On decay and blow-up of solutions for a nonlinear beam equation with double damping terms

Abstract

Abstract This paper deals with the initial boundary value problem for the nonlinear beam equation with double damping terms utt−uxxt+uxxxx+uxxxxt=g(uxx)xx,x∈Ω,t>0, $$\begin{aligned} u_{tt}-u_{xxt}+u_{xxxx}+u_{xxxxt}=g(u_{xx})_{xx}, \quad x\in\Omega, t>0, \end{aligned}$$ where Ω=(0,1) $\Omega=(0,1)$, and g(s) $g(s)$ is a given nonlinear function. We derive sufficient conditions for the blow-up of the solution to the problem by virtue of an adapted concavity method. In addition, global existence of weak solutions as well as exponential and uniform decay rates of the solution energy are established by the use of an integral inequality.

Keywords

• Nonlinear beam equation
• Initial boundary value problem
• Existence of weak solution
• Blow-up
• Energy decay