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Abstract
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We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density of the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on this, we prove that we cannot get a global classical solution no matter how small the initial data are, as long as the initial mass density is compactly supported. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for 1-D case.
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