arXiv2021
In this paper, we study a nonlinear one spatial dimensional stochastic heat equations driven by Gaussian noise: $\frac{\partial u }{\partial t}=\frac{\partial^2 u }{\partial x^2}+σ(u )\dot{W} $, where $\dot{W} $ is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac 14,\frac 12)$. We remove a critical and unnatural condition $σ(0)=0$ previously imposed in a recent paper by Hu, Huang, Lê, Nualart and Tindel. The idea is to work on a weighted space $\mathcal{Z}_{λ,T}^p$ for some power decay weight $λ(x)=c_H(1+|x|^2)^{H-1}$. We obtain the weak existence of solution. With additional decay conditions on $σ$ we obtain the existence of strong solution and the pathwise uniqueness of the strong solution. The reason to introduce the weight function is that the solution $u(t,x)$ may explode as $|x|\rightarrow \infty$ when the "diffusion coefficient" $σ(u)$ does not satisfy $σ(0)=0$ regardless of the initial condition. This motivates us to study the exact asympotics of the solution $u_{\rm add}(t,x)$ as $t$ and $x$ go to infinity when $σ(u)=1$ and when the initial condition $u_0(x)\equiv 0$. In particular, we find the exact growth of $\sup_{|x|\leq L}{|u_{\rm add}(t,x)|}$. Furthermore, we find the sharp growth rate for the Hölder coefficients, namely, $\sup_{|x|\leq L} \frac{| u_{\rm add}(t,x+h)-u_{\rm add}(t,x)|}{|h|^β}$ and $\sup_{|x|\leq L} \frac{| u_{\rm add}(t+τ,x)-u_{\rm add}(t,x)|}{τ^α}$. These results are interesting and fundamental themselves.