Abstract : Blow-up phenomena appear in various science fields. They are described by partial differential equations. It is difficult to construct a general theory for the blow-up phenomena of partial differential equations. Therefore, we need to approach them from both mathematical and numerical aspects. In this mini-symposium, we discuss blow-up time and blow-up profile from the perspectives of mathematical and numerical analysis. The purpose of this minisymposium is to bring together researchers from both mathematical and numerical analysis to discuss recent advances on blow-up phenomena.
[01625] On the convergence order of the numerical blow-up time
Author(s) :
Chien-Hong Cho (National Sun Yat-sen University)
Abstract : It is quite often that the solutions of initial-value problems become unbounded in a finite time. Such a phenomenon is called blow-up and the finite time is called the blow-up time. In this talk, we put our attention on the computation of an approximate blow-up time and its convergence order.
[01957] The blow-up curve for systems of semilinear wave equations
Author(s) :
Takiko Sasaki (Department of Mathematical Engineering, Faculty of Engineering, Musashino University and Mathematical Institute, Tohoku University)
Abstract : In this talk, we consider a blow-up curve for systems of semilinear wave equations with different propagation speeds in one space dimension. The blow-up curve has been studied from the view point of its differentiability and singularity. We show that the blow-up curve has a singular point under suitable initial conditions. We also show some numerical examples of blow-up curves.
[01774] Lifespan estimates of semilinear wave equations of derivative type with characteristic weights in one space dimension
Author(s) :
Shunsuke Kitamura (Tohoku University, Graduate School of Science)
Abstract : In this talk, I discuss the initial value problems for semilinear wave equations of derivative type with characteristic weights in one space dimension. Such equations provide basic principles on extending the general theory for nonlinear wave equations to the non-autonomous case. The results are quite different from the results of a series of joint work with Takamura, Wakasa and Morisawa about the nonlinear terms of unknown function itself.
[01740] On degenerate blow-up profiles for the semilinear heat equation
Author(s) :
Hatem Zaag (CNRS and Université Sorbonne Paris Nord)
Abstract : We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.
[03241] Collapse Versus Blowup and Global Existence in Generalized Constantin–Lax–Majda Equation with dissipation
Format : Online Talk on Zoom
Author(s) :
Pavel M Lushnikov (University of New Mexico)
David M Ambrose (Drexel University)
Michael Siegel (Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology,)
Denis A Silantyev (Department of Mathematics, University of Colorado, Colorado Springs,)
Abstract : We analyze dynamics of singularities and finite time blowup of
generalized Constantin-Lax-Majda equation for
non-potential effective motion of fluid with competing convection,
vorticity stretching and dissipation. Multiple exact solutions are found with blowups. Global existence of solutions is proven for small data in periodic case. The analytical solutions on real line allow finite-time singularity formation for
arbitrarily small data illustrating critical difference between real line and periodic
cases. Analysis is complemented by accurate numerical
simulations.
