IJPAM: Volume 32, No. 4 (2006)

EXISTENCE AND DECAY OF SOLUTIONS OF
A DAMPED KIRCHHOFF EQUATION

S.S. Souza$^1$, M. Milla Miranda$^2$
$^1$Department of Mathematics - UFPB
João Pessoa, PB, BRAZIL
e-mail: shirley@mat.ufpb.br
$^2$Institute of Mathematics - UFRJ
Caixa Postal 68530, 21945-970, Rio de Janeiro, RJ, BRAZIL
e-mail: milla@im.ufrj.br


Abstract.This paper is concerned with the study of local and global solutions of the mixed problem for the damped Kirchhoff equation

\begin{displaymath}
u''(t) + M\big(\vert A^{1/2}\,u(t)\vert^2\big) Au(t) + \delta u'(t)=0, \quad
t > 0\,,
\end{displaymath}

where $A$ is an unbounded self-adjoint operator, $A \ge 0$, of a real separable Hilbert space with norm $\vert u\vert$, $M$ a real function with $M(\la) \ge m_0 > 0$ and $\delta$ a positive real number. The exponential decay of solutions is obtained when $A \ge \be I$, $\be$ positive real number.

Received: September 25, 2006

AMS Subject Classification: 35L70, 35B35

Key Words and Phrases: Kirchhoff equation, decay of solutions, quasilinear hyperbolic equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 32
Issue: 4