Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as ∂ t t u − Δ u = f ( x , u ) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same...

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Bibliographic Details
Published inJournal of Differential Equations Vol. 241; no. 1; pp. 184 - 205
Main Authors Karageorgis, Paschalis, Strauss, Walter A.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.10.2007
Subjects
Instability
Nonlinear heat equation
Nonlinear wave equation
Steady states
Instability
Nonlinear wave equation
Nonlinear heat equation
Steady states
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Summary:We consider time-independent solutions of hyperbolic equations such as ∂ t t u − Δ u = f ( x , u ) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as ∂ t u − Δ u = f ( x , u ) . Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2007.06.006