Lp Solutions for Stochastic Evolution Equation with Nonlinear Potential

This article considers the stochastic partial differential equation $\left\{ \begin{array}{l} u_t = \frac{1}{2} u_{xx} + u^γξu(0,.) = u_0 \end{array}\right.$ \noindent where $ξ$ is a space / time white noise Gaussian random field, $γ> 1$ and $u_0$ is a non-negative initial condition independent of $ξ$ 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 $α\in [0,1)$, $\mathbb{E} \left [ \left( \int_0^\infty \int_{\mathbb{S}^1} u(t,x)^{2γ} dx dt \right)^{α/ 2 } \right ] \leq C( α) < + \infty.$ \noindent where the constant $C(α)$ arises in the Burkholder-Davis-Gundy inequality. The solution is also shown to satisfy $\mathbb{E} \left [ \int_0^T \left(\int_{\mathbb{S}^1} u (t,x)^p dx \right)^{α/ p} dt \right ] < +\infty \qquad \forall T < +\infty, \qquad p < +\infty, \qquad α\in \left (0, \frac{1}{2} \right ).$

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