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Exponential stability for wave equations with non-dissipative damping

✍ Scribed by Jaime E. Muñoz Rivera; Reinhard Racke

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
365 KB
Volume
68
Category
Article
ISSN
0362-546X

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✦ Synopsis

We consider the nonlinear wave equation u ttσ (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 L 0 a(x)dx > 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 L 0 a(x) sin 2 (π x/L) dx. 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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