Special Session 146: Nonlinear differential equations: control, delay, and boundary value problems

Standing waves for the wave equation with hyperbolic boundary conditions
Enzo Vitillaro
Dipartimento di Matematica e Informatica Universit\`a degli Studi di Perugia
Italy
Co-Author(s):    
Abstract:
We deal with standing waves for the wave equation with hyperbolic boundary conditions, posed in a bounded domain with regular boundary. These problems possess a wide literature, including the papers on Arch. Rat. Mech. Anal. (2017), J.D.E. (2018) and DCDS-S (2021) by the author. Standing waves solutions of this evolution problem, in the linear setting, turn out to be eigenfunctions for a doubly elliptic problem, which involves the Laplace operator inside the domain and the Laplace--Beltrami one at the boundary. In the talk we show how these eigenfunctions constitute a Hilbert basis of a suitable space, and we study some of their properties.