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 We consider the blow-up problem for the generalized Proudman-Johnson equation. It was proved that, for the periodic boundary condition, the solutions exist globally if $-3\leq a\leq 1$. For $a>1$, Hamada showed numerically that a blow-up solution might exist. In this paper, we would like to explore the case of $a<-3$.To this end, we propose a numerical scheme for the computation and use the method given by Cho to determine whether blow-up might occur or not. In addition, we also consider the case of the Grundy-McLaughlin boundary condition. The behavior is different to that of the periodic boundary solution in the case of $a>1$. We check numerically whether this also occur in the case of $a<-3$.Blow-up rate is computed while blow-up occurs. To verify the validity of our results analytical, we analyze by asymptotic expansion to exclude some possibilities.
 1 Introduction2 Cho's Method3 Periodic boundary condition 3.1 Numerical Scheme 3.2 Numerical result4 Grundy-McLaughlin Boundary condition 4.1 Numerical Scheme 4.2 Numerical result5 Analyzed By Asymptotic Expansion6 Conclusion
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