Global classical solutions for general quasilinear hyperbolic systems with decay initial data. (English) Zbl 0874.35068
The authors consider the initial value problem for general quasilinear hyperbolic systems, where the initial function is a “small” \(C^1\) vector function. The global existence or the blow-up phenomenon of \(C^1\) solutions to the initial value problem is investigated. This kind of problems has been studied by P. D. Lax, F. John, T. P. Lin and L. Hörmander. Recently, by introducing the concept of weak linear degeneracy, T.-T. Li et al. gave a complete result on the global existence and life-span of \(C^1\) solutions in the case of initial data with compact support. The aim of this paper is to simplify the proof and to generalize the result in the aforementioned work to the case that the initial function possesses certain decay properties as \(|x|\to\infty\). Some applications of the result to systems of equations of physical interest are presented.
Reviewer: Wang Jinghua (Beijing)
MSC:
| 35L60 | First-order nonlinear hyperbolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L45 | Initial value problems for first-order hyperbolic systems |
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