What question did this study set out to answer?

To establish the well-posedness of a space-time fractional stochastic partial differential equation (SPDE) with locally Lipschitz coefficients.

June 13, 2026

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

Key Points

To establish the well-posedness of a space-time fractional stochastic partial differential equation (SPDE) with locally Lipschitz coefficients.
Analyzed the space-time SPDE defined by ∂tβu=−(−Δ)α/2u+It1−β[b(u)+σ(u)W˙].
Proved that the equation has a unique solution under conditions on the initial condition and the growth of coefficients.
Generalized previous findings in the literature to a fractional context.
The SPDE admits a unique solution with finite moments under the prescribed conditions.
Initial condition u0 must be bounded and measurable for well-posedness to hold.
Conditions on the Lipschitz constants of truncated versions of b and σ were identified as crucial for the existence of solutions.

Abstract

In this article, we study the space-time SPDE ∂tβu=−(−Δ)α/2u+It1−βb(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.

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