IJPAM: Volume 37, No. 1 (2007)

INITIAL-BOUNDARY VALUE PROBLEM FOR A CLASS
OF NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

Yonghua Ren$^1$, Jianwen Zhang$^2$
$^{1,2}$Department of Mathematics
Taiyuan University of Technology
Taiyuan, Shanxi, 030024, P.R. CHINA
$^1$e-mail: ryh80216@sina.com.cn
$^2$e-mail: jianwenz@public.ty.sx.cn


Abstract.For a class of integro-differential equation with nonlinear damped and memory terms arising from the models of nonlinear viscoelasticity, we consider the existence of a weak solution to the initial-boundary value problem
\begin{multline*} \frac{\partial^{2}u}{\partial t^{2}} -(\frac{\partial^{2}}{\... ...^{2}})u(\tau)d\tau=0,\quad (x,y;t)\in\Omega\times(0,\infty)\,, \end{multline*}
where $\Omega$ is bounded domain of $R^{2}$, and $f,\ g$ are power like functions. By virtue of the Galerkin method combined with the a priori estimate, it is proved that under rather mild conditions on nonlinear terms and initial data the above-mentionded problem admits a weak solution.

Received: March 6, 2007

AMS Subject Classification: 26A33

Key Words and Phrases: nonlinear integro-differential equation, memory term, weak solution, initial-boundary value problem, Faedo-Galerkin method

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 37
Issue: 1