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1995 On smooth global solutions of a Kirchhoff type equation on unbounded domains
Gustavo Perla Menzala, Jardel Morais Pereira
Differential Integral Equations 8(6): 1571-1583 (1995).

Abstract

We consider the quasilinear hyperbolic equation $$ u_{tt}-M\Bigl( \int_\Omega|\text{ grad }u|^2\,dx\Bigr)\,\Delta u=0, \tag1 $$ where $x\in\Omega=\Bbb R^n$, $t$ denotes time and $M(s)$ is a smooth function satisfying $M(s)>0$ for all $s\ge 0$. We prove that there are no non-trivial ``breathers" for equation (1). Here, a "breather" means a time periodic solution which is "small" as $|x|\to +\infty$. We also present a simpler proof of the so-called Pohozaev's second conservation law for (1) solving the global Cauchy problem for "non-physical" nonlinearity $M$ arising from this conservation law.

Citation

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Gustavo Perla Menzala. Jardel Morais Pereira. "On smooth global solutions of a Kirchhoff type equation on unbounded domains." Differential Integral Equations 8 (6) 1571 - 1583, 1995.

Information

Published: 1995
First available in Project Euclid: 15 May 2013

zbMATH: 0822.35087
MathSciNet: MR1329857

Subjects:
Primary: 35L70

Rights: Copyright © 1995 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.8 • No. 6 • 1995
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