Lp Solutions for Stochastic Evolution Equation with Nonlinear Potential

This article considers the stochastic partial differential equation $$\{\begin{array}{l}{u}_t=\frac{1}{2}{u}_{xx}+u^{\gamma }\xi u(0,\mathrm{.})=u_0\end{array}$$ \noindent where $\xi$ is a space / time white noise Gaussian random field, $\gamma >1$ and $u_0$ is a non-negative initial condition independent of $\xi$ satisfying $$u_0\ge 0,\underset{n\to +\mathrm{\infty }}{\lim }\mathbb{E}\left[{({\int }_{{\mathbb{S}}^1}{u}_0(x)\wedge ndx)}^2\right]=\mathbb{E}\left[{({\int }_{{\mathbb{S}}^1}{u}_0(x)dx)}^2\right]<+\mathrm{\infty }\mathrm{.}$$ \noindent The {\em space} variable is $x\in {\mathbb{S}}^1=[0,1]$ 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 unique non-negative solution $u$ such that for all $\alpha \in [0,1)$, $$\mathbb{E}\left[{({\int }_0^{\mathrm{\infty }}{\int }_{{\mathbb{S}}^1}u(t,x{)}^{2\gamma }dxdt)}^{\alpha \mathrm{/}2}\right]\le C(\alpha )<+\mathrm{\infty }\mathrm{.}$$ \noindent where the constant $C(\alpha )$ arises in the Burkholder-Davis-Gundy inequality. The solution is also shown to satisfy $$\mathbb{E}\left[{\int }_0^T{({\int }_{{\mathbb{S}}^1}u(t,x{)}^{p}dx)}^{\alpha \mathrm{/}p}dt\right]<+\mathrm{\infty }\mathrm{\forall }T<+\mathrm{\infty },p<+\mathrm{\infty },\alpha \in (0,\frac{1}{2})\mathrm{.}$$