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