Kirchhoff systems with nonlinear source and boundary damping terms

Giuseppina Autuori; Patrizia Pucci

doi:10.3934/cpaa.2010.9.1161

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2010, Volume 9, Issue 5: 1161-1188. Doi: 10.3934/cpaa.2010.9.1161

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Kirchhoff systems with nonlinear source and boundary damping terms

Giuseppina Autuori^1, and
Patrizia Pucci^1,

1.
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, I–06123 Perugia

Received: July 31, 2009

Revised: October 31, 2009

Published: August 2010

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Abstract

In this paper we treat the question of the non--existence of global solutions, or their long time behavior, of nonlinear hyperbolic Kirchhoff systems. The main $p$--Kirchhoff operator may be affected by a perturbation which behaves like $|u|^{p-2} u$ and the systems also involve an external force $f$ and a nonlinear boundary damping $Q$. When $p=2$, we consider some problems involving a higher order dissipation term, under dynamic boundary conditions. For them we give criteria in order that $ || u(t,\cdot) ||_q\to\infty$ as $t \to\infty$ along any global solution $u=u(t,x)$, where $q$ is a parameter related to the growth of $f$ in $u$. Special subcases of $f$ and $Q$, interesting in applications, are presented in Sections 4, 5 and 6.

Keywords:

Kirchhoff systems,
nonlinear source and boundary damping terms,
non continuation,
blow up.

Mathematics Subject Classification: Primary: 35L70, 35L20; Secondary: 35Q70.

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