IJPAM: Volume 59, No. 2 (2010)

THE WELL-POSEDNESS OF THE GLOBAL SOLUTION
FOR A DAMPED EULER-BERNOULLI EQUATION

Shaoyong Lai$^1$, Benchawan Wiwatanapataphee$^2$
$^1$Department of Applied Mathematics
Southwestern University of Finance and Economics
Chengdu, 610074, P.R. CHINA
$^2$Department of Mathematics
Faculty of Science
Mahidol University
272, Rama 6 Road, Bangkok, 10400, THAILAND
e-mail: scbww@mahidol.ac.th


Abstract.In this paper, we study the well-posedness of the global solution to the following damped Euler-Bernoulli equation

\begin{displaymath}u_{tt}+au_{xxxx}+2bu_{t}+cu=f(u),\quad t\geq 0,\ \ x\in \lbrack 0,+\infty ). \end{displaymath}

For the case $f(u)=u^2$, the existence and uniqueness of the global solution to an initial value problem of the equation are established in the space $C([0,+\infty ),L^{2}([0,+\infty )))\cap C^{1}([0,+\infty ),H^{-1}([0,+\infty )))$. For the case where $f(u)$ is a polynomial, we find that the well-posedness can be established in the Sobolev space $C([0,+\infty ),H^{s}([0,+\infty )))\cap C^{1}([0,+\infty ),H^{s-1}([0,+\infty )))$ ( $s> \frac{1}{2}$).

Received: January 11, 2010

AMS Subject Classification: 00A71

Key Words and Phrases: beam equation, global solution, initial value problem

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 59
Issue: 2