$L^p$ solutions for a stochastic evolution equation with nonlinear potential
Volume 264 / 2022
Abstract
This article deals with the stochastic partial differential equation \[ \begin {cases} u_t = \frac {1}{2} u_{xx} + u^\gamma \xi ,\\ u(0,\cdot ) = u_0, \end {cases} \] where $\xi $ is a space/time white noise Gaussian random field, $\gamma \in (1,\infty )$ and the initial condition $u_0$ is a non-negative measurable mapping, independent of $\xi $ satisfying $u_0 \geq 0$ and additional conditions given in the article. The 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 non-negative solution $u$ such that for all $\alpha \in [0,1)$, \[ \mathbb {E} \Bigl [ \Bigl ( \int _0^\infty \int _{\mathbb {S}^1} u(t,x)^{2\gamma } \,dx\,dt \Bigr )^{\alpha / 2 } \Bigr ] \leq K( \alpha ) \mathbb {E} \Bigl [ \Bigl (\int _{\mathbb {S}^1} u_0(x)\,dx \Bigr )^\alpha \Bigr ] \lt \infty . \] where the finite constant $K(\alpha )$ is derived from the Burkholder–Davis–Gundy inequality constants. The solution is unique among solutions which satisfy this. Solutions are also shown to satisfy $$\mathbb {E} \Bigl [ \int _0^T \Bigl (\int _{\mathbb {S}^1} u (t,x)^p \,dx \Bigr )^{\alpha / p}\,dt \Bigr ] \lt \infty \quad \ \forall T \lt \infty ,\, 0 \lt p \lt \infty , \, \alpha \in (0, 1/2).$$
