  We present a model to show that heat propagation away from a local source
depends strongly on dimensionality, leading to dramatic localization in
low-dimensional systems. An example of such a system is a carbon nanotube
array. We further show that this localization is amplified due to a runaway
mechanism if thermal conductivity declines rapidly with temperature. Extremely
high temperatures of thousands of Kelvins and gradients of hundreds of K/{\mu}m
may thus be obtained in a conductor using a modest local power source such as a
laser pointer. This is of fundamental importance for high-efficiency energy
conversion through thermoelectric and thermionic mechanisms, as well as various
other applications.
