Chaotic behavior for the third-order partial differential equations

Abstract

In our investigation, our primary focus has been on a third-order partial differential equation, as expressed below:
avttt(y, t) + bvtt(y, t) + cvt(y, t) − ν^2 vyy(y, t) − µvyyt(y, t) = ηv(y, t).     (0.1)
This equation represents the one-dimensional variant of the Moore-Gibson-Thompson equation, which holds significance in the realms of high-intensity ultrasound and the linear vibrations of elastic structures. Notably, our study marks a substantial advancement compared to existing literature. This is particularly
evident in our revelation that when the critical parameter γ := b−aν^2/µ is negative, the equation (0.1) exhibits noteworthy characteristics. Specifically, it manifests a uniformly continuous and chaotic semigroup of bounded linear operators within the Hilbert space L^2([0, ∞), C). This discovery challenges current knowledge and provides fresh insights into the dynamics and behavior of solutions to this equation. To underscore this point, we have reformulated Problem (0.1) as a first-order problem. By presenting compelling evidence of these distinctive properties, our research not only broadens the existing knowledge base but also deepens our comprehension of the intricate phenomena associated with high-intensity ultrasound and elastic vibrations. Moreover, we have transformed the equation (0.1) into a third-order equation represented by a matrix that generalizes a C0-semigroup, effectively showcasing its chaotic nature. This progress opens up new research perspectives and prompts intriguing questions regarding the complex nature of the systems under study. Additionally, it provides opportunities for developing novel analytical methods and practical applications in the realm of high-intensity ultrasound imaging and elastic structure vibrations.