  We show that the polynomial decay rate of the heat semigroup of the Dirichlet
Laplacian in curved planar wedges equals the sum of the usual dimensional decay
rate and a multiple of the reciprocal value of the opening angle. To prove the
result, we develop the method of self-similar variables for the associated heat
equation and study the asymptotic behaviour of the transformed non-autonomous
parabolic problem for large times. We also establish an improved Hardy
inequality for the Dirichlet Laplacian in non-trivially curved wedges and state
a conjecture about an improved decay rate in this case.
