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We consider the Cauchy problem of systems of quasilinear wave equations in2-dimensional space. We assume that the propagation speeds are distinct andthat the nonlinearities contain quadratic and cubic terms of the first andsecond order derivatives of the solution. We know that if the all quadratic andcubic terms of nonlinearities satisfy $Strong$ $Null$-$condition$, then thereexists a global solution for sufficiently small initial data. In this paper, westudy about the lifespan of the smooth solution, when the cubic terms in thequasi-linear nonlinearities do not satisfy the Strong null-condition. In theproof of our claim, we use the $ghost$ $weight$ energy method and the$L^{\infty}$-$L^{\infty}$ estimates of the solution, which is slightlyimproved.

Lower Bound of the Lifespan of the Solution to Systems of Quasi-linear Wave Equations with Multiple Propagation Speeds