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Convergence properties of strongly-damped semilinear wave equations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1248)

Abstract

We consider the strongly-damped nonlinear Klein-Gordon equation

$$u_{tt} + \alpha ( - \Delta + \gamma )u_t + ( - \Delta + m^2 )u + \lambda \left| u \right|^{p - 1} u = 0$$

over a domain Ω in ℝ3. Let uα be a solution of this equation with α>0. Aviles and Sandefur show that such solutions are unique, strong, and exist globally for any p≥1 and arbitrary initial data u(0), ut(0) ε D(Δ). We establish here, in the case of a bounded Ω, the existence of a weak global solution with α=0 and a subsequence αk such that αk↓0 and lim \(\mathop {\lim }\limits_{k \to \infty } u^{\alpha _k } = v\) in C([0,T]; L2(Ω)) for any T>0. We k→∞ conclude with a few remarks concerning the difficulty of extending this result to the case Ω=ℝ3.

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Bibliography

  1. Adams, R.A., Sobolev Spaces, Academic Press, New York, 1975.

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  2. Aviles, P., and Sandefur, J., "Nonlinear second order equations with applications to partial differential equations", J. Diff. Equations, to appear.

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  3. Reed, M., Abstract Non-Linear Wave Equations, Springer-Verlag, Berlin/Heidelberg/New York, 1976.

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  4. Strauss, W., "On weak solutions of semilinear hyperbolic equations", Anais Acad. Brazil Ciencias, 42 (1970), pp. 645–651.

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© 1987 Springer-Verlag

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Avrin, J.D. (1987). Convergence properties of strongly-damped semilinear wave equations. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1248. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077410

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  • DOI: https://doi.org/10.1007/BFb0077410

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17741-8

  • Online ISBN: 978-3-540-47791-4

  • eBook Packages: Springer Book Archive