= Spectral construction of a parabolic solution
{title2=$u(t)=\sum_m e^{-t\lambda_m}(\psi,w_m)w_m$}
For a time-independent strictly positive <Dirichlet realization of an elliptic operator> and $\psi\in L^2(U)$, the <eigenfunction expansion>
$$
u(t)=\sum_m e^{-t\lambda_m}(\psi,w_m)w_m
$$
solves $u_t+Lu=0$ with homogeneous <Dirichlet boundary conditions>. For $t\geq\varepsilon>0$, every power $\lambda_m^N$ times the exponential is bounded, so the series and all its time derivatives converge in all the <Sobolev domains of powers of an elliptic Dirichlet operator>. <Elliptic regularity> and the <Sobolev embedding theorem> give smoothness up to the spatial boundary for positive time. <Parseval identity> and the <dominated convergence theorem> give $u(t)\to\psi$ in $L^2$ as $t\downarrow0$.
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