  In the Einestein-dilaton theory with a Liouville potential parameterized by
$\eta$, we find a Schwarzschild-type black hole solution. This black hole
solution, whose asymptotic geometry is described by the warped metric, is
thermodynamically stable only for $0 \le \eta < 2$. Applying the gauge/gravity
duality, we find that the dual gauge theory represents a non-conformal thermal
system with the equation of state depending on $\eta$. After turning on the
bulk vector fluctuations with and without a dilaton coupling, we calculate the
charge diffusion constant, which indicates that the life time of the quasi
normal mode decreases with $\eta$. Interestingly, the vector fluctuation with
the dilaton coupling shows that the DC conductivity increases with temperature,
a feature commonly found in electrolytes.
