WIAS Preprint No. 1150, (2006)

# Spectral mapping theorem for linear hyperbolic systems

Authors

• Lichtner, Mark

2010 Mathematics Subject Classification

• 47D03 47D06 34D09 35P20 37L10 37D10

Keywords

• Linear hyperbolic systems, estimates for spectrum and resolvent, spectral mapping theorem, $C_0$ semigroups, exponential dichotomy, invariant manifolds

DOI

10.20347/WIAS.PREPRINT.1150

Abstract

We prove spectral mapping theorem for linear hyperbolic systems of PDEs. The system is of the following form: For $0 < x < l$ and $t > 0$ $$rm(H) quad left beginarrayl displaystyle partial over partial t beginpmatrix u(t,x) v(t,x) endpmatrix + K(x) partial over partial x beginpmatrix u(t,x) v(t,x) endpmatrix + C(x) beginpmatrix u(t,x) v(t,x) endpmatrix = 0, displaystyle d over dt left [ v(t,l) - D u(t,l) right ] = F u(t,cdot) + G v(t,cdot) , displaystyle u(t,0) = E v(t,0), endarray right .$$ where $u(t,x) in C^n_1$, $v(t,x) in C^n_2$, $K(x) = mathrmdiag , left( k_i(x) right )_1 le i le n$ is a diagonal matrix of functions $k_i in C^1left( [0,l], R right)$, $k_i(x) > 0$ for $i = 1, dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, dots, n=n_1+n_2$, and $D$,$E$ are matrices. We show high frequency estimates of spectra and resolvents in terms of reduced (block)diagonal systems. Let $A$ denote the infinitesimal generator for $mathrm(H)$ which generates $C_0$ semigroup $e^At$ on $L^2 times C^n_2$. Our main result is the following spectral mapping theorem $$sigma(e^At) setminus 0 = overlinee^sigma(A)t setminus 0 .$$

Appeared in

• Proc. Amer. Math. Soc., 136 (2008) pp. 2091-2101.