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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.

The first exit problem of reaction-diffusion equations for small multiplicative Lévy noise