We consider here the so called thermistor problem. The heat produced in a conductor by an electric current leads to the system:
u t -∇·k(u)∇u=σ(u)|∇ϕ| 2 ,∇·σ(u)∇ϕ=0inΩ×(0,T),u=0,ϕ=ϕ 0 onΓ×(0,T),u(·,0)=u 0 ·(1)
Here, Ω is a smooth bounded open set of ℝ n , Γ denotes the boundary, T is some positive given number, ϕ is the electrical potential, u the temperature inside the conductor, k(u)>0 the thermal conductivity and σ(u)>0 the electrical conductivity.
We show existence of a solution to (1), and focus on the question of uniqueness and on the problem of global existence or blow up.