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 Banach space E: $$({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], U(0) = u_0.\end{array}\right.$$ Here, \({(A(t))_{t\in [0,T]}}\) are unbounded operators with domains \({(D(A(t)))_{t\in [0,T]}}\) which may be time dependent. We assume that \({(A(t))_{t\in [0,T]}}\) satisfies the conditions of Acquistapace and Terreni. The functions F and B are nonlinear functions defined on certain interpolation spaces and \({u_0\in E}\) is the initial value. W H is a cylindrical Brownian motion on a separable Hilbert space H. We assume that the Banach space E is a UMD space with type 2. Under locally Lipschitz conditions we show that there exists a unique local mild solution of (SE). If the coefficients also satisfy a linear growth condition, then it is shown that the solution exists globally. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapien, and Zabczyk, to obtain space-time regularity results for the solution U of (SE). 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-Sole and Vuillermot. This leads to several improvements of their result.