Weak Nonmild Solutions to Some SPDEs

Daniel Conus and Davar Khoshnevisan

Abstract. We study the nonlinear stochastic heat equation driven by space-time white noise in the case that the initial datum u0 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 definition of a stochastic convolution via Young-type inequalities.

Keywords. Stochastic PDEs, weak solutions, measure-valued initial conditions.

AMS Classification (2000) Primary: 60H15; Secondary: 35R60.

Support. Research supported by grants from the Swiss National Science Foundation Fellowship PBELP2-122879 (D.C.) and the US National Science Foundation grant DMS-0706728 (D.K.)

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Daniel Conus & Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A. &

Last Update: January 27, 2010
© 2010 - Daniel Conus and Davar Khoshnevisan