By Shuxing Chen

The booklet offers a finished evaluate at the concept on research of singularities for partial differential equations (PDEs). It features a summarization of the formation, improvement and major effects in this subject. a number of the author's discoveries and unique contributions also are incorporated, equivalent to the propagation of singularities of strategies to nonlinear equations, singularity index and formation of shocks

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In more general case the equation is called fully nonlinear equation. This chapter is first devoted to study the singularity analysis for semilinear equation. 1 World Scientific Book - 9in x 6in singularities Analysis of Singularities for Partial Differential Equations Theorem of propagation of 2s weak singularity In some cases the result established in Chapter 2 on singularity propagation for linear equations is also valid for semilinear equations with some additional restrictions. 1) where f (u) is a C ∞ function of u.

The first step is to decompose the operator P , so that to reduce the problem to the case for an operator of first order, while the second step is to prove the theorem for the reduced operator of first order. August 12, 2010 15:49 World Scientific Book - 9in x 6in singularities Singularity analysis for linear equations 25 Since ∇ξ pm (x, ξ) = 0, we may assume ∂pm /∂ξn = 0. Hence we may assume that pm (x, ξ) takes the form m−1 am−k (x, ξ )ξnk . 27) where λ(x, ξ ) is a homogeneous function of ξ with degree 1, qm−1 (x, ξ) is a homogeneous function of ξ with degree m − 1, and qm−1 (x0 , ξ0 ) = 0.

It means that W F (W ) does not meet any γs , hence W F (U ) also does not meet any γs . 3. 5 contains the requirement of the transversal intersection of bicharacteristics with the boundary. Indeed, γi is the integral curve of the Hamilton vector field of ξn − λi (x, ξ ). Hence ∂ dxn = (ξn − λi (x, ξ )) = 0, dt ∂ξn then the projection of γi on the base space is transversal to the boundary xn = 0. In the above theorem the operator in Eq. 40) is required to have N real eigenvalues. 41) is an initial problem in fact.