I've got some problems verifying an assumption for a parabolic PDE. Namely, let $(V,H,V^*)$ be a Gelfand-Triple, $u_0 \in V$, $\psi\colon V \to \mathbb{R}$ convex and lower-semicontinuous and $a\colon V\times V \to \mathbb{R}$ a coerzive, bounded bilinear form. We will approximate $\psi$ with it's Moreau-Enveloppe $\psi_j$ and define $\Psi_j\colon V\to V^*$ as the Gateâux-derivative of $\psi_j$. We can prove, that $\Psi_j$ is demi-continuous and maps bounded sets to bounded sets. With this preparation, we need to find a sequence $\lbrace u_{0j} \rbrace \subset V$ converging to $u_0$ and another sequence $\lbrace k_j \rbrace \subset H$, uniformly bounded, satisfying \begin{align*} a(u_{0j}, v) + \langle \Psi_j(u_{0j}), v \rangle_{V^*,V} = (k_j, v)_H \quad \forall v\in V. \end{align*}

Can someone give me a hint how to prove this? I found this in Lions & Duvant's book. Thanks.


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