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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 convexdomain, $f:[0,T]\times\mathbb{R}^n \rightarrow \mathbb{R}$ is $L^1$-integrablein time and convex in the second variable. Assuming that the initial andboundary datum $u_o:\overline{\Omega}\rightarrow \mathbb{R}$ satisfies thebounded slope condition, we prove the existence of a unique variationalsolution that is Lipschitz continuous in the space variable.

The bounded slope condition for parabolic equations with time-dependent integrands