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Instantaneous everywhere-blowup of parabolic SPDEs

Davar Khoshnevisan, University of Utah
Monday, October 16, 2023 - 2:30pm to 3:20pm
SMI 305
We consider the following stochastic heat equation 
 \$\partial_t u(t\,,x) = \frac12\partial^2_x u(t\,,x) +  b(u(t\,,x)) + \sigma(u(t\,,x))\dot{W}(t\,,x)\$
defined for \$(t\,,x)\in(0\,,\infty)\times\mathbb{R}\$, where \$\dot{W}\$ denotes space-time white noise. The function \$\sigma\$ 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 \$\int_1^\infty {\rm d}y/b(y)<\infty\$  implies that the solution almost surely blows up everywhere and instantaneously. In other words, the Osgood condition ensures that \$P\{u(t\,,x)=\infty\text{ for all } t>0 \text{ and  } x\in\mathbb{R}\}=1\$.  The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities, and the study of the spatial growth of stochastic convolutions  using techniques from the Malliavin calculus and the Poincare inequalities that were developed by Le Chen, D. Khoshnevisan, David Nualart, and Fei Pu (2021, 2022).
This is based on joint work with Mohammud Foondun and Eulalia Nualart.
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