We consider quasilinear parabolic variational–hemivariational inequalities in a cylindrical domain \documentclass[12pt]{minimal}
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\begin{document}$$Q=\Omega \times (0,\tau )$$\end{document} of the form \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q j^o(x,t, u;v-u)\,dxdt\ge 0,\ \ \forall \ v\in K, \end{aligned}$$\end{document}where \documentclass[12pt]{minimal}
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\begin{document}$$K\subset X_0=L^p(0,\tau ;W_0^{1,p}(\Omega ))$$\end{document} is some closed and convex subset, \documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} is a time-dependent quasilinear elliptic operator, and \documentclass[12pt]{minimal}
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\begin{document}$$s\mapsto j(\cdot ,\cdot ,s)$$\end{document} is assumed to be locally Lipschitz with \documentclass[12pt]{minimal}
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\begin{document}$$(s,r)\mapsto j^o(x,t, s;r)$$\end{document} denoting its generalized directional derivative at \documentclass[12pt]{minimal}
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\begin{document}$$s$$\end{document} in the direction \documentclass[12pt]{minimal}
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\begin{document}$$r$$\end{document}. The main goal of this paper is threefold: first, an existence and comparison principle is proved; second, the existence of extremal solutions within some sector of appropriately defined sub-supersolutions is shown; third, the equivalence of the above parabolic variational–hemivariational inequality with an associated multi-valued parabolic variational inequality of the form \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q \eta \, (v-u)\,dxdt\ge 0,\ \ \forall \ v\in K \end{aligned}$$\end{document}with \documentclass[12pt]{minimal}
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\begin{document}$$\eta (x,t)\in \partial j(x,t, u(x,t))$$\end{document} is established, where \documentclass[12pt]{minimal}
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\begin{document}$$s\mapsto \partial j(x,t, s)$$\end{document} denotes Clarke’s generalized gradient of the locally Lipschitz function \documentclass[12pt]{minimal}
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\begin{document}$$s\mapsto j(\cdot ,\cdot ,s)$$\end{document}.
