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

Abstract

We present unique solvability result in weighted Sobolev spaces of the equation u(t) = (au(xx) + bu(x) + cu) + xi vertical bar u vertical bar(1+lambda)<(B)over dot>, t > 0, x is an element of (0, 1) given with initial data u(0, .) = u(0) and zero boundary condition. Here lambda is an element of [0, 1/2), <(B)over dot> is a space-time white noise, and the coefficients a, b, c and xi are random functions depending on (t, x). We also obtain various interior Holder regularities and boundary behaviors of the solution. For instance, if the initial data is in appropriate L-p space, then for any small epsilon > 0 and T < infinity, almost surely where rho(x) is the distance from x to the boundary. Taking kappa down arrow lambda, one gets the maximal Holder exponents in time and space, which are 1/4 - lambda/2 - epsilon and 1/2 - lambda - epsilon respectively. Also, letting kappa up arrow 1/2, one gets better decay or behavior near the boundary. (C) 2020 Elsevier Inc. All rights reserved.