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

Let II be the circle [0,J] with the ends identified. We prove longtime existence for the following equation. u(t) = u(xx) + g(u)(W) over dot, T>0, x is an element of Pi u(0,x) = u(0)(x) Here, (W) over dot = (W) over dot(t,x) is 2-parameter white noise, and we assume that u(0)(x) is a continuous function on Pi. We show that if g(u) grows no faster than C-0(1+\u\)(gamma) for some gamma < 3/2, C-0 > 0, then this equation has a unique solution u(t,x) valid for all times t > 0.