{c}

Take $\psi\in L^2(U)$, the natural initial-data assumption needed for the requested $L^2$ convergence but not explicitly stated in this subpart. Use the genuine Dirichlet eigensystem from part (b)(iii) and put $\psi_m=(\psi,w_m)$. The <spectral construction of a parabolic solution> is
$$
\boxed{u(t,x)=\sum_{m=1}^\infty e^{-\lambda_mt}\psi_mw_m(x).}
$$
Each finite sum has zero <Dirichlet boundary condition> and satisfies $u_t+Lu=0$.

To justify all derivatives, fix $\varepsilon>0$. For any integers $k,j\geq0$, the squared $X_k$ norm of a tail of the $j$th time derivative is
$$
\sum_{m>N}\lambda_m^{k+2j}e^{-2t\lambda_m}|\psi_m|^2
\leq\sup_{\lambda\geq\lambda_1}\bigl(\lambda^{k+2j}e^{-2\varepsilon\lambda}\bigr)
\sum_{m>N}|\psi_m|^2\longrightarrow0
$$
uniformly for $t\geq\varepsilon$. Use the corrected <Sobolev domains of powers of an elliptic Dirichlet operator>, not the inaccurate unqualified assertion in the PDF. Their norm equivalence gives convergence in every ordinary $H^k$. The <Sobolev embedding theorem> then gives convergence of every desired spatial derivative by choosing $k$ large enough. Thus termwise differentiation is justified, $u\in C^\infty(U_T)$, and $u(t,\cdot)\in C^\infty(\overline U)$ for every $t>0$. The trace remains zero, and $u_t+Lu=0$ holds pointwise.

Finally <Parseval identity> gives
$$
\|u(t)-\psi\|_2^2=\sum_m(1-e^{-t\lambda_m})^2|\psi_m|^2\longrightarrow0
$$
by the <dominated convergence theorem>, since each factor tends to zero and is bounded by one. No boundary compatibility of the initial $L^2$ data is required for positive-time smoothness.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 105 / 3 / c / 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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 Incoming links: Spectral construction of a parabolic solution