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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?

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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.

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