$L^p$ solutions for stochastic evolution equation with nonlinear potential

This article considers the stochastic partial differential equation \[ \left\{ \begin{array}{l} u_t = \frac{1}{2} u_{xx} + u^\gamma \xi u(0,.) = u_0 \end{array}\right. \] \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 \[ u_0 \geq 0, \qquad \lim_{n \rightarrow +\infty} \mathbb{E} \left [ \left (\int_{\mathbb{S}^1} u_0 (x)\wedge n dx \right)^2 \right ] = \mathbb{E} \left [ \left (\int_{\mathbb{S}^1} u_0 (x) dx \right)^2 \right ]< +\infty.\] \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 $\alpha \in [0,1)$, \[\mathbb{E} \left [ \left( \int_0^\infty \int_{\mathbb{S}^1} u(t,x)^{2\gamma} dx dt \right)^{\alpha / 2 } \right ] \leq C( \alpha) < + \infty. \] \noindent where the constant $C(\alpha)$ arises in the Burkholder-Davis-Gundy inequality. The solution is also shown to satisfy \[ \mathbb{E} \left [ \int_0^T \left(\int_{\mathbb{S}^1} u (t,x)^p dx \right)^{\alpha / p} dt \right ] < +\infty \qquad \forall T < +\infty, \qquad p < +\infty, \qquad \alpha \in \left (0, \frac{1}{2} \right ). \]