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.
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Just Published
Apr 15, 2024
Matheus C. Bortolan + 2