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: ( ∂ t β + ν 2 ( − Δ ) α / 2 ) u ( t , x ) = I t γ [ λ u ( t , x ) W ˙ ( t , x ) ] t > 0 , x ∈ R d , \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 λ ≠ 0 \lambda \ne 0 and ν > 0 \nu >0 , where ∂ t β \partial ^{\beta }_t is the Caputo fractional derivative of order β ∈ ( 0 , 2 ] \beta \in (0,2] , I t γ I_{t}^{\gamma } refers to the Riemann-Liouville integral of order γ ≥ 0 \gamma \ge 0 , and ( − Δ ) α / 2 \left (-\Delta \right )^{\alpha /2} is the standard fractional/power of Laplacian with α > 0 \alpha >0 . We concentrate on the scenario where the noise W ˙ \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 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 p -th moment Lyapunov exponents. In particular, by letting β = 2 \beta = 2 , α = 2 \alpha = 2 , γ = 0 \gamma = 0 , and d = 1 d = 1 , we confirm the following standing conjecture for the stochastic wave equation: 1 t log ⁡ E [ | u ( t , x ) | p ] ≍ p 3 / 2 , for p ≥ 2 as t → ∞ . \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.