  We establish $L^p$, $2\le p\le\infty$ solvability of the Dirichlet boundary
value problem for a parabolic equation $u_t-\mbox{div}(A\nabla u)=0$ on
time-varying domains with coefficient matrix $A=(a_{ij})$ that satisfy a small
Carleson condition. The result is motivated by similar results for the elliptic
equation $\mbox{div}(A\nabla u)=0$ that were established previously.
