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

Shijie Shang, Pengyu Wang and Tusheng Zhang

Stochastic Processes and their Applications, 2026, vol. 199, issue C

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.

Keywords: Stochastic reaction-diffusion equations; Superlinear drift; Logarithmic nonlinearity; Space-time white noise; Stochastic convolution (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2026.104991

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