Exponential stability for wave equations with non-dissipative damping

Abstract We consider the nonlinear wave equation u t t − σ ( u x ) x + a ( x ) u t = 0 in a bounded interval ( 0 , L ) ⊂ R 1 . The function a is allowed to change sign, but has to satisfy a ¯ = 1 L ∫ 0 L a ( x ) d x > 0 . For this non-dissipative situation we prove the exponential stability of the corresponding linearized system for: (I) possibly large ‖ a ‖ L ∞ with small ‖ a ( ⋅ ) − a ¯ ‖ L 2 , and (II) a class of pairs ( a , L ) with possibly negative moment ∫ 0 L a ( x ) sin 2 ( π x / L ) d x . Estimates for the decay rate are also given in terms of a ¯ . Moreover, we show the global existence of smooth, small solutions to the corresponding nonlinear system if, additionally, the negative part of a is small enough.

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