  Superconductivity of transition metal dichalcogenide $1T$-TiTe$_2$ under high
pressure was investigated by the first-principles calculations. Our results
show that the superconductivity of $1T$-TiTe$_2$ exhibits very different
behavior under the hydrostatic and uniaxial pressure. The hydrostatic pressure
is harmful to the superconductivity, while the uniaxial pressure is beneficial
to the superconductivity. Superconducting transition temperature $T_C$ at
ambient pressure is 0.73 K, and it reduces monotonously under the hydrostatic
pressure to 0.32 K at 30 GPa. While the $T_C$ increases dramatically under the
uniaxial pressure along $c$ axis. The established $T_C$ of 6.34 K under the
uniaxial pressure of 17 GPa, below which the structural stability maintains, is
above the liquid helium temperature of 4.2 K. The increase of density of states
at Fermi level, the redshift of $F(\omega)$/$\alpha^2F(\omega)$ and the
softening of the acoustic modes with pressure are considered as the main
reasons that lead to the enhanced superconductivity under uniaxial pressure. In
view of the previously predicted topological phase transitions of $1T$-TiTe$_2$
under the uniaxial pressure [Phys. Rev. B 88, 155317 (2013)], we consider
$1T$-TiTe$_2$ as a possible candidate in transition metal chalcogenides for
exploring topological superconductivity.
