Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations
Abstract
In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space E with type 2, equationeq:SEabSE aligned dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dWH(t), t∈ [0,T], U(0) & = u0. aligned. equation Here (A(t))t∈ [0,T] are unbounded operators with domains (D(A(t)))t∈ [0,T] which may be time dependent. We assume that (A(t))t∈ [0,T] satisfies the conditions of Acquistapace and Terreni. The functions F and B are nonlinear functions defined on certain interpolation spaces and u0∈ E is the initial value. WH is a cylindrical Brownian motion on a separable Hilbert space H. Under Lipschitz and linear growth conditions we show that there exists a unique mild solution of eq:SEab. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk, to obtain space-time regularity results for the solution U of eq:SEab. For Hilbert spaces E we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second order stochastic partial differential equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to several improvements of their result.