### Abstract

We present unique solvability result in weighted Sobolev spaces of the equation u_{t}=(au_{xx}+bu_{x}+cu)+ξ|u|^{1+λ}B˙,t>0,x∈(0,1) given with initial data u(0,⋅)=u_{0} 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 L_{p} 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 language | English |
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Pages (from-to) | 9904-9935 |

Number of pages | 32 |

Journal | Journal of Differential Equations |

Volume | 269 |

Issue number | 11 |

DOIs | |

Publication status | Published - 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