  In this paper, we study the Cauchy-Dirichlet problem \begin{equation*}
  \left\{
  \begin{array}{ll}
  \mbox{$\partial_t u - \operatorname{div} \left( D_\xi f(t, Du)\right) = 0$ }
& \mbox{in $\Omega_T$}, \\[5pt]
  \mbox{$u = u_o$} & \mbox{on $\partial_{\mathcal{P}} \Omega _T$},\\[5pt]
  \end{array}
  \right. \end{equation*} where $\Omega \subset \mathbb{R}^n$ is a convex
domain, $f:[0,T]\times\mathbb{R}^n \rightarrow \mathbb{R}$ is $L^1$-integrable
in time and convex in the second variable. Assuming that the initial and
boundary datum $u_o:\overline{\Omega}\rightarrow \mathbb{R}$ satisfies the
bounded slope condition, we prove the existence of a unique variational
solution that is Lipschitz continuous in the space variable.
