Regularity of solutions of abstract linear evolution equations^*

In this paper, we study regularity of solutions to linear evolution equations of the form dX/dt +AX = F(t) in a Banach space H, where A is a sectorial operator in H, and A^−αF(α > 0) belongs to a weighted Hölder continuous function space. Similar results are obtained for linear evolution equations with additive noise of the form dX + AXdt = F(t)dt + G(t)dW(t) in a separable Hilbert space H, where W is a cylindrical Wiener process. Our results are applied to a model arising in neurophysiology, which has been proposed byWalsh [J.B. Walsh, An introduction to stochastic partial differential equations, École d’Été de Probabilités de Saint-Flour, XIV – 1984, Springer, Berlin, 1986, pp. 265–439].