Finite-time blow-up of a non-local stochastic parabolic problem

TL;DR

This paper investigates the conditions leading to finite-time blow-up in a non-local stochastic parabolic PDE, establishing existence, regularity, and probabilistic bounds for blow-up phenomena.

Contribution

It introduces new criteria for noise-induced blow-up, proves spatial regularity and a Hopf's lemma for solutions, and provides probabilistic bounds on blow-up events.

Findings

Noise can induce finite-time blow-up under certain conditions.

Solutions exhibit $C^{1}$ spatial regularity.

Probabilistic bounds on blow-up likelihood are derived.

Abstract

The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.