Existence and Stability for Semilinear Differential Equations

Abstract

In the previous chapter we treated linear systems. In this chapter we study the semilinear differential equation $\dot{z}(t) = A z(t) + f(z(t))$, with A the infinitesimal generator of strongly continuous semigroup. We do this for two cases, the first one in which we assume that f is Lipschitz continuous on the state space, and the second one, where we assume that A generates a holomorphic semigroup and f is Lipschitz continuous on the domain of a fractional power of A, i.e., $D(A^{\alpha })$, $\alpha \in (0,1)$. Since for non-linear systems stability is studied via Lyapunov function, this is treated in detail. Among others, an infinite-dimensional version of LaSalle’s invariance theorem is proved. The chapter ends with a set of 25 exercises and a notes and references section.