Dichotomous solutions for semilinear ill-posed equations with sectorially dichotomous operator

Abstract

In this paper, we study a class of semilinear ill-posed equations with sectorially dichotomous operator S on Banach space Z. Firstly we give a direct sum decomposition of Z, Z+⊕Z−=Z, corresponding to spectrum of S such that hyperbolic bisectorial operator S can be split into two sectorial operators S|Z+ and −S|Z− on Z+ and Z−, respectively. Then we construct the intermediate spaces between whole space Z and domain D(S) of sectorially dichotomous operator S. Following ElBialy's works, we propose the dichotomous initial condition for this semilinear ill-posed equation, and obtain the existence, uniqueness, continuous dependence on the dichotomous initial value, regularity and Zα-estimate of dichotomous solutions. As applications of the results, we give the existence and uniqueness of local solutions for an elliptic PDE in infinite cylindrical domain and an abstract semilinear ill-posed equation with non-dense domain.