Moments and asymptotics for a class of SPDEs with space-time white noise

Abstract

In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: \begin{equation*} \left(\partial^β_t+\dfracν{2}\left(-Δ\right)^{α/ 2}\right) u(t, x)= ~ I_{t}^γ\left[λu(t, x) \dot{W}(t, x)\right] \quad t>0,~ x\in\mathbb R^d, \end{equation*} where $\dot{W}$ is space-time white noise, $\alpha >0$, $\beta \in (0,2]$, $\gamma \ge 0$, $\lambda \ne 0$ and $\nu >0$. The existence and uniqueness of solution in the Itô-Skorohod sense is obtained under Dalang's condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the $p$-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the $p$-th moment Lyapunov exponents. In particular, by letting $\beta =2$, $\alpha =2$, $\gamma =0$, and $d=1$, we confirm the following standing conjecture for the stochastic wave equation: \begin{align*} t^{-1}\log\mathbb E[u(t,x)^p] \asymp p^{3/2}, \quad \text{for $p\ge 2$ as $t\to \infty$.} \end{align*} The method for the lower bounds is inspired by a recent work by Hu and Wang [HW21], where the authors focus on the space-time colored Gaussian noise.

37 pages

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