Paper detail

Finite Time Blowup of Solutions to SPDEs with Bernstein Functions of the Laplacian

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. $ϕ(-Δ)=(-Δ)^{α/2}$ ($1<α<2$).