Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models

• Published in: Advanced nonlinear studies Vol. 22; no. 1; pp. 436 - 468
• Main Authors: Yang, Chao, Rădulescu, Vicenţiu D., Xu, Runzhang, Zhang, Mingyou
• Format: Journal Article
• Language: English
• Published: De Gruyter 20.09.2022
• Subjects: 35L05; 35L35; 35L75; extensible beam equation; global existence and nonexistence; nonlinear weak damping; strong damping
• ISSN: 2169-0375; 2169-0375
• DOI: 10.1515/ans-2022-0024

For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invariant manifold via Nehari manifold. To establish the results of global well-posedness of solution, we consider the problem at three different initial energy levels, i.e., subcritical initial energy level, critical initial energy level, and arbitrarily high initial energy level. We first obtain the local existence of the solution by using the contraction mapping principle. Then, in the framework of potential well, we obtain global existence, nonexistence, and asymptotic behavior of solution for both subcritical initial energy level and critical initial energy level. In the end, we establish the global nonexistence of solution for the problem with linear weak damping and strong damping at the arbitrarily high initial energy level.