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)-?2vyy(y,t)-?yyvt (y,t) = ?v(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 ?:=b-a?2/? 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 L2 ([0,?), C). This discovery challenges current knowledge and provides fresh insights into the dynamics and behavior of solutions to this equation.