An L_p -maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

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).