摘要
In this paper we study the initial boundary value problem for the system div(σ(u)∇φ)=0,u_(t)−∆u=σ(u)|∇φ|2.This problem is known as the thermistor problem which models the electrical heating of conductors.Our assumptions onσ(u)leave open the possibility that lim inf_(u→∞)σ(u)=0,while lim sup_(u→∞)σ(u)is large.This means thatσ(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|∇φ|,|∇u|∈L∞by first establishing a uniform upper bound for eεu for some smallε.This leads to an inequality in∇φ,from which the regularity result follows.This approach enables us to avoid first proving the Holder continuity ofφin the space variables,which would have required that the elliptic coefficientσ(u)be an A2 weight.As it is known,the latter implies that lnσ(u)is“nearly bounded”.
