Boundary behavior and interior H\"older 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 $$ u_t=(au_{xx}+bu_x+cu)+ \xi |u|^{1+\lambda} {\dot{B}}, \quad\,\, t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundary data. Here $\lambda\in [0,1/2)$, $\dot{B}$ is a space-time white noise, and the coefficients $a,b,c$ and the function $\xi$ depend on $(\omega,t,x)$ and the initial data $u_0$ depends on $(\omega,x)$. More importantly, we obtain various interior H\"older 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 $$ \rho^{-1/2-\kappa}u \in C^{\frac{1}{4}-\frac{\kappa}{2}-\varepsilon, \frac{1}{2}-\kappa-\varepsilon}_{t,x}([0,T]\times (0,1)), \quad \forall\, \kappa\in (\lambda, 1/2), $$ where $\rho(x)$ is the distance from $x$ to the boundary. Taking $\kappa \downarrow \lambda$, one gets the the maximal H\"older exponents in time and space, which are $1/4-\lambda/2-\varepsilon$ and $1/2-\lambda-\varepsilon $ respectively. Also, letting $\kappa \uparrow 1/2$, one gets better decay or behavior near the boundary.