Preface
Singularity formation in non-linear evolution equations
In honour of the 60th birthday of Howard A. Levine An issue of fundamental importance in the mathematical theory of evolution equations (timedependent partial di erential equation or system of equations) is that of singularity formation. Given an initial state possessing some degree of smoothness, which evolves in time as a solution of the PDE, we wish to know whether this degree of smoothness is preserved, or whether some loss of regularity occurs, such that the solution develops singularities, or possibly ceases to exist altogether. Analysis of this phenomenon may be essential in understanding the physical processes being modelled by the PDE. There are many aspects to this that one might be interested in, such as the proper deΓΏnition of solution, what singularities are physically acceptable, continuation of solutions past singularities, asymptotic behaviour of solutions near singularities, numerical computation of solutions in the presence of singularities, and other more esoteric matters. Perhaps the most famous still unresolved problem of this kind is the Navier-Stokes equations in three-space dimensions. Given an initial state even of class C β it is known that a classical solution exists locally, and a weak solution exists globally, but it is not known whether classical solutions can develop singularities, thus ceasing to exist as classical solutions, in a ΓΏnite time.
One might very roughly divide the subject into three categories: regularity theory (proofs that singularities do not occur), propagation and interaction of singularities results (proofs that singularities persist in some respect) and theory of singularity formation (proofs that singularities must occur, and behaviour of solutions near singular points). It is this last topic which is the main theme of this special issue.
Classical examples include shock solutions of ΓΏrst-order conservation laws, focusing phenomena in eikonal (and more general Hamilton-Jacobi type) equations, and ΓΏnite time blow up in semilinear heat and wave equations. More recent developments along these lines analyse singularity formation in parabolic and hyperbolic systems, wave equations with damping, PDEs on manifolds, and degenerate parabolic equations such as the geometric heat equation arising in the theory of motion by mean curvature. More reΓΏned results about critical exponents and continuation beyond singularities have been obtained.
A conference on recent developments in this area of study was held at Iowa State University in June 2002, dedicated to the 60th birthday of Howard A. Levine. Most of the contributions in this special issue were written by speakers at that conference. β’ In the paper by A. S. Ackleh and K. Deng, results on global existence and ΓΏnite time blow-up are derived for a non-local wave equation. β’ The asymptotic behaviour of global solutions of the porous medium equation is studied in the contribution of V.A. Galaktionov. β’ The rate of blow-up of sign-changing solutions of a subcritical semilinear heat equation is established in the paper by Y. Giga, S. Matsui and S. Sasayama.