On the well-posedness of SPDEs with locally Lipschitz coefficients

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Foondun, Mohammud and Khoshnevisan, Davar and Nualart, Eulalia (2024) On the well-posedness of SPDEs with locally Lipschitz coefficients. Other. arXiv, Ithaca, NY. (https://doi.org/10.48550/arXiv.2411.09381)

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Abstract

We consider the stochastic partial differential equation, ∂tu = ½∂2xu+b(u)+σ(u)W˙, 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 and have at most linear growth. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The results naturally generalize to the case where b and σ are time dependent with uniform-in-time growth and oscillation properties. Additionally, our method can be extended to the stochastic wave equation.

ORCID iDs

Foondun, Mohammud ORCID logoORCID: https://orcid.org/0000-0001-7634-916X, Khoshnevisan, Davar and Nualart, Eulalia;
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