Classical Solutions To Quasilinear Hyperbolic Singularity Formation

This paper deals systematically with the formation of singularities for quasilinear hyperbolic systems with small and decaying initial data. Under the assumption of non-weak linear degeneracy, we obtain the relations between the life-span of classical solution to quasilinear strictly and non-strictly hyperbolic system and the index α of non-weak linear degeneracy under weaker decaying rates of the initial data. This improves some related results essentially .The main contents are as follows.The third chapter deals with a kind of quasilinear hyperbolic system with characteristic with constant multiplicity. Under the assumption that the index α is equal to zero, we obtain the blow-up result without restriction on characteristics with constant multiplicity for the C~1 solution to the Cauchy problem with intial data possessing smallness and certain decaying properties.The other three chapters consider the index α of non-weak linear degeneracy satisfying that 0 ≤ α < +∞, we obtain respectively that1) the life-span and the formation mechanism of singularities of C~1 solution to quasilinear strictly hyperbolic system with small and weaker decaying initial data;2) under the assumption that the system possesses normalized coordinates and constant characteristics are weakly linear degenerate, the life-span and the formation mechanism of singularities of C~1 solution to quasilinear hyperbolic system with small and weaker decaying initial data;3) under the assumptions that there is no restriction on characteristic with constant multiplicity and the leftmost or rightmost characteristic is simple and not weakly linearly degenerate, the blow-up result of C~1 solution to quasilinear non-strictly hyperbolic system with small and weaker decaying initial data.