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In this article, we study a class of stochastic partial differentialequations with fractional differential operators subject to sometime-independent multiplicative Gaussian noise. We derive sharp conditions,under which a unique global $L^p(\Omega)$-solution exits for all $p\ge 2$. Inthis case, we derive exact moment asymptotics following the same strategy in arecent work by Balan et al [1]. In the case when there exits only a localsolution, we determine the precise deterministic time, $T_2$, before which aunique $L^2(\Omega)$-solution exits, but after which the series correspondingto the $L^2(\Omega)$ moment of the solution blows up. By properly choosing theparameters, results in this paper interpolate the known results for bothstochastic heat and wave equations.

Interpolating the Stochastic Heat and Wave Equations with Time-independent Noise: Solvability and Exact Asymptotics