Consider the following Cauchy problem for the first order quasilinear strictly hy- perbolic system ?u ?u + A(u) = 0, ?t ?x t = 0 : u = f(x). We let M = sup |f (x)| < +∞. x∈R The main result of this paper is that under the assumption that the system is weakly linearly degenerated, there exists a positive constant ε independent of M, such that the above Cauchy problem admits a unique global C1 solution u = u(t,x) for all t ∈ R, provided that +∞ |f (x)|dx ≤ ε, ?∞ +∞ ε |f(x)|dx ≤ .
在这份报纸,我们学习多少起始的数据的整齐被需要保证下列 semilinear 波浪方程 utt 的一个本地答案的存在- u = F ( u ,杜), u ( 0 , x )= f (x) H ,( 6 ) tu ( 0 , x )= g (x) Hs-1 在 F 在有 D =的杜是二次的的地方(( 6 ) t ,( 6 ) x1 ,,( 6 ) xn )。我们证明分别地, s 的范围与 1/4 是 s n+1/2 + 如果 n=2,并且 0 如果 n = 3,和 0 如果 n 4。它与 Lindblad 的反例一致[3 ] 为 n = 3,并且主要成分是 Strichartz 估计和这些的精炼的使用。