It is possible to define a weak solution of a parabolic PDE $$u_t - Au = f$$ $$u(0) = u_0$$ as $u \in L^2(0,T;H^1)$ such that $$-\int_0^T\int_\Omega u(t)\varphi'(t) + \int_0^T\int_\Omega Au(t)\varphi(t) = \int_0^T\int_\Omega f(t)\varphi(t) + \int_{\Omega} u_0\varphi(0)$$ holds for all smooth test functions $\varphi$ with $\varphi(T)=0$.

I guess we should require $u \in C^0([0,T];L^2(\Omega))$ too for the initial condition to make sense. But since we have no control of the time derivative we cannot use the embedding $H^1 \subset C^0$ to make sense of $u$ pointwise, so how does it make sense?