(untitled) (Analysis in Theory and Applications, 2021)
In this paper we study the initial boundary value problem for the system div(σ(u) ablaφ)=0, u_t-Δ u=σ(u)| ablaφ|^2. This problem is known as the thermistor problem which models the electrical heating of conductors. Our assumptions on σ(u) leave open the possibility that _u→∞σ(u)=0, while _u→∞σ(u) is large. This means that σ(u) can oscillate wildly between 0 and a large positive number as u→ ∞. Thus our degeneracy is fundamentally different from the one that is present in porous medium type of equations. We obtain a weak solution (u, φ) with | abla φ|, | abla u|L^∞ by first establishing a uniform upper bound for e^ε u for some small ε. This leads to an inequality in ablaφ, from which the regularity result follows. This approach enables us to avoid first proving the Hölder continuity of φ in the space variables, which would have required that the elliptic coefficient σ(u) be an A_2 weight. As it is known, the latter implies that σ(u) is "nearly bounded''.