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Under the genuinely nonlinear assumptions for one dimensional $n\times n$strictly hyperbolic conservation laws, we investigate the geometric blowup ofsmooth solutions and the development of singularities when the small initialdata fulfill the generic nondegenerate condition. At first, near the uniqueblowup point we give a precise description on the space-time blowup rate of thesmooth solution and meanwhile derive the cusp singularity structure ofcharacteristic envelope. These results are established through extending thesmooth solution of the completely nonlinear blowup system across the blowuptime. Subsequently, by utilizing a new form on the resulting 1-D strictlyhyperbolic system with $(n-1)$ good components and one bad component, togetherwith the choice of an efficient iterative scheme and some involved analyses, aweak entropy shock wave starting from the blowup point is constructed.

Formation and construction of a shock wave for one dimensional $n\times n$ strictly hyperbolic conservation laws with small smooth initial data