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Description:
We prove principle of linearized stability and smooth center manifold theorem for a general class of semilinear hyperbolic systems $\mathrm{(SH)}$ in one space dimension, which are of the following form: For $0 < x < l$ and $t > 0$ $$ \mathrm{(SH)} \left \{ \begin{array}{l} {\partial \over {\partial t}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + K(x) {\partial \over {\partial x}} \begin{pmatrix} u(t,x) \\ v(t,x) \end{pmatrix} + H(x, u(t,x), v(t,x)) = 0, \\ {d \over {dt}} \left [ v(t,l) - D u(t,l) \right ] = F(u(t,\cdot),v(t,\cdot)), \\ u(t,0) = E \, v(t,0), \\ u(0,x) = u_0(x), \; v(0,x) = v_0(x), \end{array} \right . $$ where $u(t,x) \in \R^{n_1}$, $v(t,x) \in \R^{n_2}$, $K(x) = \mathrm{diag} \, \left( k_i(x) \right )_{1 \le i \le n}$ 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 $i = n_1+1, \dots, n=n_1+n_2$, and $D$,$E$ are matrices. First we prove that weak solutions to $\mathrm{(SH)}$ form a smooth semiflow in a Banach space $X$ of continuous functions under natural conditions on the nonlinearities $H$ and $F$. Then we show a spectral gap mapping theorem for linearizations of $\mathrm{(SH)}$ in the complexification of $X$, which implies that growth and spectral bound coincide. Consequently we obtain principle of linearized stability for $\mathrm{(SH)}$. Moreover, the spectral gap mapping theorem characterizes exponential dichotomy in terms of a spectral gap of the infinitesimal generator for linearized hyperbolic systems. This resolves a key problem in applying invariant manifold theory to prove smooth center manifold theorem for $\mathrm{(SH)}$.