Kirchhoff systems with nonlinear source and boundary damping terms

Pages: 1161 - 1188, Volume 9, Issue 5, September 2010

doi:10.3934/cpaa.2010.9.1161       Abstract        Full Text (392.4K)       Related Articles

Giuseppina Autuori - Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, I–06123 Perugia, Italy (email)
Patrizia Pucci - Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, I–06123 Perugia, Italy (email)

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.

Received: August 2009;      Revised: November 2009;      Published: May 2010.