A one-dimensional nonlinear hyperbolic homogeneous isotropic rigid heat conductor proposed by Coleman is analyzed using the method of weakly nonlinear geometric optics. For such a model the law of conservation of energy, the dissipation inequality, the Cattaneo's equation, and a generalized energy-entropy relation with a parabolic variation of the energy and entropy along the heat-flux axis, are postulated. First, it is shown that the model can be described by a non-homogeneous quasi-linear hyperbolic matrix partial differential equation of the first order for an unknown vector u = (θ, Q) ^T , where θ and Q are the dimensionless absolute temperature and heat-flux fields, respectively. Next, the Cauchy problem for the matrix equation with a weakly perturbed initial condition is formulated, and an asymptotic solution to the problem in terms of the amplitudes σ_α (α = 1, 2) that satisfy a pair of nonlinear first order partial differential equations, is obtained. The Cauchy problem is then solved in a closed form when the initial data are suitably restricted. Numerical examples are included.