Bob Palais

Bio: Bob Palais is an academic researcher from University of California, Berkeley. The author has contributed to research in topics: Nonlinear system & Partial differential equation. The author has an hindex of 1, co-authored 1 publications receiving 29 citations.

Blowup for nonlinear equations using a comparison principle in fourier space

Abstract: On decrit une nouvelle methode pour demontrer l'eclatement en temps fini pour une classe d'equations paraboliques semi-lineaires

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.

A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems

Abstract: We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems. It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports. It tries to give a comprehensive account of the western existing mathematicalphysical-engineering literature on this research field. RESUMEN Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calor-difusión, en particular sobre el problema de Stefan y problemas relacionados. Contiene 5869 títulos distribuidos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 informes técnicos o prepublicaciones. Se da un informe amplio de la bibliografía matemática, física y de las ingenierías existente en occidente sobre este tema de investigación. Primary Mathematics Subject Classification Number (*): 35R35, 80A22 Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20. (*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik. Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phase-change problem, Solidification, Stefan problem. Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions. Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan. Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles. El manuscrito fue recibido y aceptado en octubre de 1999. D.A. Tarzia, A bibliography on FBP. The Stefan problem, MAT Serie A, # 2 (2000). 3

Symmetric singularity formation in lubrication-type equations for interface motion

Abstract: Fourth-order degenerate diffusion equations arise in a “lubrication approximation” of a thin film or neck driven by surface tension. Numerical studies of the lubrication equation (LE) $h_t + ( h^n h_{xxx} )_x = 0$ with various boundary conditions indicate that singularity formation in which $h( x( t ),t ) \to 0$ occurs for small enough n with “anomalous” or “second type” scaling inconsistent with usual dimensional analysis.This paper considers locally symmetric or even singularities in the (LE) and in the modified lubrication equation (MLE) $h_t + h^n h_{xxxx} = 0$. Both equations have the property that entropy bounds forbid finite-time singularities when n is sufficiently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with $n < 1$ and (MLE) for all $n \in \mathbb{R}$. In the latter case, special boundary conditions that force singularity formation are required to produce singularities when n is large. Matching conditions at higher-order terms in the...