In this talk, we shall concern with the Cauchy problem of the viscous compressible fluids of Korteweg type in zero sound speed case. It is found that the linearized system admits the purely parabolic structure where acoustic waves are not available in our models, which enables us to establish the global-in-time existence and Gevrey analyticity of strong solutions in hybrid Besov spaces of $L^p$-type for $d\geq3$.  We also present a new derivation for the optimal decay of arbitrary higher order derivatives for $L^p$ solutions. This approach, based on Gevrey analyticity we established, reduces the decay problem to establishing uniform bounds on the growth of the radius of analyticity for the solution under a negative regularity. Compared to the classical time-weight energy method, this approach concerns only a regularity problem allowing us to get asymptotic behaviors without asking smallness condition for the initial assumption and is applicable to a wide range of dissipative systems.  This talk is based on joint works with professor Jiang Xu at Nanjing University of Aeronautics and Astronautics.

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