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
SN - 0022-0396
VL - 269
SP - 9904
EP - 9935
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 11
ER -