  This article studies the dynamics of a nonlinear dissipative
reaction-diffusion equation with well-separated stable states which is
perturbed by infinite-dimensional multiplicative L\'evy noise with a regularly
varying component at intensity $\epsilon>0$. The main results establish the
precise asymptotics of the first exit times and locus of the solution
$X^\epsilon$ from the domain of attraction of a deterministic stable state, in
the limit as $\epsilon\rightarrow 0$. In contrast to the exponential growth for
respective Gaussian perturbations the exit times grow essentially as a power
function of the noise intensity as $\epsilon \rightarrow 0$ with the exponent
given as the tail index $-\alpha$, $\alpha>0,$ of the L\'evy measure,
analogously to the case of additive noise in Debussche et al (2013). In this
article we substantially improve their quadratic estimate of the small jump
dynamics and derive a new exponential estimate of the stochastic convolution
for stochastic L\'evy integrals with bounded jumps based on the recent pathwise
Burkholder-Davis-Gundy inequality by Siorpaes (2018). This allows to cover
perturbations with general tail index $\alpha>0$, multiplicative noise and
perturbations of the linear heat equation. In addition, our convergence results
are probabilistically strongest possible. Finally, we infer the metastable
convergence of the process on the common time scale $t/\epsilon^\alpha$ to a
Markov chain switching between the stable states of the deterministic dynamical
system.
