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We investigate the moment asymptotics of the solution to the stochastic heatequation driven by a $(d+1)$-dimensional L\'evy space--time white noise. Unlikethe case of Gaussian noise, the solution typically has no finite moments oforder $1+2/d$ or higher. Intermittency of order $p$, that is, the exponentialgrowth of the $p$th moment as time tends to infinity, is established indimension $d=1$ for all values $p\in(1,3)$, and in higher dimensions for some$p\in(1,1+2/d)$. The proof relies on a new moment lower bound for stochasticintegrals against compensated Poisson measures. The behavior of theintermittency exponents when $p\to 1+2/d$ further indicates that intermittencyin the presence of jumps is much stronger than in equations with Gaussiannoise. The effect of other parameters like the diffusion constant or the noiseintensity on intermittency will also be analyzed in detail.

Intermittency for the stochastic heat equation with L\'evy noise