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

Abstract

We present uniqueness and existence in weighted Sobolev spaces of the equation $$ u_t=(au_{xx}+bu_x+cu)+ ξ|u|^{1+λ} {\dot{B}}, \quad\,\, t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundary data. Here $λ\in [0,1/2)$, $\dot{B}$ is a space-time white noise, and the coefficients $a,b,c$ and the function $ξ$ depend on $(ω,t,x)$ and the initial data $u_0$ depends on $(ω,x)$. More importantly, we obtain various interior Hölder regularities and boundary behaviors of the solution. For instance, if the initial data is in appropriate $L_p$ spaces, then for any small $\varepsilon>0$ and $T<\infty$, almost surely $$ ρ^{-1/2-κ}u \in C^{\frac{1}{4}-\fracκ{2}-\varepsilon, \frac{1}{2}-κ-\varepsilon}_{t,x}([0,T]\times (0,1)), \quad \forall\, κ\in (λ, 1/2), $$ where $ρ(x)$ is the distance from $x$ to the boundary. Taking $κ\downarrow λ$, one gets the the maximal Hölder exponents in time and space, which are $1/4-λ/2-\varepsilon$ and $1/2-λ-\varepsilon $ respectively. Also, letting $κ\uparrow 1/2$, one gets better decay or behavior near the boundary.