| Institution: | 1.CNRS, Institut Fourier,Université Grenoble Alpes,Grenoble,France;2.CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche,Université Paris-Diderot,Paris,France;3.Indiana University,Bloomington,USA |
| Abstract: | We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate. |
