On the spectrum of a mixed-type operator with applications to rotating wave solutions

DOI10.1007/S00526-022-02351-ZarXiv2204.05824MaRDI QIDQ6396318

Publication date: 12 April 2022

Abstract: We study rotating wave solutions of the nonlinear wave equation

left{ �egin{aligned} partial_{t}^2 v - Delta v + m v & = |v|^{p-2} v quad && ext{in

m a t h b b R i m e s m a t h b f B

} \ v & = 0 && ext{on

m a t h b b R i m e s p a r t i a l m a t h b f B

} end{aligned} ight.

where

2 < p < i n f t y

,

m i n m a t h b b R

and

m a t h b f B s u b s e t m a t h b b R^{2}

denotes the unit disk. If the angular velocity

a l p h a

of the rotation is larger than

1

, this leads to a semilinear boundary value problem on

m a t h b f B

involving a mixed-type operator, whose spectrum is related to the zeros of Bessel functions and could generally be badly behaved. Based on new estimates for these zeros, we find values of

a l p h a

such that the spectrum only consists of eigenvalues with finite multiplicity and has no accumulation point. Combined with suitable spectral estimates, this allows us to formulate an appropriate indefinite variational setting and find ground state solutions of the reduced equation for

p i n (2, 4)

. Using a minimax characterization of the ground state energy, we ultimately show that these ground states are nonradial and thus yield nontrivial rotating waves, provided

m

is sufficiently large.

Mathematics Subject Classification ID

Variational methods involving nonlinear operators (47J30) Initial-boundary value problems for second-order hyperbolic equations (35L20) Periodic solutions to PDEs (35B10) Estimates of eigenvalues in context of PDEs (35P15) Variational methods applied to PDEs (35A15) Asymptotic distributions of eigenvalues in context of PDEs (35P20) Bessel and Airy functions, cylinder functions, ({}_0F_1) (33C10) Symmetries, invariants, etc. in context of PDEs (35B06) Second-order semilinear hyperbolic equations (35L71)

This page was built for publication: On the spectrum of a mixed-type operator with applications to rotating wave solutions

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6396318)