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We study the Cauchy problem for multi-dimensional compressible radiationhydrodynamics equations with vacuum. First, we present some sufficientconditions on the blow-up of smooth solutions in multi-dimensional space. Then,we obtain the invariance of the support of density for the smooth solutionswith compactly supported initial mass density by the property of the systemunder the vacuum state. Based on the above-mentioned results, we prove that wecannot get a global classical solution, no matter how small the initial dataare, as long as the initial mass density is of compact support. Finally, wewill see that some of the results that we obtained are still valid for theisentropic flows with degenerate viscosity coefficients as well as 1-D case.

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum