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

We consider the non-local problem, u(t) + u(x) = lambda f(u)/(integral(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. It is found that, for f decreasing with integral(0)(infinity) f(s)ds < infinity, blow-up occurs if lambda is too large for a steady state to exist or if the initial condition is too big. If f is increasing with integral(0)(infinity) ds/f(s) < infinity blow-up is also possible. If f is increasing with integral(0)(infinity) ds/f(s) = infinity or decreasing with integral(0)(infinity) f(s)ds = infinity the solution is global. Some special cases with particular forms of f are discussed to illustrate what the solution can do.