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We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in  R    N  . For time  t  ≥ 0, we can define a functional  H ( t ) associated with the solution of the equations and some testing function  f . When the pressure function  P  of the governing equations is of the form  P  =  Kρ    γ  , where  ρ  is the density function,  K  is a constant, and  γ  > 1, we can show that the nontrivial  C   1  solutions with nonslip boundary condition will blow up in finite time if  H (0) satisfies some initial functional conditions defined by the integrals of  f . Examples of the testing functions include  r    N −1 ln( r  + 1),  r    N −1   e    r  ,  r    N −1 ( r   3  − 3 r   2  + 3 r  +  ε ),  r    N −1 sin(( π /2)( r / R )), and  r    N −1 sinh  r . The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.

Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions