  In this paper, existence of a strong global solution for all finite time is
derived for the Kirchhoff's model of parabolic type. Based on exponential
weight function, some new regularity results which reflect the exponential
decay property are obtained for the exact solution. For the related dynamics,
existence of a global attractor is shown to hold for the problem, when the non-
homogeneous forcing function is either independent of time or in
$L^{\infty}(L^2)$. With finite element Galerkin method applied in spatial
direction keeping time variable continuous, a semidiscrete scheme is analyzed
and it is, further, established that the semi-discrete system has a global
discrete attractor. Optimal error estimates in $L^{\infty}(H^1_0)$-norm are
derived which are valid uniformly in time. Further, based on a Backward Euler
method, a completely discrete scheme is developed and error estimates are
derived. It is further observed that in case $f = 0$ or $f =O(e^{-{\gamma}_0
t})$ with ${\gamma}_0 > 0,$ the discrete solutions and also error estimates
decay exponentially. Finally, some numerical experiments are discussed which
confirm our theoretical findings.
