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 $$u_t=(a{u}_{xx}+b{u}_x+cu)+\xi |u{|}^{1+\lambda }\dot{B},t>0,x\in (0,1)$$ 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 Hölder regularities and boundary behaviors of the solution. For instance, if the initial data is in appropriate $L_p$ spaces, then for any small $\epsilon >0$ and $T<\infty$, almost surely $${\rho }^{-1/2-\kappa }u\in C_{t,x}^{\frac{1}{4}-\frac{\kappa }{2}-\epsilon ,\frac{1}{2}-\kappa -\epsilon }([0,T]\times (0,1)),\forall \kappa \in (\lambda ,1/2),$$ where $\rho (x)$ is the distance from $x$ to the boundary. Taking $\kappa \downarrow \lambda$, one gets the the maximal Hölder exponents in time and space, which are $1/4-\lambda /2-\epsilon$ and $1/2-\lambda -\epsilon$ respectively. Also, letting $\kappa \uparrow 1/2$, one gets better decay or behavior near the boundary.

29 pages

References in corpus (1)

• Super-Brownian motion as the unique strong solution to an SPDE