The problem Of blow-up in nonlinear parabolic equations

Abstract

The course aims at presenting an introduction to the subject of singularity formation in nonlinear evolution problems usually known as blowup. In short, we are interested in the situation where, starting from a smooth initial configuration, and after a first period of classical evolution, the solution (or in some cases its derivatives) becomes infinite in finite time due to the cumulative effect of the nonlinearities. We concentrate on problems involving differential equations of parabolic type, or systems of such equations.A first part of the course introduces the subject and discusses the classical questions addressed by the blow-up theory. We propose a list of main questions that extends and hopefully updates on the existing literature. We also introduce extinction problems as a parallel subject.In the main bulk of the paper we describe in some detail the developments in which we have been involved in recent years, like rates of growth and pattern formation before blow-up, the characterization of complete blow-up, the occurrence of instantaneous blow-up (i.e., immediately after the initial moment) and the construction of transient blow-up patterns (peaking solutions), as well as similar questions for extinction.In a final part we have tried to give an idea of interesting lines of current research. The survey concludes with an extensive list of references. Due to the varied and intense activity in the field both aspects are partial, and reflect necessarily the authors' tastes.

Chiral magnetic skyrmions with arbitrary topological charge

“…The pressure from the shell becomes too high and leads to the shrinking [68] of the internal π-skyrmions. This may result in blow-up behaviour of the solutions [69] with an increasing magnetic field. From that one may conclude that in the case of such a critical phenomena, a certain solution of higher energy branch becomes the minimizer for corresponding Q, see inset in Fig.…”

On the Computation of Extinction Time for Some Nonlinear Parabolic Equations

“…Let Ω be a bounded domain in 2 » . Consider the initial value problem ] [ , in 0, q u u u t ∂ = ∆ − Ω × ∞ ∂ (21)…”

“…Since the appearance of the pioneering work of Kalashnikov [1], extinction phenomenon in nonlinear parabolic equations has been studied extensively by many authors [2] [3]. Particular emphasis has been placed on the question as to the existence of extinction time [4]- [7].…”

Global Existence and Blowup for a Reaction-Diffusion System with Nonlocal Boundary Condition

“…with Dirichlet, Neumanns or Robin boundary condition, which can be used to describe heat propagation on the boundary of container (see [2,4,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the literatures cited therein). Specially, when ( , V), ( , V) have the form…”