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On the finite space blow up of the solutions of the Swift–Hohenberg equation

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Abstract

The aim of this paper is to study the finite space blow up of the solutions for a class of fourth order differential equations. Our results answer a conjecture by Gazzola and Pavani (Arch Ration Mech Anal 207(2):717–752, 2013) and they have implications on the nonexistence of beam oscillation given by traveling wave profile at low speed propagation.

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Notes

  1. In the case of \(k\le 0\) it is easy to verify that \(\Phi \) is a convex function.

  2. In the case of \(k\le 0\), the inequality \(\Phi (m_j) < \Phi (z_j)\) is always satisfied. As a consequence \(N_1 = \emptyset \).

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Authors and Affiliations

  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, C.P. 668, São Carlos, SP, CEP 13560-970, Brazil

    Vanderley Ferreira Jr. & Ederson Moreira dos Santos

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  1. Vanderley Ferreira Jr.
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  2. Ederson Moreira dos Santos
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Correspondence to Vanderley Ferreira Jr..

Additional information

Communicated by P. Rabinowitz.

V. Ferreira Jr. is supported by FAPESP #2012/23741-3 grant. E. Moreira dos Santos is partially supported by CNPq #309291/2012-7 grant and FAPESP #2014/03805-2 grant.

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Ferreira, V., Moreira dos Santos, E. On the finite space blow up of the solutions of the Swift–Hohenberg equation. Calc. Var. 54, 1161–1182 (2015). https://doi.org/10.1007/s00526-015-0821-6

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