  We consider the heat equation for monolayer two-dimensional materials in the
presence of heat flow into a substrate and Joule heating due to electrical
current. We compare devices including a nanowire of constant width and a bow
tie (or wedge) constriction of varying width, and we derive approximate
one-dimensional heat equations for them; a bow tie constriction is described by
the modified Bessel equation of zero order. We compare steady state analytic
solutions of the approximate equations with numerical results obtained by a
finite element method solution of the two-dimensional equation. Using these
solutions, we describe the role of thermal conductivity, thermal boundary
resistance with the substrate and device geometry. The temperature in a device
at fixed potential difference will remain finite as the width shrinks, but will
diverge for fixed current, logarithmically with width for the bow tie as
compared to an inverse square dependence in a nanowire.
