On Kirchhoff's Model of Parabolic Type
For researchers in PDEs and numerical analysis, this provides rigorous theoretical and numerical analysis for a specific parabolic model, but the results are incremental.
The paper proves existence of strong global solutions and a global attractor for Kirchhoff's parabolic model, and derives optimal error estimates for semidiscrete and fully discrete schemes, with exponential decay under certain conditions.
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 $f =O(e^{-γ_0 t})$ with $γ_0 > 0,$ the discrete solutions and also error estimates decay exponentially. Finally, some numerical experiments are discussed which confirm our theoretical findings.