TY - JOUR

T1 - Boundary behavior and interior Hölder regularity of the solution to nonlinear stochastic partial differential equation driven by space-time white noise

AU - Han, Beom Seok

AU - Kim, Kyeong Hun

N1 - Funding Information:
The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324).
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/11/15

Y1 - 2020/11/15

N2 - We present unique solvability result in weighted Sobolev spaces of the equation ut=(auxx+bux+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 Lp 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.

AB - We present unique solvability result in weighted Sobolev spaces of the equation ut=(auxx+bux+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 Lp 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.

KW - Boundary behavior

KW - Interior Hölder regularity

KW - Nonlinear stochastic partial differential equations

KW - Space-time white noise

UR - http://www.scopus.com/inward/record.url?scp=85087793183&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2020.07.002

DO - 10.1016/j.jde.2020.07.002

M3 - Article

AN - SCOPUS:85087793183

VL - 269

SP - 9904

EP - 9935

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 11

ER -