Global existence of classical solutions to the hyperbolic geometry flow with time-dependent dissipation

In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation

$\frac{{?}^2{g}_{ij}}{?t^2}+\frac{\mu }{{\left(1+t\right)}^{\lambda }}\frac{?g_{ij}}{?t}=-2R_{ij},$

on Riemann surface. On the basis of the energy method, for 0 < λ ≤ 1, μ > λ + 1, we show that there exists a global solution g_ij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t,x) of the solution metric g_ij remains uniformly bounded.