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Long-time existence for signed solutions of the heat equation with a noise term

Summary.

Let ? be the circle [0,J] with the ends identified. We prove long-time existence for the following equation.

Here, =(t,x) is 2-parameter white noise, and we assume that u 0(x) is a continuous function on ?. We show that if g(u) grows no faster than C 0(1+|u|)γ for some γ<3/2, C 0>0, then this equation has a unique solution u(t,x) valid for all times t>0.

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Received: 27 November 1996 / In revised form: 28 July 1997

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Mueller, C. Long-time existence for signed solutions of the heat equation with a noise term. Probab Theory Relat Fields 110, 51–68 (1998). https://doi.org/10.1007/s004400050144

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  • AMS Mathematics Subject Classification (1991): Primary 60H15; secondary 35R60