The geometric blowup of smooth solutions to 1D quasilinear strictly hyperbolic systems with large variational initial data*

Abstract For the first order 1D n × n quasilinear strictly hyperbolic system ∂ t u + F ( u ) ∂ x u = 0 with u ( x , 0 ) = ε u 0 ( x ) , where ɛ > 0 is small, u 0 ( x ) ≢ 0 and u 0 ( x ) ∈ C 0 2 ( R ) , when at least one eigenvalue of n × n real matrix F(u) is genuinely nonlinear, it is well-known that on the finite blowup time T ɛ , the derivatives ∂ t , x u blow up while the solution u keeps small. For the 1D scalar equation or 2 × 2 strictly hyperbolic system (corresponding to n = 1 , 2 ), if the smooth solution u blows up in finite time, then the blowup mechanism has been well understood (i.e. only the blowup of ∂ t , x u happens). In the present paper, for the 1D n × n ( n ⩾ 3 ) strictly hyperbolic system and large variational initial data u ( x , 0 ) without compact supports, where u ( x , 0 ) is small but its derivatives are large, we investigate the smallness of the solution u itself, the blowup mechanism and the detailed singularity behaviours of ∂ t , x u near the blowup point. Our results are based on the efficient decomposition of u and the involved analysis along the different characteristic directions, the suitable introduction of the modulated coordinates and the global weighted energy estimates together with the characteristics method.