On growth and instability for semilinear evolution equations: an abstract approach

TL;DR

This paper introduces an abstract framework for analyzing growth and instability in semilinear evolution equations with compact nonlinearities, showing that such perturbations can be effectively treated as linear for growth analysis.

Contribution

It presents a novel approach to handle nonlinear growth and instability by reducing the problem to linear analysis under certain conditions.

Findings

Exponential lower bounds for solutions with specific initial conditions.

Compact nonlinear perturbations can be analyzed as linear when considering solution growth.

Unbounded resolvent on a vertical line in the right half-plane influences growth behavior.

Abstract

We propose a new approach to the study of (nonlinear) growth and instability for semilinear evolution equations with compact nonlinearities. We show, in particular, that compact nonlinear perturbations of a linear evolution equation can be treated as linear ones as far as the growth of their solutions is concerned. We obtain exponential lower bounds of solutions for initial values from a dense set if, e.g., the resolvent of the generator is unbounded on a vertical line in the right halfplane.