  We are concerned with a system of equations in $\mathbb{R}^{d}(d\geq2)$
governing the evolution of isothermal, viscous and compressible fluids of
Korteweg type, that can be used as a phase transition model. In the case of
zero sound speed $P'(\rho^{\ast})=0$, it is found that the linearized system
admits the \textit{purely} parabolic structure, which enables us to establish
the global-in-time existence and Gevrey analyticity of strong solutions in
hybrid Besov spaces of $L^p$-type. Precisely, if the full viscosity coefficient
and capillary coefficient satisfy $\bar{\nu}^2\geq4\bar{\kappa}$, then the
acoustic waves are not available in compressible fluids. Consequently, the
prior $L^2$ bounds on the low frequencies of density and velocity could be
improved to the general $L^p$ version with $1\leq p\leq d$ if $d\geq2$. The
proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and
composition of functions.
