This paper studies the asymptotic behavior of the diffractive interaction of pulses with two linear phases of length ε ≤ 1 in 2 × 2 semilinear strictly hyperbolic system with constant coefficiients. By a formal analysis, it derives problems of profiles in the expansion of the pulse like solution with respect to the length s, and obtains that the leading profile satisfies a nonlinear Schrodinger type system. The problems of profiles are solved, and the formal expansion is justified. It is observed that there is interaction between two phases starting at the third order profiles.
By the subsuper solutions method, the explosive supersolutions and explosive subsol utions are obtained and the exsistence of explosive solutions is proved on a bounded domain for a class of nonlinear elliptic problems.Then, the exsitence of an entire large solution is proved by the perturbed method.
The uniqueness and existence of BV-solutions for Cauchy problem of the form are proved.
ZHAO Junning & ZHAN Huashui Department of Mathematics, Xiamen University, Xiamen 361005, China School of Science, Jimei University, Xiamen 361000, China
We focus on the blow-up phenomena of Cauchy problem for the Camassa-Holm equation. Blow-up can occur only in the form of wave-breaking, i.e. the solution is bounded but its slope becomes unbounded in finite time. We proved that there is such a point that its slope becomes infinite exactly at breaking time. We also gave the precise blow-up rate and the blow-up set.
