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Convergence of the strongly damped nonlinear Klein-Gordon equation in ℝn with radial symmetry

  • Joel D. Avrin (a1)

Synopsis

We consider the strongly-damped Klein–Gordon equation in ℝ3 in the case where the initial data possess radial symmetry. With the latter assumption we are able to extend the result of [2] which assumed a bounded spatial domain. Specifically, we construct a global weak solution v of theundamped equation for high powers p which can be approximated arbitrarily closely (for small α) by the global strong solutions of the damped equation found by Aviles and Sandefur [1].

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1Aviles, P. and Sandefur, J.. Nonlinear second order equations with applications to partial differential equations. J. Differential Equations 58 (1985), 404427.
2Avrin, J.. Convergence properties of the nonlinear Klein-Gordon equation. J. Differential Equations 67 (1987), 243255.
3Engler, H.. Existence of radially symmetric solutions of strongly-damped wave equations (to appear).
4Ginebre, J. and Velo, G.. The global Cauchy problem for the non linear Klein-Gordon equation. Math. Z. 189 (1985), 487505.
5Reed, M.. Abstract Non-Linear Wave Equations (Berlin: Springer, 1976).
6Strauss, W. A.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149162.
7Strauss, W.. On weak solutions of semilinear hyperbolic equations. An. Acad. Brazil Cienc. 42 (1970), 645651.

Convergence of the strongly damped nonlinear Klein-Gordon equation in ℝn with radial symmetry

  • Joel D. Avrin (a1)

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