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Conus, Daniel; Khoshnevisan, Davar

Weak nonmild solutions to some SPDEs. (English) Zbl 1259.60067

Ill. J. Math. 54, No. 4, 1329-1341 (2010).
Summary: We study the nonlinear stochastic heat equation driven by space-time white noise in the case where the initial datum \(u_{0}\) is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instead that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a construction of a generalized stochastic convolution via Young-type inequalities.

Cited in 7 Documents

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations

Keywords:

weak solution; nonlinear stochastic heat equation; space-time white noise; generalized stochastic convolution
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\textit{D. Conus} and \textit{D. Khoshnevisan}, Ill. J. Math. 54, No. 4, 1329--1341 (2010; Zbl 1259.60067)
Full Text: arXiv Euclid
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