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

Abstract

Let double-struck I sign be the circle [0, J] with the ends identified. We prove long-time existence for the following equation. ut = uxx + g(u)Ẇ , t > 0, x ∈ double-struck I sign u(0, x) = u0(x) Here, Ẇ = Ẇ(t, x) is 2-parameter white noise, and we assume that u0(x) is a continuous function on double-struck I sign. We show that if g(u) grows no faster than C0(1 + |u|)γfor some γ < 3/2, C0 > 0, then this equation has a unique solution u(t, x) valid for all times t > 0.