  We used one-dimensional coupled sine-Gordon equations combined with heat
diffusion equations to numerically investigate the thermal and electromagnetic
properties of a $300\,\mu\mathrm{m}$ long intrinsic Josephson junction stack
consisting of $N = 700$ junctions. The junctions in the stack are combined to
$M$ segments where we assume that inside a segment all junctions behave
identically. Most simulations are for $M = 20$. For not too high bath
temperatures there is the appearence of a hot spot at high bias currents. In
terms of electromagnetic properties, robust standing wave patterns appear in
the current density and electric field distributions. These patterns come
together with vortex/antivortex lines across the stack that correspond to $\pi$
kink states, discussed before in the literature for a homogeneous temperature
distribution in the stack. We also discuss scaling of the thermal and
electromagnetic properties with $M$, on the basis of simulations with $M$
between 10 and 350.
