Sturm’s Theorems on Zero Sets in Nonlinear Parabolic Equations

Victor A. Galaktionov () and Petra J. Harwin ()
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Victor A. Galaktionov: University of Bath, Department of Mathematical Sciences
Petra J. Harwin: University of Bath, Department of Mathematical Sciences

A chapter in Sturm-Liouville Theory, 2005, pp 173-199 from Springer

Abstract: 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.

Keywords: Sturm theorems; multiple zeros; nonlinear parabolic equations; intersection comparison; blow-up; asymptotic behavior (search for similar items in EconPapers)
Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-7643-7359-7_8

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DOI: 10.1007/3-7643-7359-8_8

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