  We study a family of non-linear stochastic heat equations in (1+1)
dimensions, driven by the generator of a L\'evy process and space-time white
noise. We assume that the underlying L\'evy process has finite exponential
moments in a neighborhood of the origin and that the initial condition has
exponential decay at infinity. Then we prove that under natural conditions on
the non-linearity: (i) The absolute moments of the solution to our stochastic
heat equation grow exponentially with time; and (ii) The distances to the
origin of the farthest high peaks of those moments grow exactly linearly with
time. Very little else seems to be known about the location of the high peaks
of the solution to the non-linear stochastic heat equation under the present
setting. Finally, we show that these results extend to the stochastic wave
equation driven by Laplacian.
