Nonlinear second order equations with applications to partial differential equations

Abstract

where A and B are (usually unbounded) linear operators in a Banach space. These problems arise often in the study of partial differential equations. As is usual we control a nonlinear perturbation (possibly involving spatial derivatives) by the linear terms, which contain higher order spatial derivatives. But contrary to the usual method of reducing the problem to a first order system in some “energy” norm space (as in [9, 5]), we use the factoring method of [lo]. This method allows the equation to be written as an integral equation containing a double integral involving the nonlinearity, reflecting the fact that the equation is second order. There are several advantages to this approach which will be illustrated fully in the examples. First, we show that the equation is locally well posed if (loosely) the nonlinearity satisfies the local Lipschitz condition