Abstract
The energy decay and blow-up of a solution for a Kirchhoff equation with dynamic boundary condition are considered. With the help of Nakao’s inequality and a stable set, the energy decay of the solution is given. By the convexity inequality lemma and an unstable set, the sufficient condition of blow-up of the solution with negative and small positive initial energy are obtained, respectively.
Highlights
This problem is based on the equation l utt + uxxxx – α + β u x(x, t) dx uxx =, ( )
Which was proposed by Woinowsky-Krieger [, ] as a model for a vibrating beam with hinged ends, where u(x, t) is the lateral displacement at the space coordinate x and time t
We find that the set of initial data such that the solution of problem ( )-( ) is decay, is smaller than the potential well in [ ]
Summary
1 Introduction The aim of this article is to study the energy decay and blow-up of a solution of the following Kirchhoff-type equation with nonlinear dynamic boundary condition: utt + ut + uxxxx – M ux uxx = , < x < l, t > , ( ) Equation ( ) was studied by many authors such as Dickey [ ], Ball Rivera [ ], Tucsnak [ ], Kouemou Patchen [ ], Aassila [ ], Oliveira and Lima [ ]; Wu and Tsai [ ] considered the following beam equation: utt + α u – M u u + g(ut) = f (u), in ⊂ Rn. They obtained the existence and uniqueness, as well as decay estimates, of global solutions and blow-up of solutions for the initial boundary value problem of equation ( ) through various approaches and assumptive conditions.
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