The paper deals with weak solvability of the following time dependent thermistor problem with degenerate thermal conductivity \[ \begin{cases} u_t -\nabla\cdot (a(u)\nabla u)=\sigma(u)|\nabla\varphi|^2 &\text{ in }Q=\Omega\times (0,T),\\ \nabla\cdot (\sigma(u)\nabla \varphi)=0 &\text{ in }Q,\\ u=0 &\text{ on }\partial\Omega\times(0,T),\\ \varphi=\varphi_0 &\text{ on }\partial\Omega\times(0,T),\\ u(\cdot,0)=u_0 &\text{ in }\Omega. \end{cases} \] {}.