  We consider the heat equation in the presence of compactly supported magnetic
field in the plane. We show that the magnetic field leads to an improvement of
the decay rate of the heat semigroup by a polynomial factor with power
proportional to the distance of the total magnetic flux to the discrete set of
flux quanta. The proof employs Hardy-type inequalities due to Laptev and Weidl
for the two-dimensional magnetic Schroedinger operator and the method of
self-similar variables and weighted Sobolev spaces for the heat equation. A
careful analysis of the asymptotic behaviour of the heat equation in the
similarity variables shows that the magnetic field asymptotically degenerates
to an Aharonov-Bohm magnetic field with the same total magnetic flux, which
leads asymptotically to the gain on the polynomial decay rate in the original
physical variables. Since no assumptions about the symmetry of the magnetic
field are made in the present work, it confirms that the recent results of
Kovarik about large-time asymptotics of the heat kernel of magnetic
Schroedinger operators with radially symmetric field hold in greater
generality.
