Lp solutions for a stochastic evolution equation with nonlinear potential

This article deals with the stochastic partial differential equation ⎧⎩⎨ut=12uxx+uγξ,u(0,⋅)=u0, where ξ is a space/time white noise Gaussian random field, γ∈(1,∞) and the initial condition u0 is a non-negative measurable mapping, independent of ξ satisfying u0≥0 and additional conditions given in the article. The space variable is x∈S1=[0,1] with the identification 0=1. The definition of the stochastic term, taken in the sense of Walsh, will be made clear in the article. The result is that there exists a non-negative solution u such that for all α∈[0,1), E[(∫∞0∫S1u(t,x)2γdxdt)α/2]≤K(α)E[(∫S1u0(x)dx)α]<∞. where the finite constant K(α) is derived from the Burkholder–Davis–Gundy inequality constants. The solution is unique among solutions which satisfy this. Solutions are also shown to satisfy E[∫T0(∫S1u(t,x)pdx)α/pdt]<∞ ∀T<∞,0<p<∞,α∈(0,1/2).