A Spectral Gap Mapping Theorem and Smooth Invariant Center Manifolds for Semilinear Hyperbolic Systems

Mark Lichtner

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 158

MSC 2000

35L50 Boundary value problems for hyperbolic systems of first-order PDE

47D03 Groups and semigroups of linear operators

35L60 Nonlinear first-order PDE of hyperbolic type

47D06 One-parameter semigroups and linear evolution equations

37L05 General theory, nonlinear semigroups, evolution equations

35B30 Dependence of solutions of PDE on initial and boundary data, parameters

35Q60 Equations of electromagnetic theory and optics

37D10 Invariant manifold theory

Abstract
Although the spectral mapping property in general fails it is shown that a ``spectral gap mapping theorem'', which characterizes exponential dichotomy, holds for a general class of semilinear hyperbolic systems of PDEs in a Banach space $X$ of continuous functions. This resolves a key problem on existence and smoothness of invariant manifolds for semilinear hyperbolic systems. The system is of the following form: For $0 < x < l$ and $t > 0$ $$ \mathrm{(SH)} \left \{ \begin{array}{l} {\partial \over {\partial t}} \begin{pmatrix} u(t,x) \\ v(t,x) \\ w(t,x) \end{pmatrix} + K(x) {\partial \over {\partial x}} \begin{pmatrix} u(t,x) \\ v(t,x) \\ w(t,x) \end{pmatrix} + H(x, u(t,x), v(t,x), w(t,x)) = 0, \\ {d \over {dt}} \left [ v(t,l) - D u(t,l) \right ] = F(u(t,\cdot),v(t,\cdot)), \\ u(t,0) = E \, v(t,0), \\ u(0,x) = u_0(x), \; v(0,x) = v_0(x), \; w(0,x) = w_0(x), \end{array} \right . $$ where $u(t,x) \in \R^{n_1}$, $v(t,x) \in \R^{n_2}$ and $w(t,x) \in \R^{n_3}$, $K(x) = \mathrm{diag} \, \left( k_i(x) \right)_{i = 1, \dots, n}$ is a diagonal matrix of functions $k_i \in C^1\left( [0,l], \R \right)$, $k_i(x) > 0$ for $i = 1, \dots, n_1$ and $k_i(x) < 0$ for $i = n_1+1, \dots n_1+n_2$, $k_i \equiv 0$ for $i = n_1+n_2+1, \dots, n_1+n_2+n_3 = n$, and $D$ and $E$ are matrices. It is shown that weak solutions to $\mathrm{(SH)}$ form a smooth semiflow in $X$ under natural conditions on $H$ and $F$. For linearizations of $\mathrm{(SH)}$ estimates of spectra and resolvents in terms of reduced diagonal and blockdiagonal systems are given. Using these estimates and theory of Kaashoek, Lunel and Latushkin a spectral gap mapping theorem for linearizations of $\mathrm{(SH)}$ in the ``small'' Banach space $X$ is proven: An open spectral gap of the generator is mapped exponentially to an open spectral gap of the semigroup and vice versa. Hence, a phenomenon like in Renardys counterexample cannot appear for linearizations of $\mathrm{(SH)}$. Existence of smooth center manifolds for $\mathrm{(SH)}$ is shown by applying the above results and general theory on persistence and smoothness of invariant manifolds, obtained by Bates, Lu and Zeng, in the Banach space $X$. The results are applied to traveling wave models of semiconductor laser dynamics. For such models mode approximations (ODE systems which approximately describe the dynamics on center manifolds) are derived and justified. Global existence and smooth dependence of nonautonomous traveling wave models with more general solutions, which possess jumps, are considered, and mode approximations are derived for such nonautonomous models. In particular the theory applies to stability and bifurcation analysis for Turing models with correlated random walk. Moreover, the class $\mathrm{(SH)}$ includes neutral and retarded functional differential equations.