  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}+\sigma(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 $\sigma(0)=0$ previously imposed in a recent paper by Hu, Huang,
L\^{e}, Nualart and Tindel. The idea is to work on a weighted space
$\mathcal{Z}_{\lambda,T}^p$ for some power decay weight
$\lambda(x)=c_H(1+|x|^2)^{H-1}$. We obtain the weak existence of solution. With
additional decay conditions on $\sigma$ 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" $\sigma(u)$ does not
satisfy $\sigma(0)=0$ 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 $\sigma(u)=1$ 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 H\"older
coefficients, namely, $\sup_{|x|\leq L} \frac{| u_{{\rm add}}(t,x+h)-u_{{\rm
add}}(t,x)|}{|h|^\beta}$ and $\sup_{|x|\leq L} \frac{| u_{{\rm
add}}(t+\tau,x)-u_{{\rm add}}(t,x)|}{\tau^\alpha}$. These results are
interesting and fundamental themselves.
