Instability of steady states for nonlinear wave and heat equations

2006/11/18 by Paschalis Karageorgis, Walter A. Strauss, Karageorgis, Paschalis +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.math/0611559

arxiv created 2006/11/18 · openalex publication_date 2006/11/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider time-independent solutions of hyperbolic equations such as \d_ttu -Δu= 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 \d_t u -Δu= f(x,u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.

Citations

• Existence of solitary waves in higher dimensions
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
• On the focusing critical semi-linear wave equation

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