Asymptotic Behaviour of Positive Solutions of Nonlinear Volterra Equations for Heat Flow

摘要

This report considers nonlinear heat flow in a homogeneous bar of unit length of a material with memory with the ends of the rod maintained at zero temperature and with the history of temperature prescribed for time t less than or = 0. For such materials the internal energy and heat flux are functionals (rather than functions) of the temperature and of the gradient of temperature respectively. Application of the law of balance of heat leads to a nonlinear Volterra integrodifferential equation, together with appropriate boundary and initial conditions, which model the physical problem. This initial boundary value problem, which cannot be solved explicitly and which is difficult to analyse, can be transformed by standard methods to the general equation (V) given in the Abstract. The resulting kernel b can be expressed in terms of the internal energy and heat flux relaxation functions. These are presumed to be known for the physical problem. The operator A in (V) is a nonlinear differential operator together with boundary conditions, and the forcing term f in (V) depends on the given initial temperature distribution, the given external heat supply, and the given history of temperature.