Toward well-posedness of space-time fractional SPDEs with locally Lipschitz coefficients

In this article, we study the space-time SPDE ∂tβu=−(−Δ)α/2u+It1−β[b(u)+σ(u)W˙], where u=u(t,x) is defined for (t,x)∈R+×R,β∈(0,1),α∈(0,2) and 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 σ with linear growth. We prove that this SPDE is well-posed under the assumptions that the initial condition u0 is bounded and measurable, and the functions b and σ are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants of the truncated versions of b and σ. Our results generalize the work of Foondun et al.to a space-time fractional setting.