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 ^{\beta }_t+\dfrac {\nu }{2}\left (-\Delta \right )^{\alpha / 2}\right ) u(t, x) = \: I_{t}^{\gamma }\left [\lambda u(t, x) \dot {W}(t, x)\right ] \quad t>0,\: x\in \mathbb {R}^d, \end{equation*} with constants $\lambda \ne 0$ and $\nu >0$, where $\partial ^{\beta }_t$ is the Caputo fractional derivative of order $\beta \in (0,2]$, $I_{t}^{\gamma }$ refers to the Riemann-Liouville integral of order $\gamma \ge 0$, and $\left (-\Delta \right )^{\alpha /2}$ is the standard fractional/power of Laplacian with $\alpha >0$. We concentrate on the scenario where the noise $\dot {W}$ is the space-time white noise. 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*} \frac {1}{t}\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 of Hu and Wang, where the authors focus on the space-time colored Gaussian noise case.