In this talk, we consider the following one dimensional (in space variable) nonlinear stochastic heat equation driven by the Gaussian noise which is white in time and fractional in space:
$$ \frac{\partial u(t,x)}{\partial t}=\frac{\partial^2 u(t,x)}{\partial x^2}+\sigma(u(t,x))\dot{W}(t,x), $$
where \$W(t,x) \$ is a centered Gaussian process with covariance given by
$$ {\bf E} [ W(t, x)W(s, y)] =\frac 12 \left( |x|^{2H}+|y|^{2H}-|x-y|^{2H} \right) (s\wedge t).$$
Here the Hurst parameter \$H\$ is between 1/4 and 1/2 and \$\dot{W}(t,x)=\frac{\partial^2 W}{\partial t\partial x}\$. We remove the technical condition \$\sigma(0)=0\$ previously assumed. The idea is to introduce a weight for the solution.
When $\sigma(t,u)$ is a constant the solution is a Gaussian random field and we obtain the bound of the solution \$\sup_{0\le t\le T, |x|\le L} |u(t,x)|\$ when \$T\$ and \$L\$ goes to infinity. This is a joint work with Xiong Wang.