Nonlinear Beam Equation with Indefinite Damping
نویسنده
چکیده
We consider the nonlinear beam equation utt + a(x)ut − f(ux)x + uxxxx = 0 in a bounded interval (0, 1) ⊂ R. The equation has an indefinite damping term, i.e., with a damping function a = a(x) possibly changing sign. For this non-dissipative situation we prove the exponential stability of the corresponding linearized system provided ā = ∫ 1 0 a(x)dx > 0 and ‖ a− ā ‖L2≤ τ , for τ small enough. We shall also demonstrate that the system has the spectrum determined growth (SDG) property for the constant case a ≡ ā. Moreover, we show the global existence of the solution to the corresponding nonlinear system. To our knowledge, this paper is the first to deal with a fourth-order nonlinear evolution equations with indefinite damping. Keyword: exponential stability; indefinite damping; non-dissipative system MSC 2000: 35G25; 35B40
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