Abstract. We consider a non-local equation with initial and boundary conditions of the form2 , which models the temperature when an electric current flows through a moving material with negligible thermal conductivity and time-dependent velocity. The potential difference across the material is fixed but the electrical resistivity f (u) varies significantly with temperature. It is found that for f (u) 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 large enough. On the other hand, if f (u) is decreasing with ∞ 0 f (s) ds = ∞, then it is proved that u(x, t) is a global-in-time and unbounded solution.
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