$L^2$-solutions to stochastic reaction-diffusion equations with superlinear drifts driven by space-time white noise^

Abstract

Consider the following stochastic reaction-diffusion equation with logarithmic superlinear coefficient b, driven by space-time white noise W: $$u_t(t,x)=(1/2)u_{xx}(t,x)+b(u(t,x))+\ddot{I}ƒ(u(t,x))W(dt,dx)$$ for $t>0$ and $x\in [0,1]$, with initial condition $$u(0,x)=u_0(x)$$ for $x\in [0,1]$, where $u_0\in L^2[0,1]$. In this paper, we establish existence and uniqueness of probabilistically strong solutions in $C(R_{+},L^2[0,1])$. Our result resolves a problem from [Ann. Probab. 47 (2019) 519-559] and provides an alternative proof of the non-blowup of $L^2[0,1]$ solutions from the same reference. We use new Gronwall-type inequalities. Due to nonlinearity, we work with first order moments, requiring precise estimates of the stochastic convolution.