Instability of steady states for nonlinear wave and heat equations |
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Authors: | Paschalis Karageorgis Walter A. Strauss |
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Affiliation: | a School of Mathematics, Trinity College, Dublin 2, Ireland b Department of Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, RI 02912, USA |
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Abstract: | We consider time-independent solutions of hyperbolic equations such as ∂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 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. |
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Keywords: | Nonlinear heat equation Nonlinear wave equation Instability Steady states |
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