On the Well-Posedness of Stochastic Partial Differential Equations with Locally Lipschitz Coefficients

• Authors: Mohammud Foondun, Davar Khoshnevisan and Eulàlia Nualart
• Journal of Theoretical Probability, Vol. 39, No. 2, 23, June 2026

We consider the stochastic partial differential equation (SPDE) (Formula presented.) where u=u(t,x) is defined for (t,x)∈(0,∞)×R and W˙ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u(0) is bounded and measurable, and b and σ are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.