Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics

In this article, we study a class of stochastic partial differential equations with fractional operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which unique global $$L^p(\Omega )$$ -solution exists for all $$p\ge 2$$ . case, exact moment asymptotics following the same strategy as that in recent work by Balan et al. (Inst Henri Poincaré Probab Stat. To appear, 2021). case when there only local solution, determine precise deterministic time, $$T_2$$ , before $$L^2(\Omega exists, but after series corresponding solution blows up. By properly choosing parameters, results paper interpolate known both heat and wave equations.