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A regularity theory for stochastic partial differential equations with a super-linear diffusion coefficient and a spatially homogeneous colored noise

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  • Choi, Jae-Hwan
  • Han, Beom-Seok

Abstract

Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise F and its super-linear diffusion coefficient: du=(aijuxixj+biuxi+cu)dt+ξ|u|1+λdF,(t,x)∈(0,∞)×Rd, where λ≥0 and the coefficients depend on (ω,t,x). The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of λ, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of F and the spatial dimension d.

Suggested Citation

  • Choi, Jae-Hwan & Han, Beom-Seok, 2021. "A regularity theory for stochastic partial differential equations with a super-linear diffusion coefficient and a spatially homogeneous colored noise," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 1-30.
    • Handle: RePEc:eee:spapps:v:135:y:2021:i:c:p:1-30
    DOI: 10.1016/j.spa.2021.01.006
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    References listed on IDEAS

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    1. Dawson, Donald A. & Salehi, Habib, 1980. "Spatially homogeneous random evolutions," Journal of Multivariate Analysis, Elsevier, vol. 10(2), pages 141-180, June.
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    Cited by:

    1. Ke, Yue & Zhu, Linhe & Wu, Peng & Shi, Lei, 2022. "Dynamics of a reaction-diffusion rumor propagation model with non-smooth control," Applied Mathematics and Computation, Elsevier, vol. 435(C).

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