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Convergence properties of the strongly damped nonlinear Klein-Gordon equation. (English) Zbl 0625.35058
The author studies the damped nonlinear Klein-Gordon equation \[ (1)\quad u_{tt}(x,t)+\alpha (-\Delta +\gamma)u_ t(x,t)+(-\Delta +m^ 2)u(x,t)+\lambda u(x,t)| u(x,t)|^{p-1}=0\quad x\in \Omega,\quad t\in {\mathbb{R}} \] with the initial and boundary conditions \[ (2)\quad u(x,0)=f(x)\quad for\quad x\in \Omega,\quad u_ t(x,0)=g(x)\quad and\quad u(x,t)=0\quad for\quad x\in \partial \Omega,\quad t\in {\mathbb{R}} \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ 3\) with smooth boundary \(\partial \Omega\); \(\alpha\), \(\gamma\), m, p, \(\lambda\) are constants with \(\alpha,\lambda >0\); \(\gamma\),m\(\geq 0\) and \(p\geq 1\). Moreover f and g are smooth assigned functions.
It is known that for any \(\alpha >0\) there exists a unique global strong solution \(u_{\alpha}\) of (1), (2) [see P. Aviles and J. Sandefur, J. Differ. Equations 58, 404-427 (1985; Zbl 0572.34004)]. In this paper the author assumes \(p>3\) and proves that there exists a sequence \(\{\alpha_ k\}\) of positive, real numbers such that \(\alpha_ k\to 0\) as \(k\to \infty\) and a global weak solution v of (1), (2) with \(\alpha =0\), such that \(u_{\alpha_ k}\to v\) in \((([0,T],L^ 2(\Omega))\) for all \(T>0\) a convenient summary of the undamped case (i.e. \(\alpha =0)\) can be found by M. Reed [“Abstract nonlinear wave equations”, Lect. Notes Math. 507 (1976; Zbl 0317.35002)].
Reviewer: D.Fortunato

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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References:
[1] Adams, R.A, Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Aviles, P; Sandefur, J, Nonlinear second-order equations with applications to partial differential equations, J. differential equations, 58, 404-427, (1985) · Zbl 0572.34004
[3] Reed, M, Abstract non-linear wave equations, (1976), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0317.35002
[4] Strauss, W, On weak solutions of semilinear hyperbolic equations, Anais acad. Brazil ci√™ncias, 42, 645-651, (1970) · Zbl 0217.13104
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