Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations

TL;DR

This paper studies the nonlinear beam equations with rotational inertia, proving the existence of attractors and analyzing how their behavior changes as rotational inertia effects diminish, highlighting the continuity of attractors with respect to inertia parameter.

Contribution

It demonstrates the continuity of global and exponential attractors for nonlinear beam equations as rotational inertia effects fade, a novel insight into their qualitative behavior.

Findings

Existence of compact global attractors and exponential attractors.

Continuity of attractors with respect to the rotational inertia parameter.

Behavioral differences between cases with and without rotational inertia.

Abstract

In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter alpha and consider the difference of behavior between the case alpha in (0,1] and the case alpha=0. The main novel contribution of this paper is to show the continuity of global attractors and exponential attractors with respect to alpha in some sense.