Fredholm operators, semigroups and the asymptotic and boundary behavior of solutions of PDEs

Every semigroup T on a Banach space X can be used to define elements u X of exponential type relative to T by requiring that u=T(s)v for some s>0 and v X. Let then X and Y be Banach spaces in which the exponential type is characterized by the semigroups T and S, respectively, and let be Fredholm. It is shown that if L satisfies some compatibility conditions with respect to T and S and if f Y has exponential type, then every solution u X of Lu=f has exponential type as well. When L is a differential operator, it is often compatible in this sense (and in suitable spaces) with semigroups that embody an asymptotic or boundary behavior. This yields a way to study such a behavior in solutions of PDEs, which is technically simple, very general and delivers rather sharp results. Furthermore, this approach is easily generalized to the nonlinear setting.