Behaviour of a non-local reactive convective problem modelling ohmic heating of foods

Abstract

We consider the non-local problem, u t +u x =λ+(u)/(∫(u)/(∫ 0 1 f(u)dx) 2 , 0 < x < 1, which models the temperature when an electric current flows through a moving material with negligible thermal conductivity. The potential difference across the material is fixed but the electrical resistivity f(u) varies with temperature. For f decreasing with ∫∞ 0 f(s)ds < ∞, blow-up occurs if λ is too large for a steady state to exist or if the initial condition is too big. If f is increasing with ∫∞ 0 ds/f(s) < ∞ blow-up is also possible. If f is increasing with ∫∞ 0 ds/f(s) = ∞ or decreasing with ∫∞ 0 f(s)ds = ∞ the solution is global. Some special cases with particular forms of f are discussed to illustrate what the solution can do