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In this paper, we are concerned with the convergence rates of the globalstrong solution to constant equilibrium state for the compressible viscoelasticfluids in the whole space. We combine both analysis about Green's matrix methodand energy estimate method to get optimal time decay rate in critical Besovspace framework. Our result imply the optimal $L^{2}$-time decay rate and onlyneed the initial data to be small in critical Besov space which have very lowregularity compared with traditional Sobolev space.

Optimal Time Decay Rate for the Compressible Viscoelastic Equations in Critical Spaces