  In this paper, we consider the compressible fluid model of Korteweg type in a
critical case where the derivative of pressure equals to $0$ at the given
constant state. It is shown that the system admits a unique, global strong
solution for small initial data in the maximal $L_p$-$L_q$ regularity class. As
a result, we also prove the decay estimates of the solutions to the nonliner
problem. In order to obtain the global well-posedness for the critical case, we
show $L_p$-$L_q$ decay properties of solutions to the linearized equations
under an additional assumption for a low frequencies.
