Abstract
In this paper, we consider a fourth-order suspension bridge equation with nonlinear damping term \(|u_t|^{m-2}u_t\) and source term \(|u|^{p-2}u\). We give necessary and sufficient condition for global existence and energy decay results without considering the relation between m and p. Moreover, when \(p>m\), we give sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time \(T_{max}\) is also established. It worth to mention that our obtained results extend the recent results of Wang (J Math Anal Appl 418(2):713–733, 2014) to the nonlinear damping case.
Highlights
Suspension bridge means a bridge takes the cable, which is hanged by tower and anchored on both sides, as an upper structure of the main load-bearing elements
Suspension bridge is suitable for the valley, rivers and other natural barrier regions
The construction method of suspension bridge is mostly used in modern bridges
Summary
Suspension bridge means a bridge takes the cable (or steel chain), which is hanged by tower and anchored on both sides (or both ends of the bridge), as an upper structure of the main load-bearing elements. We study the following fourth-order suspension bridge equation with nonlinear damping and source terms: (1.1). The following wave equations with nonlinear damping and source terms have been extensively studied and many results concerning the existence and nonexistence were established: utt − ∆u + a|ut|m−2ut = b|u|p−2u, (x, t) ∈ Ω × (0, T ),. In [23, 24], Levine first considered the interaction between the damping and the source terms in the linear damping case (m = 2) He showed that solutions with negative initial energy blow up in finite time. Our purpose is to investigate the global existence, energy decay and finite time blow-up of solutions of initial-boundary value problem (1.1)-(1.2). Compared to the results given in [39], the lower bound of the blow-up time Tmax of the present paper is a new content
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