We present unique solvability result in weighted Sobolev spaces of the equationut=(auxx+bux+cu)+ξ|u|1+λB˙,t>0,x∈(0,1) given with initial data u(0,⋅)=u0 and zero boundary condition. Here λ∈[0,1/2), B˙ is a space-time white noise, and the coefficients a,b,c and ξ are random functions depending on (t,x).We also obtain various interior Hölder regularities and boundary behaviors of the solution. For instance, if the initial data is in appropriate Lp space, then for any small ε>0 and T<∞, almost surely(1)ρ−1/2−κu∈Ct,x14−κ2−ε,12−κ−ε([0,T]×(0,1)),∀κ∈(λ,1/2), where ρ(x) is the distance from x to the boundary. Taking κ↓λ, one gets the maximal Hölder exponents in time and space, which are 1/4−λ/2−ε and 1/2−λ−ε respectively. Also, letting κ↑1/2, one gets better decay or behavior near the boundary.