Abstract
We consider the strongly-damped nonlinear Klein-Gordon equation
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
Adams, R.A., Sobolev Spaces, Academic Press, New York, 1975.
Aviles, P., and Sandefur, J., "Nonlinear second order equations with applications to partial differential equations", J. Diff. Equations, to appear.
Reed, M., Abstract Non-Linear Wave Equations, Springer-Verlag, Berlin/Heidelberg/New York, 1976.
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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