## Abstract

We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: du=(a¯ij(ω,t)uxixj+f)dt+gkdwtk,t∈(0,T);u(0,·)=0,where T∈ (0 , ∞) , w^{k}(k= 1 , 2 , …) are independent Wiener processes, (a¯ ^{ij}(ω, t)) is a (predictable) nonnegative symmetric matrix valued stochastic process such that κ|ξ|2≤a¯ij(ω,t)ξiξj≤K|ξ|2∀(ω,t,ξ)∈Ω×(0,T)×Rdfor some κ, K∈ (0 , ∞) , f∈Lp((0,T)×Rd,dt×dx;Lr(Ω,F,dP)),and g,gx∈Lp((0,T)×Rd,dt×dx;Lr(Ω,F,dP;l2))with 2 ≤ r≤ p< ∞ and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate ∫0T∫Rd(E[|u(t,x)|r])p/rdxdt+∫0T∫Rd(E[|uxx(t,x)|r])p/rdxdt≤N(∫0T∫Rd(E[|f(t,x)|r])p/rdxdt+∫0T∫Rd(E[|g(t,x)|l2r])p/rdxdt+∫0T∫Rd(E[|gx(t,x)|l2r])p/rdxdt),where N is a positive constant depending only on d, p, r, κ, K, and T. As an application, for the solution u to (1), the rth moment m^{r}(t, x) : = E| u(t, x) | ^{r} is in the parabolic Sobolev space Wp/r1,2((0,T)×Rd).

Original language | English |
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Pages (from-to) | 278-316 |

Number of pages | 39 |

Journal | Stochastics and Partial Differential Equations: Analysis and Computations |

Volume | 10 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2022 Mar |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Maximal regularity moment estimate
- Stochastic partial differential equations
- Zero initial evolution equation

## ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics

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