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Sturm’s Theorems on Zero Sets in Nonlinear Parabolic Equations

  • Chapter
Sturm-Liouville Theory

Abstract

We present a survey on applications of Sturm’s theorems on zero sets for linear parabolic equations, established in 1836, to various problems including reaction-diffusion theory, curve shortening and mean curvature flows, symplectic geometry, etc. The first Sturm theorem, on nonincrease in time of the number of zeros of solutions to one-dimensional heat equations, is shown to play a crucial part in a variety of existence, uniqueness and asymptotic problems for a wide class of quasilinear and fully nonlinear equations of parabolic type. The survey covers a number of the results obtained in the last twenty-five years and establishes links with earlier ones and those in the ODE area.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Victor A. Galaktionov

  2. Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047, Moscow, Russia

    Victor A. Galaktionov

  3. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Petra J. Harwin

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Editors and Affiliations

  1. Section de Physique, Université de Genève, 24, quai Ernest-Ansermet, 1211, Genève 4, Switzerland

    Werner O. Amrein

  2. Mathematisches Institut, Universität München, Theresienstrasse 39, D-80333, München, Germany

    Andreas M. Hinz

  3. Department of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK

    David P. Pearson

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Galaktionov, V.A., Harwin, P.J. (2005). Sturm’s Theorems on Zero Sets in Nonlinear Parabolic Equations. In: Amrein, W.O., Hinz, A.M., Pearson, D.P. (eds) Sturm-Liouville Theory. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7359-8_8

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