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

Abstract

The blowup in finite time of solutions to SPDEs \begin{equation*} \partial_tu_t(x)=-\phi(-\Delta)u_t(x) +\sigma(u_t(x))\dot{\xi}(t,x), \quad t>0,x\in\mathbb{R}^d, \end{equation*} { is} investigated, where $\dot{\xi}$ could be either a white noise or a colored noise and $\phi:(0,\infty)\to (0,\infty)$ is a Bernstein function. The sufficient conditions on $\sigma$, $\dot{\xi}$ 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. $\phi(-\Delta)=(-\Delta)^{\alpha/2}$ ($1<\alpha<2$).