Instantaneous everywhere-blowup of parabolic SPDEs
Foondun, Mohammud and Khoshnevisan, Davar and Nualart, Eulalia (2023) Instantaneous everywhere-blowup of parabolic SPDEs. Other. arXiv, Ithaca, New York. (https://doi.org/10.48550/arXiv.2305.08458)
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Abstract
We consider the following stochastic heat equation ∂_u u(t,x)=1/2 ∂_x^2 u(t,x)+b(u(t,x))+σ(u(t,x))W ̇(t,x) defined for (t , x) ∈ (0 ,∞) × R, where ˙W denotes space-time white noise. The function σ is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function b is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition ∫_1^∞▒ dy/(b(y)) implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that P{u(t , x) = ∞ for all t > 0 and x ∈ R} = 1. The main ingredientsof the proof involve a hitting-time bound for a class of differential inequalities (Remark 4.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al [3, 4].
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Item type: Monograph(Other) ID code: 85551 Dates: DateEvent15 May 2023PublishedSubjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 18 May 2023 09:49 Last modified: 11 Nov 2024 16:07 URI: https://strathprints.strath.ac.uk/id/eprint/85551