Some non-existence results for a class of stochastic partial differential equations

TL;DR

This paper establishes non-existence results for certain stochastic partial differential equations driven by Gaussian noise, focusing on conditions under which global solutions do not exist, thereby extending previous theoretical understanding.

Contribution

It provides new non-existence theorems for a class of SPDEs with stable process generators and Gaussian noise, under specific conditions on the coefficients and initial data.

Findings

Proves non-existence of global solutions under certain conditions.

Extends previous non-existence results to broader classes of SPDEs.

Highlights the impact of the noise and operator properties on solution existence.

Abstract

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \sigma (u_t(x))\dot F(t,x)\quad{t>0}\quad\text{and}\quad x\in R^d. \end{equation*} The operator $\mathcal{L}$ is the generator of a strictly stable process and $\dot F$ is the random forcing term which is assumed to be Gaussian. Under some additional conditions, most notably on $\sigma$ and the initial condition, we show non-existence of global random field solutions. Our results are new and complement earlier works.