On Kirchhoff's Model of Parabolic Type

TL;DR

This paper establishes the existence, regularity, and long-term behavior of solutions to Kirchhoff's parabolic model, introduces finite element and backward Euler schemes with error analysis, and confirms results through numerical experiments.

Contribution

It provides new regularity results, proves existence of global attractors, and derives optimal error estimates for both semi-discrete and fully discrete schemes for Kirchhoff's model.

Findings

Existence of strong global solutions for all finite times.

Global attractors exist under certain conditions.

Error estimates decay exponentially when forcing is zero or decays exponentially.

Abstract

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.