Space-time fractional SPDEs with locally Lipschitz coefficients: well-posedness

Abstract

In this article, we study the space-time SPDE $${\partial }_t^{\beta }u=-(-\Delta {)}^{\alpha /2}u+I_t^{1-\beta }[b(u)+\sigma (u)\dot{W}],$$ where $u=u(t,x)$ is defined for $(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R},$ $\beta \in (0,1),\alpha \in (0,2)$ and $\dot{W}$ denotes a space-time white noise. It has long been conjectured that this equation has a unique solution with finite moments under the minimal assumptions of locally Lipschitz coefficients $b$ and $\sigma$ with linear growth. We prove that this SPDE is well-posed under the assumptions that the initial condition $u_0$ is bounded and measurable, and the functions $b$ and $\sigma$ are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants on the truncated versions of $b$ and $\sigma$. Our results generalize the work of Foondun et al.(2025) to a space-time fractional setting.