Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems

Abstract

We study the initial boundary value problem of linear homogeneous wave equation with dynamic boundary condition. We aim to prove the finite time blow-up of the solution at critical energy level or high energy level with the nonlinear damping term on boundary in control systems.

Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data

“…Ruying Xue considered problem (1) with the nonlinearity f (u) = u k+1 , k = 1, 2, • • • , and proved the global existence for the small initial data and k > 4. The problem (1) with much more general nonlinear case f (u) = ±|u| p and |u| p−1 u was then considered in [15], and also by the potential well theory [4], [9], [8], [22], in the sub-critical initial energy case, i.e. E(0) ≤ d, the global and non-global solutions were classified in term of initial data.…”

Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

“…where h (u) ∼ [u + 1] + − 1 describes restoring force due to the hangers and external forces including gravity and [u] + stands for its positive part. In other papers [5,6,19,20], this kind of problem was also investigated under the Navier boundary condition. More recently, Ferrero and Gazzola [8] considered the boundaries of a plate Ω = (0, π) × (−l, l) which represents the roadway of a suspension bridge.…”

“…In order to consider the effect of internal friction on a system, Ferrero and Gazzola [8] studied the following linear damped wave equation u tt + ∆ 2 u + δu t + h (x, y, u) = f (x, y, t) , (x, y, t) ∈ Ω × (0, T ) , (5) with the initial boundary data conditions (2) and (3), where h (•, •, s) is a Lipschitz function and increasing with respect to s ∈ R. They proved the existence and uniqueness of global solutions, but did not give any non-global results due to the linear source term. It is worth mentioning the work on an extensible beam model similar to the model equation in this paper, the so-called polyharmonic Kirchhoff equation was considered in [1], to show the global nonexistence and a priori estimates for the life span of solution.…”

“…In Theorem 6.6, the condition µ > 0 is essential. When the model not contain a damping term, even for the classical wave equation of form u tt − ∆u = |u| p−2 u or u tt − ∆u = |u| p−2 u ln |u| for p 2 (See [3,4,5,6,14,15,17,19,25,31], due to the lack of the invariant sets, the global existence of solution when E (0) > 0 is still unsolved within potential well's framework. Although we have full confidence and reason to believe that there exist such initial data satisfying E (0) > d and finite time.…”