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

Beom Seok Han, Kyeong Hun Kim

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish
Pages (from-to)9904-9935
Number of pages32
JournalJournal of Differential Equations
Volume269
Issue number11
DOIs
Publication statusPublished - 2020 Nov 15

Keywords

  • Boundary behavior
  • Interior Hölder regularity
  • Nonlinear stochastic partial differential equations
  • Space-time white noise

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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