Existence and regularity results for semilinear stochastic time-tempered fractional wave equations with multiplicative Gaussian noise and additive fractional Gaussian noise

To model wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and fractional Gaussian noise, because of the potential fluctuations of the external sources. We first give a representation of the mild solution and some stability estimates for the homogeneous problem, and then prove the existence and uniqueness of the mild solution by using fixed point theorem. Finally, the decay and regularity theory of the solution are provided.