Boundary behavior and interior Hölder regularity of the solution to nonlinear stochastic partial differential equation driven by space-time white noise

We present unique solvability result in weighted Sobolev spaces of the equation u_t=(au_xx+bu_x+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 L_p space, then for any small ε>0 and T<∞, almost surely [Formula presented] 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.