Life Spans For Weakly Decaying Classical Solutions Of First-order Quasilinear Hyperbolic Systems

In this paper,we consider the life-span of classical solution to the Cauchy prob-lem for general first order quasilinear strictly hyperbolic systems with totally weak linearly degenerate characteristic fields as follows:where u=(u1,u2…,un)T is an unknown vector-valued function of(t,X),A(u)=(aij(u))is an n×n matrix,and uij(u)∈C2(Rn)(ij=1,…n).If the system has at least one character which is genuinely nonlinear,according to[6][14],for any v>0 and even for φ(x)which has compact support,the life-span is ε-1.If the system totally weak linearly degenerate,it will admit global and almost global solution for v>1 and v=1,respectively,see[19]and[9].In the present paper,we study the lifespan estimation of the classical solution of the Cauchy problem with weakly linearly degenerate first-order quasilinear hyperbolic systems with weakly decaying initial data.With the help of the John’s formula on the decomposition of waves and normal coordinates,and using the characteristic method,by establishing the uniform a priori estimate on the Cl norm of the solution on its existence domain,we prove that when 0<v<1 the lifespan of the solution is(ε-1)1/1-v.