论文标题
用Laplacian的Bernstein功能对SPDE的解决方案的有限时间爆炸
Finite Time Blowup of Solutions to SPDEs with Bernstein Functions of the Laplacian
论文作者
论文摘要
SPDES \ begin {equation*} \ partial_tu_t（x）= - ϕ（-Δ）u_t（x）的有限解决方案的爆炸 +σ（u_t（x））\dotξ（t，x）， \ quad t> 0，x \ in \ mathbb {r}^d，\ end {equation*} {is}进行了研究，其中$ \ dotT配给了$ \ dotT配给白噪声或颜色噪声或$ ϕ：（0，\ infty）\ to（0，\ infty）\ to（0，\ infty）$是伯恩斯坦的功能。讨论了$σ$，$ \ dot耳的足够条件以及暗示全局解决方案不存在的初始值。本文的结果概括了``Foondun，M.，Liu，W。和Nane，E。 J.微分方程，266（5）（2019），2575--2596。''，其中考虑了分数laplacian案例，即$ ϕ（-Δ）=（ - δ）^{α/2} $（$ 1 <α<2 $）。
The blowup in finite time of solutions to SPDEs \begin{equation*} \partial_tu_t(x)=-ϕ(-Δ)u_t(x) +σ(u_t(x))\dotξ(t,x), \quad t>0,x\in\mathbb{R}^d, \end{equation*} { is} investigated, where $\dotξ$ could be either a white noise or a colored noise and $ϕ:(0,\infty)\to (0,\infty)$ is a Bernstein function. The sufficient conditions on $σ$, $\dotξ$ and the initial value that imply the non-existence of the global solution are discussed. The results in this paper generalise those in ``Foondun, M., Liu, W. and Nane, E. Some non-existence results for a class of stochastic partial differential equations. J. Differential Equations, 266 (5) (2019), 2575--2596.'', where the fractional Laplacian case was considered, i.e. $ϕ(-Δ)=(-Δ)^{α/2}$ ($1<α<2$).
