Consider the following problem: we seek $u\in W^{1,2}(0,T; V, H)$ such that $$\langle u'(t), w_j \rangle_H+a(u(t), w_j)=\langle b(t), w_j\rangle_{V^*\times V},\qquad u(0)=u_0.$$ In here, $V\subset H \subset V^*$ is an evolution triple, $a\colon V\times V\to\mathbb{R}$ is a bilinear, bounded and strongly positive map. Moreover, we are given $u_0\in H$ and $\{w_1, w_2,...\}$ is a basis in $V$ and finally $b\in L^2(0,T; V^*)$.
Can we treat this system as a linear system of ordinary equations and solve it applying Generalized Picard's theorem (or something similar)? I was thinking about the embedding $$W^{1,2}(0,T;V,H)\subset C([0,T], H).$$ But this means that (only!) if $u\in W^{1,2}(0,T; V, H)$, then there exists a uniquely determined continuous function $u_1\colon [0,T]\to H$ which coincides almost everywhere on $[0,T]$ with the initial function $u$.