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We consider nonlinear parabolic SPDEs of the form $\partial_t u=\Delta u +\lambda \sigma(u)\dot w$ on the interval $(0, L)$, where $\dot w$ denotesspace-time white noise, $\sigma$ is Lipschitz continuous. Under Dirichletboundary conditions and a linear growth condition on $\sigma$, we show that theexpected $L^2$-energy is of order $\exp[\text{const}\times\lambda^4]$ as$\lambda\rightarrow \infty$. This significantly improves a recent result ofKhoshnevisan and Kim. Our method is very different from theirs and it allows usto arrive at the same conclusion for the same equation but with Neumannboundary condition. This improves over another result of Khoshnevisan and Kim.

Remarks on non-linear noise excitability of some stochastic heat equations