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Some properties of a class of stochastic heat and wave equations with multiplicative Gaussian noises

We study stochastic partial differential equations of the form Lu = ∆u+σ(u)N˙ with particular initial conditions. Here L denotes a first or second order partial differential operator, N˙ a Gaussian noise, ∆ the Laplacian operator, and σ : R → R a Lipschitz continuous function. In the first case we choose Lu := ∂tu, we study the stochastic fractional heat equation of the form ∂tu(t, x) = ∆ α/2u(t, x) +λ σ(u(t, x))N˙ for t > 0, x ∈ B(0, R) ⊂ R d . Here ∆ α/2 is the infinitesimal generator of a symmetric α-stable process killed at the boundary of the ball B(0, R), λ is a positive parameter called the level noise and N˙ denotes space-time white noise when d = 1 or white-colored noise in the case of Riesz Kernel of order β when d ≥ 1. We start to show the existence and uniqueness of solutions, the main task is to study how the second moment of the solution u(t, x) and excitation index of the solution grows as λ tends to infinity for a fixed t > 0. This study was initiated by [KK13] and [KK15]. Our results are significant extensions of those results and that of [FJ14]. In the second case we choose Lu := ∂ttu+2η∂tu, and study the stochastic damped wave equation of the form ∂ttu(t, x)+2η∂tu(t, x) = ∆u(t, x)+σ(u(t, x))W˙ for t > 0, x ∈ R, where η represents the positive damping parameter and W˙ space-time white noise. We study second moment and p-th moment of solution to show the intermittency properties, and existence and uniqueness of solutions will be proved. The study of the intermittency properties in stochastic partial differential equations was initiated by [FK09], our results are significant extension of results in [CJKS13]. In the end, we also show the result of excitation index for the stochastic damped wave equation.