In this talk, we solve heat equations on a cylinder with nonzero lateral data. The lateral boundary of the cylinder is decomposed into two portions separated by a time-varying, nonsmooth interface. On each portion, we impose Dirichlet data from the parabolic Riesz potential space $L^1_p$ or Neumann data from the Sobolev space $L_p$, respectively. Based on several joint works with Jongkeun Choi (Busan national university) and Hongjie Dong (Brown university).