On non-existence of global solutions to a class of stochastic heat equations

TL;DR

This paper investigates nonlinear stochastic heat equations driven by space-time white noise, demonstrating conditions under which solutions' second moments blow up in finite time, highlighting the impact of noise on solution behavior.

Contribution

It establishes finite-time blow-up conditions for second moments of solutions to certain nonlinear stochastic heat equations, extending previous deterministic and stochastic analyses.

Findings

Second moments blow up in finite time under specific conditions.

Noise can influence the occurrence of blow-up in solutions.

Results extend understanding of stochastic heat equations with white noise.

Abstract

We consider nonlinear parabolic SPDEs of the form $\partial_t u=-(-\Delta)^{\alpha/2} u + b(u) +\sigma(u)\dot w$, where$\dot w$ denotes space-time white noise. The functions $b$ and $\sigma$ are both locally Lipschitz continuous. Under some suitable conditions on the parameters of these SPDEs, we show that the second moment of their solutions blow up in finite time. This complements recent works of Khoshnevisan and his coauthors; see for instance Foondun and Khoshnevisan (2009), Foondun and Khoshnevisan (to appear, 2012) and Conus, Joseph, and Khoshnevisan, as well as those of Chow (Chow, 2009, and Chow, 2011). Furthermore, upon comparing our stochastic equations with their deterministic counterparts, we find that our results indicates that the presence of noise might affect the occurrence of blow-up.