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

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  • Kavallaris, Nikos I.
  • Yan, Yubin

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 C1-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.

Suggested Citation

  • Kavallaris, Nikos I. & Yan, Yubin, 2020. "Finite-time blow-up of a non-local stochastic parabolic problem," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5605-5635.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:9:p:5605-5635
    DOI: 10.1016/j.spa.2020.04.002
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    References listed on IDEAS

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    1. Dozzi, Marco & López-Mimbela, José Alfredo, 2010. "Finite-time blowup and existence of global positive solutions of a semi-linear SPDE," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 767-776, June.
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    Cited by:

    1. Liang, Fei & Zhang, Yidi & Zhao, Shuangshuang, 2024. "Blow-up solution to an abstract non-local stochastic heat equation with Lévy noise," Statistics & Probability Letters, Elsevier, vol. 206(C).

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