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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 whitenoise. We assume that the underlying L\'evy process has finite exponentialmoments in a neighborhood of the origin and that the initial condition hasexponential decay at infinity. Then we prove that under natural conditions onthe non-linearity: (i) The absolute moments of the solution to our stochasticheat equation grow exponentially with time; and (ii) The distances to theorigin of the farthest high peaks of those moments grow exactly linearly withtime. Very little else seems to be known about the location of the high peaksof the solution to the non-linear stochastic heat equation under the presentsetting. Finally, we show that these results extend to the stochastic waveequation driven by Laplacian.

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations