Principle of linearized stability and smooth center manifold theorem for semilinear hyperbolic systems
- Lichtner, Mark
2010 Mathematics Subject Classification
- 35L40 35B30 37C05 34D09 37C75 37L05 37L10 47D03 47D06 37D10 35L05 35L40 35L50 35L60
- Semilinear hyperbolic systems, spectral mapping theorem, semigroups, exponential dichotomy, center manifolds, smooth dependence on data, stability
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 $t > 0$ $$ mathrm(SH) left beginarrayl partial over partial t beginpmatrix u(t,x) v(t,x) endpmatrix + K(x) partial over partial x beginpmatrix u(t,x) v(t,x) endpmatrix + H(x, u(t,x), v(t,x)) = 0, d over dt left [ v(t,l) - D u(t,l) right ] = F(u(t,cdot),v(t,cdot)), u(t,0) = E , v(t,0), u(0,x) = u_0(x), ; v(0,x) = v_0(x), endarray right . $$ where $u(t,x) in R^n_1$, $v(t,x) in R^n_2$, $K(x) = mathrmdiag , left( k_i(x) right )_1 le i le n$ 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. 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 $mathrm(SH)$.