Asymptotic stability for equilibria of nonlinear semiflows with applications to rotating viscoelastic rods

This work establishes results on global existence and stability of solutions of quasilinear equations near an equlibrium point whose spectrum lies in the strict left half plane, which extend those of Sobolevksii and Potier-Ferry. These results are then applied to the dynamical stability of moving viscoelastic rods.; The following result, which has the form of a linearization principle for fixed points of semiflows in a Banach space, is proved under some modest continuity conditions.; If the linearized system at a fixed point has eigenvalues all with negative real parts, then this fixed point is locally asymptotically stable and there is global existence of solutions with initial conditions in a neighborhood of the fixed point.; The result is applied to the fixed points of evolution equations in a Banach space. Making use of Sobolevskii's results on Cauchy problems for equations of parabolic type in a Banach space, we find conditions on evolution equations which guarantee the asymptotic stability of their fixed points.; Simo, Posbergh and Marsden analysed the stability of relative equlibria of dissipationless geometriacally exact rods. In their work only sufficient conditions for formal stability of relative equilibria were given due to the difficulty arising from the quasilinear nature of the equations and associated shock formation. In this work we show that the presence of dissipation along with formal stability ensures the asymptotic stability for a relative equilibria and the global existence of smooth solutions in a neighborhood.; Using the Cosserat two-director rod model, the equations of motion for geometrically exact hyperelastic rods moving under viscoelastic damping is written in evolution form du/dt = G(u). Applying our result on fixed points of evolution equations in a Banach space to viscoelastic geometrically exact rods, we prove the following result.; In the presence of viscoelastic dissipation, the relative equilibria of hyperelastic rods are asymptotically stable if they are formally stable. In addition, the solutions to the equations of motion in the neighborhood of such a relative equilibrium exist for all positive time, and decay exponentially to the relative equilibrium.