In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals $0$ at a given constant state. We show that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. Consequently, we also prove the decay estimates of the solutions to the nonlinear problem. 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 low frequencies.
"The global well-posedness of the compressible fluid model of Korteweg type for the critical case." Differential Integral Equations 34 (5/6) 245 - 264, May/June 2021.