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L2-Solutions to stochastic reaction-diffusion equations with superlinear drifts driven by space-time white noise

Author

Listed:
  • Shang, Shijie
  • Wang, Pengyu
  • Zhang, Tusheng

Abstract

Consider the following stochastic reaction-diffusion with logarithmic superlinear coefficient b driven by space-time white noise W,{∂tu(t,x)=12∂xxu(t,x)+b(u(t,x))+σ(u(t,x))W(dt,dx),t>0,x∈[0,1].u(t,0)=u(t,1)=0.u(0,x)=u0(x),x∈[0,1],where the initial value u0 ∈ L2([0, 1]). In this paper, we establish the existence and uniqueness of probabilistically strong solutions in C(R+,L2([0,1])) to this equation. Our result not only resolves a recent problem posed in [Ann. Probab. 47 (2019), no. 1, 519–559], but also provides an alternative proof of the non-blowup of L2([0, 1]) solutions obtained in [Ann. Probab. 47 (2019), no. 1, 519-559]. Our approach crucially exploits some new Gronwall-type inequalities that we derive. Moreover, because of the nature of the nonlinearity, we are forced to work with the first order moment of the solutions, which requires precise first order moment estimates of the stochastic convolution.

Suggested Citation

  • Shang, Shijie & Wang, Pengyu & Zhang, Tusheng, 2026. "L2-Solutions to stochastic reaction-diffusion equations with superlinear drifts driven by space-time white noise," Stochastic Processes and their Applications, Elsevier, vol. 199(C).
    • Handle: RePEc:eee:spapps:v:199:y:2026:i:c:s0304414926001237
    DOI: 10.1016/j.spa.2026.104991
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    References listed on IDEAS

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    1. Yue Li & Shijie Shang & Jianliang Zhai, 2024. "Large Deviation Principle for Stochastic Reaction–Diffusion Equations with Superlinear Drift on $$\mathbb {R}$$ R Driven by Space–Time White Noise," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3496-3539, November.
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