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? 



Checking initial data in parabolic PDE with no control on time derivative
The fundament solution approximates the $\delta$ function as $t\to0^+$. Thus if we represent the solution as an integral, it converges to the initial value as $t\to 0^+$ in $L^2$. This is similar to the situation of approximations to the identity using mollifier.


